^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ This eBook Is Provided By www.PlentyofeBooks.net Plenty of eBooks (Free eBooks & Tutorials) is a free eBooks links library where you can find and download free books in almost any category without registering. For More Free eBooks & Tutorials Visit www.PlentyofeBooks.net Uploaded By samsexy98 Enjoy...!!! ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Data Mining Third Edition This page intentionally left blank AMSTERDAM • BOSTON • HEIDELBERG • LONDON NEW YORK • OXFORD • PARIS • SAN DIEGO SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO Morgan Kaufmann Publishers is an imprint of Elsevier Data Mining Practical Machine Learning Tools and Techniques Third Edition Ian H. Witten Eibe Frank Mark A. Hall Morgan Kaufmann Publishers is an imprint of Elsevier 30 Corporate Drive, Suite 400, Burlington, MA 01803, USA This book is printed on acid-free paper. Copyright © 2011 Elsevier Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. Library of Congress Cataloging-in-Publication Data Witten, I. H. (Ian H.) Data mining : practical machine learning tools and techniques.—3rd ed. / Ian H. Witten, Frank Eibe, Mark A. Hall. p. cm.—(The Morgan Kaufmann series in data management systems) ISBN 978-0-12-374856-0 (pbk.) 1. Data mining. I. Hall, Mark A. II. Title. QA76.9.D343W58 2011 006.3′12—dc22 2010039827 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. For information on all Morgan Kaufmann publications, visit our website at www.mkp.com or www.elsevierdirect.com Printed in the United States 11 12 13 14 15 10 9 8 7 6 5 4 3 2 1 Working together to grow libraries in developing countries www.elsevier.com | www.bookaid.org | www.sabre.org Contents LIST OF FIGURES..................................................................................................xv LIST OF TABLES...................................................................................................xix PREFACE................................................................................................................xxi Updated and Revised Content............................................................................xxv Second Edition................................................................................................xxv Third Edition..................................................................................................xxvi ACKNOWLEDGMENTS.....................................................................................xxix ABOUT THE AUTHORS...................................................................................xxxiii PART I INTRODUCTION TO DATA MINING CHAPTER 1 What’s It All About?................................................................. 3 1.1 Data Mining and Machine Learning...............................................3 Describing Structural Patterns.........................................................5 Machine Learning............................................................................7 Data Mining.....................................................................................8 1.2 Simple Examples: The Weather Problem and Others.....................9 The Weather Problem......................................................................9 Contact Lenses: An Idealized Problem.........................................12 Irises: A Classic Numeric Dataset.................................................13 CPU Performance: Introducing Numeric Prediction....................15 Labor Negotiations: A More Realistic Example...........................15 Soybean Classification: A Classic Machine Learning Success.....19 1.3 Fielded Applications......................................................................21 Web Mining...................................................................................21 Decisions Involving Judgment......................................................22 Screening Images...........................................................................23 Load Forecasting............................................................................24 Diagnosis........................................................................................25 Marketing and Sales......................................................................26 Other Applications.........................................................................27 1.4 Machine Learning and Statistics...................................................28 1.5 Generalization as Search ..............................................................29 1.6 Data Mining and Ethics.................................................................33 Reidentification..............................................................................33 Using Personal Information...........................................................34 Wider Issues...................................................................................35 1.7 Further Reading.............................................................................36 v vi Contents CHAPTER 2 Input: Concepts, Instances, and Attributes.............................. 39 2.1 What’s a Concept?.........................................................................40 2.2 What’s in an Example?..................................................................42 Relations.........................................................................................43 Other Example Types.....................................................................46 2.3 What’s in an Attribute?..................................................................49 2.4 Preparing the Input........................................................................51 Gathering the Data Together..........................................................51 ARFF Format.................................................................................52 Sparse Data....................................................................................56 Attribute Types...............................................................................56 Missing Values...............................................................................58 Inaccurate Values...........................................................................59 Getting to Know Your Data...........................................................60 2.5 Further Reading.............................................................................60 CHAPTER 3 Output: Knowledge Representation......................................... 61 3.1 Tables.............................................................................................61 3.2 Linear Models................................................................................62 3.3 Trees...............................................................................................64 3.4 Rules...............................................................................................67 Classification Rules........................................................................69 Association Rules...........................................................................72 Rules with Exceptions...................................................................73 More Expressive Rules..................................................................75 3.5 Instance-Based Representation......................................................78 3.6 Clusters...........................................................................................81 3.7 Further Reading.............................................................................83 CHAPTER 4 Algorithms: The Basic Methods.............................................. 85 4.1 Inferring Rudimentary Rules.........................................................86 Missing Values and Numeric Attributes........................................87 Discussion......................................................................................89 4.2 Statistical Modeling.......................................................................90 Missing Values and Numeric Attributes .......................................94 Naïve Bayes for Document Classification....................................97 Discussion......................................................................................99 4.3 Divide-and-Conquer: Constructing Decision Trees......................99 Calculating Information...............................................................103 Highly Branching Attributes........................................................105 Discussion....................................................................................107 Contents vii 4.4 Covering Algorithms: Constructing Rules..................................108 Rules versus Trees.......................................................................109 A Simple Covering Algorithm.....................................................110 Rules versus Decision Lists.........................................................115 4.5 Mining Association Rules............................................................116 Item Sets.......................................................................................116 Association Rules.........................................................................119 Generating Rules Efficiently........................................................122 Discussion....................................................................................123 4.6 Linear Models..............................................................................124 Numeric Prediction: Linear Regression......................................124 Linear Classification: Logistic Regression..................................125 Linear Classification Using the Perceptron.................................127 Linear Classification Using Winnow...........................................129 4.7 Instance-Based Learning..............................................................131 Distance Function........................................................................131 Finding Nearest Neighbors Efficiently........................................132 Discussion....................................................................................137 4.8 Clustering.....................................................................................138 Iterative Distance-Based Clustering............................................139 Faster Distance Calculations........................................................139 Discussion....................................................................................141 4.9 Multi-Instance Learning...............................................................141 Aggregating the Input..................................................................142 Aggregating the Output...............................................................142 Discussion....................................................................................142 4.10 Further Reading...........................................................................143 4.11 Weka Implementations.................................................................145 CHAPTER 5 Credibility: Evaluating What’s Been Learned......................... 147 5.1 Training and Testing....................................................................148 5.2 Predicting Performance................................................................150 5.3 Cross-Validation...........................................................................152 5.4 Other Estimates............................................................................154 Leave-One-Out Cross-Validation.................................................154 The Bootstrap...............................................................................155 5.5 Comparing Data Mining Schemes...............................................156 5.6 Predicting Probabilities................................................................159 Quadratic Loss Function..............................................................160 Informational Loss Function........................................................161 Discussion....................................................................................162 viii Contents 5.7 Counting the Cost........................................................................163 Cost-Sensitive Classification.......................................................166 Cost-Sensitive Learning...............................................................167 Lift Charts....................................................................................168 ROC Curves.................................................................................172 Recall–Precision Curves..............................................................174 Discussion....................................................................................175 Cost Curves .................................................................................177 5.8 Evaluating Numeric Prediction....................................................180 5.9 Minimum Description Length Principle......................................183 5.10 Applying the MDL Principle to Clustering.................................186 5.11 Further Reading...........................................................................187 PART II ADVANCED DATA MINING CHAPTER 6 Implementations: Real Machine Learning Schemes............... 191 6.1 Decision Trees..............................................................................192 Numeric Attributes.......................................................................193 Missing Values.............................................................................194 Pruning.........................................................................................195 Estimating Error Rates.................................................................197 Complexity of Decision Tree Induction......................................199 From Trees to Rules.....................................................................200 C4.5: Choices and Options..........................................................201 Cost-Complexity Pruning............................................................202 Discussion....................................................................................202 6.2 Classification Rules......................................................................203 Criteria for Choosing Tests..........................................................203 Missing Values, Numeric Attributes............................................204 Generating Good Rules................................................................205 Using Global Optimization..........................................................208 Obtaining Rules from Partial Decision Trees.............................208 Rules with Exceptions.................................................................212 Discussion....................................................................................215 6.3 Association Rules.........................................................................216 Building a Frequent-Pattern Tree................................................216 Finding Large Item Sets..............................................................219 Discussion....................................................................................222 6.4 Extending Linear Models............................................................223 Maximum-Margin Hyperplane....................................................224 Nonlinear Class Boundaries........................................................226 Contents ix Support Vector Regression..........................................................227 Kernel Ridge Regression.............................................................229 Kernel Perceptron �����������������������������������������������������������������������231 Multilayer Perceptrons.................................................................232 Radial Basis Function Networks.................................................241 Stochastic Gradient Descent........................................................242 Discussion....................................................................................243 6.5 Instance-Based Learning..............................................................244 Reducing the Number of Exemplars...........................................245 Pruning Noisy Exemplars............................................................245 Weighting Attributes....................................................................246 Generalizing Exemplars...............................................................247 Distance Functions for Generalized Exemplars.....................................................................................248 Generalized Distance Functions..................................................249 Discussion....................................................................................250 6.6 Numeric Prediction with Local Linear Models...........................251 Model Trees.................................................................................252 Building the Tree.........................................................................253 Pruning the Tree...........................................................................253 Nominal Attributes.......................................................................254 Missing Values ����������������������������������������������������������������������������254 Pseudocode for Model Tree Induction........................................255 Rules from Model Trees..............................................................259 Locally Weighted Linear Regression...........................................259 Discussion....................................................................................261 6.7 Bayesian Networks......................................................................261 Making Predictions......................................................................262 Learning Bayesian Networks.......................................................266 Specific Algorithms......................................................................268 Data Structures for Fast Learning...............................................270 Discussion....................................................................................273 6.8 Clustering.....................................................................................273 Choosing the Number of Clusters...............................................274 Hierarchical Clustering................................................................274 Example of Hierarchical Clustering............................................276 Incremental Clustering.................................................................279 Category Utility ��������������������������������������������������������������������������284 Probability-Based Clustering.......................................................285 The EM Algorithm.......................................................................287 Extending the Mixture Model.....................................................289 x Contents Bayesian Clustering.....................................................................290 Discussion....................................................................................292 6.9 Semisupervised Learning.............................................................294 Clustering for Classification........................................................294 Co-training...................................................................................296 EM and Co-training.....................................................................297 Discussion....................................................................................297 6.10 Multi-Instance Learning...............................................................298 Converting to Single-Instance Learning......................................298 Upgrading Learning Algorithms..................................................300 Dedicated Multi-Instance Methods..............................................301 Discussion....................................................................................302 6.11 Weka Implementations.................................................................303 CHAPTER 7 Data Transformations........................................................... 305 7.1 Attribute Selection.......................................................................307 Scheme-Independent Selection....................................................308 Searching the Attribute Space.....................................................311 Scheme-Specific Selection...........................................................312 7.2 Discretizing Numeric Attributes..................................................314 Unsupervised Discretization........................................................316 Entropy-Based Discretization......................................................316 Other Discretization Methods......................................................320 Entropy-Based versus Error-Based Discretization......................320 Converting Discrete Attributes to Numeric Attributes................322 7.3 Projections....................................................................................322 Principal Components Analysis...................................................324 Random Projections.....................................................................326 Partial Least-Squares Regression................................................326 Text to Attribute Vectors..............................................................328 Time Series..................................................................................330 7.4 Sampling......................................................................................330 Reservoir Sampling......................................................................330 7.5 Cleansing......................................................................................331 Improving Decision Trees............................................................332 Robust Regression.......................................................................333 Detecting Anomalies....................................................................334 One-Class Learning.....................................................................335 7.6 Transforming Multiple Classes to Binary Ones..........................338 Simple Methods...........................................................................338 Error-Correcting Output Codes...................................................339 Ensembles of Nested Dichotomies..............................................341 Contents xi 7.7 Calibrating Class Probabilities....................................................343 7.8 Further Reading...........................................................................346 7.9 Weka Implementations.................................................................348 CHAPTER 8 Ensemble Learning.............................................................. 351 8.1 Combining Multiple Models........................................................351 8.2 Bagging........................................................................................352 Bias–Variance Decomposition.....................................................353 Bagging with Costs......................................................................355 8.3 Randomization.............................................................................356 Randomization versus Bagging...................................................357 Rotation Forests...........................................................................357 8.4 Boosting.......................................................................................358 AdaBoost......................................................................................358 The Power of Boosting................................................................361 8.5 Additive Regression.....................................................................362 Numeric Prediction......................................................................362 Additive Logistic Regression......................................................364 8.6 Interpretable Ensembles...............................................................365 Option Trees.................................................................................365 Logistic Model Trees...................................................................368 8.7 Stacking........................................................................................369 8.8 Further Reading...........................................................................371 8.9 Weka Implementations.................................................................372 Chapter 9 Moving on: Applications and Beyond.................................... 375 9.1 Applying Data Mining.................................................................375 9.2 Learning from Massive Datasets.................................................378 9.3 Data Stream Learning..................................................................380 9.4 Incorporating Domain Knowledge..............................................384 9.5 Text Mining..................................................................................386 9.6 Web Mining.................................................................................389 9.7 Adversarial Situations..................................................................393 9.8 Ubiquitous Data Mining..............................................................395 9.9 Further Reading...........................................................................397 PART III THE WEKA DATA MINING WORKBENCH CHAPTER 10 Introduction to Weka........................................................... 403 10.1 What’s in Weka?..........................................................................403 10.2 How Do You Use It?...................................................................404 10.3 What Else Can You Do?..............................................................405 10.4 How Do You Get It?....................................................................406 xii Contents CHAPTER 11 The Explorer........................................................................ 407 11.1 Getting Started.............................................................................407 Preparing the Data.......................................................................407 Loading the Data into the Explorer.............................................408 Building a Decision Tree.............................................................410 Examining the Output..................................................................411 Doing It Again.............................................................................413 Working with Models..................................................................414 When Things Go Wrong..............................................................415 11.2 Exploring the Explorer................................................................416 Loading and Filtering Files.........................................................416 Training and Testing Learning Schemes.....................................422 Do It Yourself: The User Classifier.............................................424 Using a Metalearner.....................................................................427 Clustering and Association Rules................................................429 Attribute Selection.......................................................................430 Visualization.................................................................................430 11.3 Filtering Algorithms.....................................................................432 Unsupervised Attribute Filters.....................................................432 Unsupervised Instance Filters......................................................441 Supervised Filters.........................................................................443 11.4 Learning Algorithms....................................................................445 Bayesian Classifiers.....................................................................451 Trees.............................................................................................454 Rules.............................................................................................457 Functions......................................................................................459 Neural Networks..........................................................................469 Lazy Classifiers............................................................................472 Multi-Instance Classifiers............................................................472 Miscellaneous Classifiers.............................................................474 11.5 Metalearning Algorithms.............................................................474 Bagging and Randomization........................................................474 Boosting.......................................................................................476 Combining Classifiers..................................................................477 Cost-Sensitive Learning...............................................................477 Optimizing Performance..............................................................478 Retargeting Classifiers for Different Tasks.................................479 11.6 Clustering Algorithms..................................................................480 11.7 Association-Rule Learners...........................................................485 11.8 Attribute Selection.......................................................................487 Attribute Subset Evaluators.........................................................488 Contents xiii Single-Attribute Evaluators.........................................................490 Search Methods............................................................................492 CHAPTER 12 The Knowledge Flow Interface............................................. 495 12.1 Getting Started.............................................................................495 12.2 Components.................................................................................498 12.3 Configuring and Connecting the Components............................500 12.4 Incremental Learning...................................................................502 CHAPTER 13 The Experimenter................................................................ 505 13.1 Getting Started.............................................................................505 Running an Experiment...............................................................506 Analyzing the Results..................................................................509 13.2 Simple Setup................................................................................510 13.3 Advanced Setup...........................................................................511 13.4 The Analyze Panel.......................................................................512 13.5 Distributing Processing over Several Machines..........................515 CHAPTER 14 The Command-Line Interface................................................ 519 14.1 Getting Started.............................................................................519 14.2 The Structure of Weka.................................................................519 Classes, Instances, and Packages.................................................520 The weka.core Package................................................................520 The weka.classifiers Package.......................................................523 Other Packages.............................................................................525 Javadoc Indexes...........................................................................525 14.3 Command-Line Options...............................................................526 Generic Options...........................................................................526 Scheme-Specific Options.............................................................529 CHAPTER 15 Embedded Machine Learning............................................... 531 15.1 A Simple Data Mining Application.............................................531 MessageClassifier()......................................................................536 updateData()................................................................................536 classifyMessage().........................................................................537 CHAPTER 16 Writing New Learning Schemes........................................... 539 16.1 An Example Classifier.................................................................539 buildClassifier()...........................................................................540 makeTree()....................................................................................540 computeInfoGain().......................................................................549 classifyInstance().........................................................................549 xiv Contents toSource().....................................................................................550 main()...........................................................................................553 16.2 Conventions for Implementing Classifiers..................................555 Capabilities...................................................................................555 CHAPTER 17 Tutorial Exercises for the Weka Explorer.............................. 559 17.1 Introduction to the Explorer Interface.........................................559 Loading a Dataset........................................................................559 The Dataset Editor.......................................................................560 Applying a Filter..........................................................................561 The Visualize Panel.....................................................................562 The Classify Panel.......................................................................562 17.2 Nearest-Neighbor Learning and Decision Trees.........................566 The Glass Dataset........................................................................566 Attribute Selection.......................................................................567 Class Noise and Nearest-Neighbor Learning..............................568 Varying the Amount of Training Data.........................................569 Interactive Decision Tree Construction.......................................569 17.3 Classification Boundaries.............................................................571 Visualizing 1R..............................................................................571 Visualizing Nearest-Neighbor Learning......................................572 Visualizing Naïve Bayes..............................................................573 Visualizing Decision Trees and Rule Sets...................................573 Messing with the Data.................................................................574 17.4 Preprocessing and Parameter Tuning..........................................574 Discretization...............................................................................574 More on Discretization................................................................575 Automatic Attribute Selection.....................................................575 More on Automatic Attribute Selection......................................576 Automatic Parameter Tuning.......................................................577 17.5 Document Classification..............................................................578 Data with String Attributes..........................................................579 Classifying Actual Documents....................................................580 Exploring the StringToWordVector Filter....................................581 17.6 Mining Association Rules............................................................582 Association-Rule Mining.............................................................582 Mining a Real-World Dataset......................................................584 Market Basket Analysis...............................................................584 REFERENCES................................................................................................ 587 INDEX.......................................................................................................... 607 xv Figure 1.1 Rules for the contact lens data. 12 Figure 1.2 Decision tree for the contact lens data. 13 Figure 1.3 Decision trees for the labor negotiations data. 18 Figure 2.1 A family tree and two ways of expressing the sister-of relation. 43 Figure 2.2 ARFF file for the weather data. 53 Figure 2.3 Multi-instance ARFF file for the weather data. 55 Figure 3.1 A linear regression function for the CPU performance data. 62 Figure 3.2 A linear decision boundary separating Iris setosas from Iris versicolors. 63 Figure 3.3 Constructing a decision tree interactively. 66 Figure 3.4 Models for the CPU performance data. 68 Figure 3.5 Decision tree for a simple disjunction. 69 Figure 3.6 The exclusive-or problem. 70 Figure 3.7 Decision tree with a replicated subtree. 71 Figure 3.8 Rules for the iris data. 74 Figure 3.9 The shapes problem. 76 Figure 3.10 Different ways of partitioning the instance space. 80 Figure 3.11 Different ways of representing clusters. 82 Figure 4.1 Pseudocode for 1R. 86 Figure 4.2 Tree stumps for the weather data. 100 Figure 4.3 Expanded tree stumps for the weather data. 102 Figure 4.4 Decision tree for the weather data. 103 Figure 4.5 Tree stump for the ID code attribute. 105 Figure 4.6 Covering algorithm. 109 Figure 4.7 The instance space during operation of a covering algorithm. 110 Figure 4.8 Pseudocode for a basic rule learner. 114 Figure 4.9 Logistic regression. 127 Figure 4.10 The perceptron. 129 Figure 4.11 The Winnow algorithm. 130 Figure 4.12 A kD-tree for four training instances. 133 Figure 4.13 Using a kD-tree to find the nearest neighbor of the star. 134 Figure 4.14 Ball tree for 16 training instances. 136 Figure 4.15 Ruling out an entire ball (gray) based on a target point (star) and its current nearest neighbor. 137 Figure 4.16 A ball tree. 141 Figure 5.1 A hypothetical lift chart. 170 Figure 5.2 Analyzing the expected benefit of a mailing campaign. 171 Figure 5.3 A sample ROC curve. 173 Figure 5.4 ROC curves for two learning schemes. 174 Figure 5.5 Effect of varying the probability threshold. 178 Figure 6.1 Example of subtree raising. 196 List of Figures xvi List of Figures Figure 6.2 Pruning the labor negotiations decision tree. 200 Figure 6.3 Algorithm for forming rules by incremental reduced-error pruning. 207 Figure 6.4 RIPPER. 209 Figure 6.5 Algorithm for expanding examples into a partial tree. 210 Figure 6.6 Example of building a partial tree. 211 Figure 6.7 Rules with exceptions for the iris data. 213 Figure 6.8 Extended prefix trees for the weather data. 220 Figure 6.9 A maximum-margin hyperplane. 225 Figure 6.10 Support vector regression. 228 Figure 6.11 Example datasets and corresponding perceptrons. 233 Figure 6.12 Step versus sigmoid. 240 Figure 6.13 Gradient descent using the error function w2 + 1. 240 Figure 6.14 Multilayer perceptron with a hidden layer. 241 Figure 6.15 Hinge, squared, and 0 – 1 loss functions. 242 Figure 6.16 A boundary between two rectangular classes. 248 Figure 6.17 Pseudocode for model tree induction. 255 Figure 6.18 Model tree for a dataset with nominal attributes. 256 Figure 6.19 A simple Bayesian network for the weather data. 262 Figure 6.20 Another Bayesian network for the weather data. 264 Figure 6.21 The weather data. 270 Figure 6.22 Hierarchical clustering displays. 276 Figure 6.23 Clustering the weather data. 279 Figure 6.24 Hierarchical clusterings of the iris data. 281 Figure 6.25 A two-class mixture model. 285 Figure 6.26 DensiTree showing possible hierarchical clusterings of a given dataset. 291 Figure 7.1 Attribute space for the weather dataset. 311 Figure 7.2 Discretizing the temperature attribute using the entropy method. 318 Figure 7.3 The result of discretizing the temperature attribute. 318 Figure 7.4 Class distribution for a two-class, two-attribute problem. 321 Figure 7.5 Principal components transform of a dataset. 325 Figure 7.6 Number of international phone calls from Belgium, 1950–1973. 333 Figure 7.7 Overoptimistic probability estimation for a two-class problem. 344 Figure 8.1 Algorithm for bagging. 355 Figure 8.2 Algorithm for boosting. 359 Figure 8.3 Algorithm for additive logistic regression. 365 Figure 8.4 Simple option tree for the weather data. 366 Figure 8.5 Alternating decision tree for the weather data. 367 Figure 9.1 A tangled “web.” 391 Figure 11.1 The Explorer interface. 408 Figure 11.2 Weather data. 409 Figure 11.3 The Weka Explorer. 410 List of Figures xvii Figure 11.4 Using J4.8. 411 Figure 11.5 Output from the J4.8 decision tree learner. 412 Figure 11.6 Visualizing the result of J4.8 on the iris dataset. 415 Figure 11.7 Generic Object Editor. 417 Figure 11.8 The SQLViewer tool. 418 Figure 11.9 Choosing a filter. 420 Figure 11.10 The weather data with two attributes removed. 422 Figure 11.11 Processing the CPU performance data with M5′. 423 Figure 11.12 Output from the M5′ program for numeric prediction. 425 Figure 11.13 Visualizing the errors. 426 Figure 11.14 Working on the segment-challenge data with the User Classifier. 428 Figure 11.15 Configuring a metalearner for boosting decision stumps. 429 Figure 11.16 Output from the Apriori program for association rules. 430 Figure 11.17 Visualizing the iris dataset. 431 Figure 11.18 Using Weka’s metalearner for discretization. 443 Figure 11.19 Output of NaiveBayes on the weather data. 452 Figure 11.20 Visualizing a Bayesian network for the weather data (nominal version). 454 Figure 11.21 Changing the parameters for J4.8. 455 Figure 11.22 Output of OneR on the labor negotiations data. 458 Figure 11.23 Output of PART for the labor negotiations data. 460 Figure 11.24 Output of SimpleLinearRegression for the CPU performance data. 461 Figure 11.25 Output of SMO on the iris data. 463 Figure 11.26 Output of SMO with a nonlinear kernel on the iris data. 465 Figure 11.27 Output of Logistic on the iris data. 468 Figure 11.28 Using Weka’s neural-network graphical user interface. 470 Figure 11.29 Output of SimpleKMeans on the weather data. 481 Figure 11.30 Output of EM on the weather data. 482 Figure 11.31 Clusters formed by DBScan on the iris data. 484 Figure 11.32 OPTICS visualization for the iris data. 485 Figure 11.33 Attribute selection: specifying an evaluator and a search method. 488 Figure 12.1 The Knowledge Flow interface. 496 Figure 12.2 Configuring a data source. 497 Figure 12.3 Status area after executing the configuration shown in Figure 12.1. 497 Figure 12.4 Operations on the Knowledge Flow components. 500 Figure 12.5 A Knowledge Flow that operates incrementally. 503 Figure 13.1 An experiment. 506 Figure 13.2 Statistical test results for the experiment in Figure 13.1. 509 Figure 13.3 Setting up an experiment in advanced mode. 511 Figure 13.4 An experiment in clustering. 513 xviii List of Figures Figure 13.5 Rows and columns of Figure 13.2. 514 Figure 14.1 Using Javadoc. 521 Figure 14.2 DecisionStump, a class of the weka.classifiers.trees package. 524 Figure 15.1 Source code for the message classifier. 532 Figure 16.1 Source code for the ID3 decision tree learner. 541 Figure 16.2 Source code produced by weka.classifiers.trees.Id3 for the weather data. 551 Figure 16.3 Javadoc for the Capability enumeration. 556 Figure 17.1 The data viewer. 560 Figure 17.2 Output after building and testing the classifier. 564 Figure 17.3 The decision tree that has been built. 565 xix Table 1.1 Contact Lens Data 6 Table 1.2 Weather Data 10 Table 1.3 Weather Data with Some Numeric Attributes 11 Table 1.4 Iris Data 14 Table 1.5 CPU Performance Data 16 Table 1.6 Labor Negotiations Data 17 Table 1.7 Soybean Data 20 Table 2.1 Iris Data as a Clustering Problem 41 Table 2.2 Weather Data with a Numeric Class 42 Table 2.3 Family Tree 44 Table 2.4 Sister-of Relation 45 Table 2.5 Another Relation 47 Table 3.1 New Iris Flower 73 Table 3.2 Training Data for the Shapes Problem 76 Table 4.1 Evaluating Attributes in the Weather Data 87 Table 4.2 Weather Data with Counts and Probabilities 91 Table 4.3 A New Day 92 Table 4.4 Numeric Weather Data with Summary Statistics 95 Table 4.5 Another New Day 96 Table 4.6 Weather Data with Identification Codes 106 Table 4.7 Gain Ratio Calculations for Figure 4.2 Tree Stumps 107 Table 4.8 Part of Contact Lens Data for which astigmatism = yes 112 Table 4.9 Part of Contact Lens Data for which astigmatism = yes and tear production rate = normal 113 Table 4.10 Item Sets for Weather Data with Coverage 2 or Greater 117 Table 4.11 Association Rules for Weather Data 120 Table 5.1 Confidence Limits for Normal Distribution 152 Table 5.2 Confidence Limits for Student’s Distribution with 9 Degrees of Freedom 159 Table 5.3 Different Outcomes of a Two-Class Prediction 164 Table 5.4 Different Outcomes of a Three-Class Prediction 165 Table 5.5 Default Cost Matrixes 166 Table 5.6 Data for a Lift Chart 169 Table 5.7 Different Measures Used to Evaluate the False Positive versus False Negative Trade-Off 176 Table 5.8 Performance Measures for Numeric Prediction 180 Table 5.9 Performance Measures for Four Numeric Prediction Models 182 Table 6.1 Preparing Weather Data for Insertion into an FP-Tree 217 Table 6.2 Linear Models in the Model Tree 257 Table 7.1 First Five Instances from CPU Performance Data 327 Table 7.2 Transforming a Multiclass Problem into a Two-Class One 340 List of Tables xx List of Tables Table 7.3 Nested Dichotomy in the Form of a Code Matrix 342 Table 9.1 Top 10 Algorithms in Data Mining 376 Table 11.1 Unsupervised Attribute Filters 433 Table 11.2 Unsupervised Instance Filters 441 Table 11.3 Supervised Attribute Filters 444 Table 11.4 Supervised Instance Filters 444 Table 11.5 Classifier Algorithms in Weka 446 Table 11.6 Metalearning Algorithms in Weka 475 Table 11.7 Clustering Algorithms 480 Table 11.8 Association-Rule Learners 486 Table 11.9 Attribute Evaluation Methods for Attribute Selection 489 Table 11.10 Search Methods for Attribute Selection 490 Table 12.1 Visualization and Evaluation Components 499 Table 14.1 Generic Options for Learning Schemes 527 Table 14.2 Scheme-Specific Options for the J4.8 Decision Tree Learner 528 Table 16.1 Simple Learning Schemes in Weka 540 Table 17.1 Accuracy Obtained Using IBk, for Different Attribute Subsets 568 Table 17.2 Effect of Class Noise on IBk, for Different Neighborhood Sizes 569 Table 17.3 Effect of Training Set Size on IBk and J48 570 Table 17.4 Training Documents 580 Table 17.5 Test Documents 580 Table 17.6 Number of Rules for Different Values of Minimum Confidence and Support 584 xxi The convergence of computing and communication has produced a society that feeds on information. Yet most of the information is in its raw form: data. If data is char- acterized as recorded facts, then information is the set of patterns, or expectations, that underlie the data. There is a huge amount of information locked up in data- bases—information that is potentially important but has not yet been discovered or articulated. Our mission is to bring it forth. Data mining is the extraction of implicit, previously unknown, and potentially useful information from data. The idea is to build computer programs that sift through databases automatically, seeking regularities or patterns. Strong patterns, if found, will likely generalize to make accurate predictions on future data. Of course, there will be problems. Many patterns will be banal and uninteresting. Others will be spurious, contingent on accidental coincidences in the particular dataset used. And real data is imperfect: Some parts will be garbled, some missing. Anything that is discovered will be inexact: There will be exceptions to every rule and cases not covered by any rule. Algorithms need to be robust enough to cope with imperfect data and to extract regularities that are inexact but useful. Machine learning provides the technical basis of data mining. It is used to extract information from the raw data in databases—information that is expressed in a comprehensible form and can be used for a variety of purposes. The process is one of abstraction: taking the data, warts and all, and inferring whatever structure under- lies it. This book is about the tools and techniques of machine learning that are used in practical data mining for finding, and describing, structural patterns in data. As with any burgeoning new technology that enjoys intense commercial atten- tion, the use of data mining is surrounded by a great deal of hype in the technical— and sometimes the popular—press. Exaggerated reports appear of the secrets that can be uncovered by setting learning algorithms loose on oceans of data. But there is no magic in machine learning, no hidden power, no alchemy. Instead, there is an identifiable body of simple and practical techniques that can often extract useful information from raw data. This book describes these techniques and shows how they work. We interpret machine learning as the acquisition of structural descriptions from examples. The kind of descriptions that are found can be used for prediction, expla- nation, and understanding. Some data mining applications focus on prediction: They forecast what will happen in new situations from data that describe what hap- pened in the past, often by guessing the classification of new examples. But we are equally—perhaps more—interested in applications where the result of “learning” is an actual description of a structure that can be used to classify examples. This struc- tural description supports explanation and understanding as well as prediction. In our experience, insights gained by the user are of most interest in the majority of practical data mining applications; indeed, this is one of machine learning’s major advantages over classical statistical modeling. Preface xxii Preface The book explains a wide variety of machine learning methods. Some are peda- gogically motivated: simple schemes that are designed to explain clearly how the basic ideas work. Others are practical: real systems that are used in applications today. Many are contemporary and have been developed only in the last few years. A comprehensive software resource has been created to illustrate the ideas in this book. Called the Waikato Environment for Knowledge Analysis, or Weka1 for short, it is available as Java source code at www.cs.waikato.ac.nz/ml/weka. It is a full, industrial-strength implementation of essentially all the techniques that are covered in this book. It includes illustrative code and working implementations of machine learning methods. It offers clean, spare implementations of the simplest techniques, designed to aid understanding of the mechanisms involved. It also provides a work- bench that includes full, working, state-of-the-art implementations of many popular learning schemes that can be used for practical data mining or for research. Finally, it contains a framework, in the form of a Java class library, that supports applications that use embedded machine learning and even the implementation of new learning schemes. The objective of this book is to introduce the tools and techniques for machine learning that are used in data mining. After reading it, you will understand what these techniques are and appreciate their strengths and applicability. If you wish to experiment with your own data, you will be able to do this easily with the Weka software. The book spans the gulf between the intensely practical approach taken by trade books that provide case studies on data mining and the more theoretical, principle- driven exposition found in current textbooks on machine learning. (A brief descrip- tion of these books appears in the Further Reading section at the end of Chapter 1.) This gulf is rather wide. To apply machine learning techniques productively, you need to understand something about how they work; this is not a technology that you can apply blindly and expect to get good results. Different problems yield to different techniques, but it is rarely obvious which techniques are suitable for a given situation: You need to know something about the range of possible solutions. And we cover an extremely wide range of techniques. We can do this because, unlike many trade books, this volume does not promote any particular commercial software or approach. We include a large number of examples, but they use illustrative data- sets that are small enough to allow you to follow what is going on. Real datasets are far too large to show this (and in any case are usually company confidential). Our datasets are chosen not to illustrate actual large-scale practical problems but to help you understand what the different techniques do, how they work, and what their range of application is. The book is aimed at the technically aware general reader who is interested in the principles and ideas underlying the current practice of data mining. It will also 1Found only on the islands of New Zealand, the weka (pronounced to rhyme with “Mecca”) is a flightless bird with an inquisitive nature. Preface xxiii be of interest to information professionals who need to become acquainted with this new technology, and to all those who wish to gain a detailed technical understanding of what machine learning involves. It is written for an eclectic audience of informa- tion systems practitioners, programmers, consultants, developers, information tech- nology managers, specification writers, patent examiners, and curious lay people, as well as students and professors, who need an easy-to-read book with lots of illustra- tions that describes what the major machine learning techniques are, what they do, how they are used, and how they work. It is practically oriented, with a strong “how to” flavor, and includes algorithms, code, and implementations. All those involved in practical data mining will benefit directly from the techniques described. The book is aimed at people who want to cut through to the reality that underlies the hype about machine learning and who seek a practical, nonacademic, unpretentious approach. We have avoided requiring any specific theoretical or mathematical knowledge, except in some sections that are marked by a box around the text. These contain optional material, often for the more technically or theoretically inclined reader, and may be skipped without loss of continuity. The book is organized in layers that make the ideas accessible to readers who are interested in grasping the basics, as well as accessible to those who would like more depth of treatment, along with full details on the techniques covered. We believe that consumers of machine learning need to have some idea of how the algorithms they use work. It is often observed that data models are only as good as the person who interprets them, and that person needs to know something about how the models are produced to appreciate the strengths, and limitations, of the technol- ogy. However, it is not necessary for all users to have a deep understanding of the finer details of the algorithms. We address this situation by describing machine learning methods at successive levels of detail. The book is divided into three parts. Part I is an introduction to data mining. The reader will learn the basic ideas, the topmost level, by reading the first three chapters. Chapter 1 describes, through examples, what machine learning is and where it can be used; it also provides actual practical applications. Chapters 2 and 3 cover the different kinds of input and output, or knowledge representation, that are involved—different kinds of output dictate different styles of algorithm. Chapter 4 describes the basic methods of machine learning, simplified to make them easy to comprehend. Here, the principles involved are conveyed in a variety of algorithms without getting involved in intricate details or tricky implementation issues. To make progress in the application of machine learning techniques to particular data mining problems, it is essential to be able to measure how well you are doing. Chapter 5, which can be read out of sequence, equips the reader to evaluate the results that are obtained from machine learning, addressing the sometimes complex issues involved in performance evaluation. Part II introduces advanced techniques of data mining. At the lowest and most detailed level, Chapter 6 exposes in naked detail the nitty-gritty issues of implement- ing a spectrum of machine learning algorithms, including the complexities that are necessary for them to work well in practice (but omitting the heavy mathematical xxiv Preface machinery that is required for a few of the algorithms). Although many readers may want to ignore such detailed information, it is at this level that the full, working, tested Java implementations of machine learning schemes are written. Chapter 7 describes practical topics involved with engineering the input and output to machine learning—for example, selecting and discretizing attributes—while Chapter 8 covers techniques of “ensemble learning,” which combine the output from different learning techniques. Chapter 9 looks to the future. The book describes most methods used in practical machine learning. However, it does not cover reinforcement learning because that is rarely applied in practical data mining; nor does it cover genetic algorithm approache, because these are really an optimization technique, or relational learning and inductive logic pro- gramming because they are not very commonly used in mainstream data mining applications. Part III describes the Weka data mining workbench, which provides implementa- tions of almost all of the ideas described in Parts I and II. We have done this in order to clearly separate conceptual material from the practical aspects of how to use Weka. At the end of each chapter in Parts I and II are pointers to related Weka algorithms in Part III. You can ignore these, or look at them as you go along, or skip directly to Part III if you are in a hurry to get on with analyzing your data and don’t want to be bothered with the technical details of how the algorithms work. Java has been chosen for the implementations of machine learning techniques that accompany this book because, as an object-oriented programming language, it allows a uniform interface to learning schemes and methods for pre- and postpro- cessing. We chose it over other object-oriented languages because programs written in Java can be run on almost any computer without having to be recompiled, having to go through complicated installation procedures, or—worst of all—having to change the code itself. A Java program is compiled into byte-code that can be executed on any computer equipped with an appropriate interpreter. This interpreter is called the Java virtual machine. Java virtual machines—and, for that matter, Java compilers—are freely available for all important platforms. Of all programming languages that are widely supported, standardized, and extensively documented, Java seems to be the best choice for the purpose of this book. However, executing a Java program is slower than running a corresponding program written in languages like C or C++ because the virtual machine has to translate the byte-code into machine code before it can be executed. This penalty used to be quite severe, but Java implementations have improved enormously over the past two decades, and in our experience it is now less than a factor of two if the virtual machine uses a just-in-time compiler. Instead of translating each byte-code individually, a just-in-time compiler translates whole chunks of byte-code into machine code, thereby achieving significant speedup. However, if this is still too slow for your application, there are compilers that translate Java programs directly into machine code, bypassing the byte-code step. Of course, this code cannot be executed on other platforms, thereby sacrificing one of Java’s most important advantages. Preface xxv UPDATED AND REVISED CONTENT We finished writing the first edition of this book in 1999, the second edition in early 2005, and now, in 2011, we are just polishing this third edition. How things have changed over the past decade! While the basic core of material remains the same, we have made the most opportunities to both update it and to add new material. As a result the book has close to doubled in size to reflect the changes that have taken place. Of course, there have also been errors to fix, errors that we had accumulated in our publicly available errata file (available through the book’s home page at http://www.cs.waikato.ac.nz/ml/weka/book.html). Second Edition The major change in the second edition of the book was a separate part at the end that included all the material on the Weka machine learning workbench. This allowed the main part of the book to stand alone, independent of the workbench, which we have continued in this third edition. At that time, Weka, a widely used and popular feature of the first edition, had just acquired a radical new look in the form of an interactive graphical user interface—or, rather, three separate interactive interfaces—which made it far easier to use. The primary one is the Explorer interface, which gives access to all of Weka’s facilities using menu selection and form filling. The others are the Knowledge Flow interface, which allows you to design configurations for streamed data processing, and the Experimenter interface, with which you set up automated experiments that run selected machine learning algorithms with different parameter settings on a corpus of datasets, collect performance statistics, and perform signifi- cance tests on the results. These interfaces lower the bar for becoming a practicing data miner, and the second edition included a full description of how to use them. It also contained much new material that we briefly mention here. We extended the sections on rule learning and cost-sensitive evaluation. Bowing to popular demand, we added information on neural networks: the perceptron and the closely related Winnow algorithm, and the multilayer perceptron and the backpropagation algorithm. Logistic regression was also included. We described how to implement nonlinear decision boundaries using both the kernel perceptron and radial basis function networks, and also included support vector machines for regression. We incorporated a new section on Bayesian networks, again in response to readers’ requests and Weka’s new capabilities in this regard, with a description of how to learn classifiers based on these networks and how to implement them efficiently using AD-trees. The previous five years (1999–2004) had seen great interest in data mining for text, and this was reflected in the introduction of string attributes in Weka, multino- mial Bayes for document classification, and text transformations. We also described efficient data structures for searching the instance space: kD-trees and ball trees for finding nearest neighbors efficiently and for accelerating distance-based clustering. We described new attribute selection schemes, such as race search and the use of xxvi Preface support vector machines, and new methods for combining models such as additive regression, additive logistic regression, logistic model trees, and option trees. We also covered recent developments in using unlabeled data to improve classification, including the co-training and co-EM methods. Third Edition For this third edition, we thoroughly edited the second edition and brought it up to date, including a great many new methods and algorithms. Our basic philosophy has been to bring the book and the Weka software even closer together. Weka now includes implementations of almost all the ideas described in Parts I and II, and vice versa—pretty well everything currently in Weka is covered in this book. We have also included far more references to the literature: This third edition practically triples the number of references that were in the first edition. As well as becoming far easier to use, Weka has grown beyond recognition over the last decade, and has matured enormously in its data mining capabilities. It now incorporates an unparalleled range of machine learning algorithms and related tech- niques. This growth has been partly stimulated by recent developments in the field and partly user-led and demand-driven. This puts us in a position where we know a lot about what actual users of data mining want, and we have capitalized on this experience when deciding what to include in this book. As noted earlier, this new edition is split into three parts, which has involved a certain amount of reorganization. More important, a lot of new material has been added. Here are a few of the highlights. Chapter 1 includes a section on web mining, and, under ethics, a discussion of how individuals can often be “reidentified” from supposedly anonymized data. A major addition describes techniques for multi-instance learning, in two new sections: basic methods in Section 4.9 and more advanced algorithms in Section 6.10. Chapter 5 contains new material on interactive cost–benefit analysis. There have been a great number of other additions to Chapter 6: cost-complexity pruning, advanced associ- ation-rule algorithms that use extended prefix trees to store a compressed version of the dataset in main memory, kernel ridge regression, stochastic gradient descent, and hierarchical clustering methods. The old chapter Engineering the Input and Output has been split into two: Chapter 7 on data transformations (which mostly concern the input) and Chapter 8 on ensemble learning (the output). To the former we have added information on partial least-squares regression, reservoir sampling, one-class learning, decomposing multiclass classification problems into ensembles of nested dichotomies, and calibrating class probabilities. To the latter we have added new material on randomization versus bagging and rotation forests. New sections on data stream learning and web mining have been added to the last chapter of Part II. Part III, on the Weka data mining workbench, contains a lot of new information. Weka includes many new filters, machine learning algorithms, and attribute selection algorithms, and many new components such as converters for different file formats and parameter optimization algorithms. Indeed, within each of these categories Weka Preface xxvii contains around 50% more algorithms than in the version described in the second edition of this book. All these are documented here. In response to popular demand we have given substantially more detail about the output of the different classifiers and what it all means. One important change is the inclusion of a brand new Chapter 17 that gives several tutorial exercises for the Weka Explorer interface (some of them quite challenging), which we advise new users to work though to get an idea of what Weka can do. This page intentionally left blank xxix Acknowledgments Writing the acknowledgments is always the nicest part! A lot of people have helped us, and we relish this opportunity to thank them. This book has arisen out of the machine learning research project in the Computer Science Department at the Uni- versity of Waikato, New Zealand. We received generous encouragement and assis- tance from the academic staff members early on in that project: John Cleary, Sally Jo Cunningham, Matt Humphrey, Lyn Hunt, Bob McQueen, Lloyd Smith, and Tony Smith. Special thanks go to Geoff Holmes, the project leader and source of inspira- tion, and Bernhard Pfahringer, both of whom also had significant input into many different aspects of the Weka software. All who have worked on the machine learn- ing project here have contributed to our thinking: We would particularly like to mention early students Steve Garner, Stuart Inglis, and Craig Nevill-Manning for helping us to get the project off the ground in the beginning, when success was less certain and things were more difficult. The Weka system that illustrates the ideas in this book forms a crucial component of it. It was conceived by the authors and designed and implemented principally by Eibe Frank, Mark Hall, Peter Reutemann, and Len Trigg, but many people in the machine learning laboratory at Waikato made significant early contributions. Since the first edition of this book, the Weka team has expanded considerably: So many people have contributed that it is impossible to acknowledge everyone properly. We are grateful to Remco Bouckaert for his Bayes net package and many other contribu- tions, Lin Dong for her implementations of multi-instance learning methods, Dale Fletcher for many database-related aspects, James Foulds for his work on multi- instance filtering, Anna Huang for information bottleneck clustering, Martin Gütlein for his work on feature selection, Kathryn Hempstalk for her one-class classifier, Ashraf Kibriya and Richard Kirkby for contributions far too numerous to list, Niels Landwehr for logistic model trees, Chi-Chung Lau for creating all the icons for the Knowledge Flow interface, Abdelaziz Mahoui for the implementation of K*, Stefan Mutter for association-rule mining, Malcolm Ware for numerous miscellaneous contributions, Haijian Shi for his implementations of tree learners, Marc Sumner for his work on speeding up logistic model trees, Tony Voyle for least-median-of- squares regression, Yong Wang for Pace regression and the original implementation of M5′, and Xin Xu for his multi-instance learning package, JRip, logistic regression, and many other contributions. Our sincere thanks go to all these people for their dedicated work, and also to the many contributors to Weka from outside our group at Waikato. Tucked away as we are in a remote (but very pretty) corner of the southern hemisphere, we greatly appreciate the visitors to our department who play a crucial role in acting as sounding boards and helping us to develop our thinking. We would like to mention in particular Rob Holte, Carl Gutwin, and Russell Beale, each of whom visited us for several months; David Aha, who although he only came for a few days did so at an early and fragile stage of the project and performed a great xxx Acknowledgments service by his enthusiasm and encouragement; and Kai Ming Ting, who worked with us for two years on many of the topics described in Chapter 8 and helped to bring us into the mainstream of machine learning. More recent visitors include Arie Ben- David, Carla Brodley, and Stefan Kramer. We would particularly like to thank Albert Bifet, who gave us detailed feedback on a draft version of the third edition, most of which we have incorporated. Students at Waikato have played a significant role in the development of the project. Many of them are in the above list of Weka contributors, but they have also contributed in other ways. In the early days, Jamie Littin worked on ripple-down rules and relational learning. Brent Martin explored instance-based learning and nested instance-based representations, Murray Fife slaved over relational learning, and Nadeeka Madapathage investigated the use of functional languages for express- ing machine learning algorithms. More recently, Kathryn Hempstalk worked on one-class learning and her research informs part of Section 7.5; likewise, Richard Kirkby’s research on data streams informs Section 9.3. Some of the exercises in Chapter 17 were devised by Gabi Schmidberger, Richard Kirkby, and Geoff Holmes. Other graduate students have influenced us in numerous ways, particularly Gordon Paynter, YingYing Wen, and Zane Bray, who have worked with us on text mining, and Quan Sun and Xiaofeng Yu. Colleagues Steve Jones and Malika Mahoui have also made far-reaching contributions to these and other machine learning projects. We have also learned much from our many visiting students from Freiburg, including Nils Weidmann. Ian Witten would like to acknowledge the formative role of his former students at Calgary, particularly Brent Krawchuk, Dave Maulsby, Thong Phan, and Tanja Mitrovic, all of whom helped him develop his early ideas in machine learning, as did faculty members Bruce MacDonald, Brian Gaines, and David Hill at Calgary, and John Andreae at the University of Canterbury. Eibe Frank is indebted to his former supervisor at the University of Karlsruhe, Klaus-Peter Huber, who infected him with the fascination of machines that learn. On his travels, Eibe has benefited from interactions with Peter Turney, Joel Martin, and Berry de Bruijn in Canada; Luc de Raedt, Christoph Helma, Kristian Kersting, Stefan Kramer, Ulrich Rückert, and Ashwin Srinivasan in Germany. Mark Hall thanks his former supervisor Lloyd Smith, now at Missouri State University, who exhibited the patience of Job when his thesis drifted from its original topic into the realms of machine learning. The many and varied people who have been part of, or have visited, the machine learning group at the University of Waikato over the years deserve a special thanks for their valuable insights and stimulating discussions. Rick Adams and David Bevans of Morgan Kaufmann have worked hard to shape this book, and Marilyn Rash, our project manager, has made the process go very smoothly. We would like to thank the librarians of the Repository of Machine Learn- ing Databases at the University of California, Irvine, whose carefully collected datasets have been invaluable in our research. Acknowledgments xxxi Our research has been funded by the New Zealand Foundation for Research, Science, and Technology and the Royal Society of New Zealand Marsden Fund. The Department of Computer Science at the University of Waikato has generously sup- ported us in all sorts of ways, and we owe a particular debt of gratitude to Mark Apperley for his enlightened leadership and warm encouragement. Part of the first edition was written while both authors were visiting the University of Calgary, Canada, and the support of the Computer Science department there is gratefully acknowledged, as well as the positive and helpful attitude of the long-suffering students in the machine learning course, on whom we experimented. Part of the second edition was written at the University of Lethbridge in Southern Alberta on a visit supported by Canada’s Informatics Circle of Research Excellence. Last, and most of all, we are grateful to our families and partners. Pam, Anna, and Nikki were all too well aware of the implications of having an author in the house (“Not again!”), but let Ian go ahead and write the book anyway. Julie was always supportive, even when Eibe had to burn the midnight oil in the machine learning lab, and Immo and Ollig provided exciting diversions. Bernadette too was very supportive, somehow managing to keep the combined noise output of Charlotte, Luke, Zach, and Kyle to a level that allowed Mark to concentrate. Among us, we hail from Canada, England, Germany, Ireland, New Zealand, and Samoa: New Zealand has brought us together and provided an ideal, even idyllic, place to do this work. This page intentionally left blank xxxiii Ian H. Witten is a professor of computer science at the University of Waikato in New Zealand. His research interests include language learning, information retrieval, and machine learning. He has published widely, including several books: Managing Gigabytes (1999), Data Mining (2005), Web Dragons (2007), and How to Build a Digital Library (2003). He is a Fellow of the ACM and of the Royal Society of New Zealand. He received the 2004 IFIP Namur Award, a biennial honor accorded for “outstanding contribution with international impact to the awareness of social implications of information and communication technology,” and (with the rest of the Weka team) received the 2005 SIGKDD Service Award for “an outstanding contribution to the data mining field.” In 2006, he received the Royal Society of New Zealand Hector Medal for “an outstanding contribution to the advancement of the mathematical and information sciences,” and in 2010 was officially inaugurated as a “World Class New Zealander” in research, science, and technology. Eibe Frank lives in New Zealand with his Samoan spouse and two lovely boys, but originally hails from Germany, where he received his first degree in computer science from the University of Karlsruhe. He moved to New Zealand to pursue his Ph.D. in machine learning under the supervision of Ian H. Witten, and joined the Department of Computer Science at the University of Waikato as a lecturer on completion of his studies. He is now an associate professor at the same institution. As an early adopter of the Java programming language, he laid the groundwork for the Weka software described in this book. He has contributed a number of publica- tions on machine learning and data mining to the literature and has refereed for many conferences and journals in these areas. Mark A. Hall was born in England but moved to New Zealand with his parents as a young boy. He now lives with his wife and four young children in a small town situated within a hour’s drive of the University of Waikato. He holds a bachelor’s degree in computing and mathematical sciences and a Ph.D. in computer science, both from the University of Waikato. Throughout his time at Waikato, as a student and lecturer in computer science and more recently as a software developer and data mining consultant for Pentaho, an open-source business intelligence software company, Mark has been a core contributor to the Weka software described in this book. He has published a number of articles on machine learning and data mining and has refereed for conferences and journals in these areas. About the Authors This page intentionally left blank PART IIntroduction to Data Mining This page intentionally left blank 3Data Mining: Practical Machine Learning Tools and Techniques Copyright © 2011 Elsevier Inc. All rights of reproduction in any form reserved. CHAPTER 1 What’s It All About? Human in vitro fertilization involves collecting several eggs from a woman’s ovaries, which, after fertilization with partner or donor sperm, produce several embryos. Some of these are selected and transferred to the woman’s uterus. The challenge is to select the “best” embryos to use—the ones that are most likely to survive. Selec- tion is based on around 60 recorded features of the embryos—characterizing their morphology, oocyte, and follicle, and the sperm sample. The number of features is large enough to make it difficult for an embryologist to assess them all simultane- ously and correlate historical data with the crucial outcome of whether that embryo did or did not result in a live child. In a research project in England, machine learn- ing has been investigated as a technique for making the selection, using historical records of embryos and their outcome as training data. Every year, dairy farmers in New Zealand have to make a tough business deci- sion: which cows to retain in their herd and which to sell off to an abattoir. Typically, one-fifth of the cows in a dairy herd are culled each year near the end of the milking season as feed reserves dwindle. Each cow’s breeding and milk production history influences this decision. Other factors include age (a cow nears the end of its pro- ductive life at eight years), health problems, history of difficult calving, undesirable temperament traits (kicking or jumping fences), and not being pregnant with calf for the following season. About 700 attributes for each of several million cows have been recorded over the years. Machine learning has been investigated as a way of ascertaining what factors are taken into account by successful farmers—not to automate the decision but to propagate their skills and experience to others. Life and death. From Europe to the Antipodes. Family and business. Machine learning is a burgeoning new technology for mining knowledge from data, a technology that a lot of people are starting to take seriously. 1.1 DATA MINING AND MACHINE LEARNING We are overwhelmed with data. The amount of data in the world and in our lives seems ever-increasing—and there’s no end in sight. Omnipresent computers make it too easy to save things that previously we would have trashed. Inexpensive disks and online storage make it too easy to postpone decisions about what to do with all 4 CHAPTER 1 What’s It All About? this stuff—we simply get more memory and keep it all. Ubiquitous electronics record our decisions, our choices in the supermarket, our financial habits, our comings and goings. We swipe our way through the world, every swipe a record in a database. The World Wide Web (WWW) overwhelms us with information; mean- while, every choice we make is recorded. And all of these are just personal choices— they have countless counterparts in the world of commerce and industry. We could all testify to the growing gap between the generation of data and our understanding of it. As the volume of data increases, inexorably, the proportion of it that people understand decreases alarmingly. Lying hidden in all this data is information— potentially useful information—that is rarely made explicit or taken advantage of. This book is about looking for patterns in data. There is nothing new about this. People have been seeking patterns in data ever since human life began. Hunters seek patterns in animal migration behavior, farmers seek patterns in crop growth, politi- cians seek patterns in voter opinion, and lovers seek patterns in their partners’ responses. A scientist’s job (like a baby’s) is to make sense of data, to discover the patterns that govern how the physical world works and encapsulate them in theories that can be used for predicting what will happen in new situations. The entrepre- neur’s job is to identify opportunities—that is, patterns in behavior that can be turned into a profitable business—and exploit them. In data mining, the data is stored electronically and the search is automated—or at least augmented—by computer. Even this is not particularly new. Economists, statisticians, forecasters, and communication engineers have long worked with the idea that patterns in data can be sought automatically, identified, validated, and used for prediction. What is new is the staggering increase in opportunities for finding patterns in data. The unbridled growth of databases in recent years, databases for such everyday activities as customer choices, brings data mining to the forefront of new business technologies. It has been estimated that the amount of data stored in the world’s databases doubles every 20 months, and although it would surely be difficult to justify this figure in any quantitative sense, we can all relate to the pace of growth qualitatively. As the flood of data swells and machines that can undertake the searching become commonplace, the opportunities for data mining increase. As the world grows in complexity, overwhelming us with the data it generates, data mining becomes our only hope for elucidating hidden patterns. Intelligently analyzed data is a valuable resource. It can lead to new insights, and, in commercial settings, to competitive advantages. Data mining is about solving problems by analyzing data already present in databases. Suppose, to take a well-worn example, the problem is fickle customer loyalty in a highly competitive marketplace. A database of customer choices, along with customer profiles, holds the key to this problem. Patterns of behavior of former customers can be analyzed to identify distinguishing characteristics of those likely to switch products and those likely to remain loyal. Once such characteristics are found, they can be put to work to identify present customers who are likely to jump ship. This group can be targeted for special treatment, treatment too costly to apply to the customer base as a whole. More positively, the same techniques can be used 1.1 Data Mining and Machine Learning 5 to identify customers who might be attracted to another service the enterprise pro- vides, one they are not presently enjoying, to target them for special offers that promote this service. In today’s highly competitive, customer-centered, service- oriented economy, data is the raw material that fuels business growth—if only it can be mined. Data mining is defined as the process of discovering patterns in data. The process must be automatic or (more usually) semiautomatic. The patterns discovered must be meaningful in that they lead to some advantage, usually an economic one. The data is invariably present in substantial quantities. And how are the patterns expressed? Useful patterns allow us to make nontrivial predictions on new data. There are two extremes for the expression of a pattern: as a black box whose innards are effectively incomprehensible, and as a transparent box whose construction reveals the structure of the pattern. Both, we are assuming, make good predictions. The difference is whether or not the patterns that are mined are represented in terms of a structure that can be examined, reasoned about, and used to inform future decisions. Such patterns we call structural because they capture the decision structure in an explicit way. In other words, they help to explain something about the data. Now, again, we can say what this book is about: It is about techniques for finding and describing structural patterns in data. Most of the techniques that we cover have developed within a field known as machine learning. But first let us look at what structural patterns are. Describing Structural Patterns What is meant by structural patterns? How do you describe them? And what form does the input take? We will answer these questions by way of illustration rather than by attempting formal, and ultimately sterile, definitions. There will be plenty of examples later in this chapter, but let’s examine one right now to get a feeling for what we’re talking about. Look at the contact lens data in Table 1.1. It gives the conditions under which an optician might want to prescribe soft contact lenses, hard contact lenses, or no contact lenses at all; we will say more about what the individual features mean later. Each line of the table is one of the examples. Part of a structural description of this information might be as follows: If tear production rate = reduced then recommendation = none Otherwise, if age = young and astigmatic = no then recommendation = soft Structural descriptions need not necessarily be couched as rules such as these. Deci- sion trees, which specify the sequences of decisions that need to be made along with the resulting recommendation, are another popular means of expression. This example is a very simplistic one. For a start, all combinations of possible values are represented in the table. There are 24 rows, representing three possible 6 CHAPTER 1 What’s It All About? Table 1.1 Contact Lens Data Age Spectacle Prescription Astigmatism Tear Production Rate Recommended Lenses young myope no reduced none young myope no normal soft young myope yes reduced none young myope yes normal hard young hypermetrope no reduced none young hypermetrope no normal soft young hypermetrope yes reduced none young hypermetrope yes normal hard pre-presbyopic myope no reduced none pre-presbyopic myope no normal soft pre-presbyopic myope yes reduced none pre-presbyopic myope yes normal hard pre-presbyopic hypermetrope no reduced none pre-presbyopic hypermetrope no normal soft pre-presbyopic hypermetrope yes reduced none pre-presbyopic hypermetrope yes normal none presbyopic myope no reduced none presbyopic myope no normal none presbyopic myope yes reduced none presbyopic myope yes normal hard presbyopic hypermetrope no reduced none presbyopic hypermetrope no normal soft presbyopic hypermetrope yes reduced none presbyopic hypermetrope yes normal none values of age and two values each for spectacle prescription, astigmatism, and tear production rate (3 × 2 × 2 × 2 = 24). The rules do not really generalize from the data; they merely summarize it. In most learning situations, the set of examples given as input is far from complete, and part of the job is to generalize to other, new examples. You can imagine omitting some of the rows in the table for which the tear production rate is reduced and still coming up with the rule If tear production rate = reduced then recommendation = none This would generalize to the missing rows and fill them in correctly. Second, values are specified for all the features in all the examples. Real-life datasets invariably contain examples in which the values of some features, for some reason or other, are unknown—for example, measurements were not taken or were lost. Third, the preceding rules classify the examples correctly, whereas often, because of errors or noise in the data, misclassifications occur even on the data that is used to create the classifier. Machine Learning Now that we have some idea of the inputs and outputs, let’s turn to machine learn- ing. What is learning, anyway? What is machine learning? These are philosophical questions, and we will not be too concerned with philosophy in this book; our emphasis is firmly on the practical. However, it is worth spending a few moments at the outset on fundamental issues, just to see how tricky they are, before rolling up our sleeves and looking at machine learning in practice. Our dictionary defines “to learn” as • To get knowledge of something by study, experience, or being taught. • To become aware by information or from observation • To commit to memory • To be informed of or to ascertain • To receive instruction These meanings have some shortcomings when it comes to talking about computers. For the first two, it is virtually impossible to test whether learning has been achieved or not. How do you know whether a machine has got knowledge of something? You probably can’t just ask it questions; even if you could, you wouldn’t be testing its ability to learn but its ability to answer questions. How do you know whether it has become aware of something? The whole question of whether computers can be aware, or conscious, is a burning philosophical issue. As for the last three meanings, although we can see what they denote in human terms, merely committing to memory and receiving instruction seem to fall far short of what we might mean by machine learning. They are too passive, and we know that computers find these tasks trivial. Instead, we are interested in improvements in performance, or at least in the potential for performance, in new situations. You can commit something to memory or be informed of something by rote learning without being able to apply the new knowledge to new situations. In other words, you can receive instruction without benefiting from it at all. Earlier we defined data mining operationally, as the process of discovering pat- terns, automatically or semiautomatically, in large quantities of data—and the pat- terns must be useful. An operational definition can be formulated in the same way for learning: • Things learn when they change their behavior in a way that makes them perform better in the future This ties learning to performance rather than knowledge. You can test learning by observing present behavior and comparing it with past behavior. This is a much more objective kind of definition and appears to be far more satisfactory. 1.1 Data Mining and Machine Learning 7 8 CHAPTER 1 What’s It All About? But still there’s a problem. Learning is a rather slippery concept. Lots of things change their behavior in ways that make them perform better in the future, yet we wouldn’t want to say that they have actually learned. A good example is a comfort- able slipper. Has it learned the shape of your foot? It has certainly changed its behavior to make it perform better as a slipper! Yet we would hardly want to call this learning. In everyday language, we often use the word training to denote a mindless kind of learning. We train animals and even plants, although it would be stretching the word a bit to talk of training objects such as slippers, which are not in any sense alive. But learning is different. Learning implies thinking and purpose. Something that learns has to do so intentionally. That is why we wouldn’t say that a vine has learned to grow around a trellis in a vineyard—we’d say it has been trained. Learning without purpose is merely training. Or, more to the point, in learning the purpose is the learner’s, whereas in training it is the teacher’s. Thus, on closer examination the second definition of learning, in operational, performance-oriented terms, has its own problems when it comes to talking about computers. To decide whether something has actually learned, you need to see whether it intended to, whether there was any purpose involved. That makes the concept moot when applied to machines because whether artifacts can behave pur- posefully is unclear. Philosophical discussions of what is really meant by learning, like discussions of what is really meant by intention or purpose, are fraught with difficulty. Even courts of law find intention hard to grapple with. Data Mining Fortunately, the kind of learning techniques explained in this book do not present these conceptual problems—they are called machine learning without really presup- posing any particular philosophical stance about what learning actually is. Data mining is a topic that involves learning in a practical, nontheoretical sense. We are interested in techniques for finding and describing structural patterns in data, as a tool for helping to explain that data and make predictions from it. The data will take the form of a set of examples, such as customers who have switched loyalties, for instance, or situations in which certain kinds of contact lenses can be prescribed. The output takes the form of predictions about new examples—a prediction of whether a particular customer will switch or a prediction of what kind of lens will be prescribed under given circumstances. But because this book is about finding and describing patterns in data, the output may also include an actual description of a structure that can be used to classify unknown examples. As well as performance, it is helpful to supply an explicit representation of the knowledge that is acquired. In essence, this reflects both definitions of learning considered above: the acquisition of knowledge and the ability to use it. Many learning techniques look for structural descriptions of what is learned— descriptions that can become fairly complex and are typically expressed as sets of rules, such as the ones described previously or the decision trees described later in this chapter. Because they can be understood by people, these descriptions serve to 1.2 Simple Examples: The Weather and Other Problems 9 explain what has been learned—in other words, to explain the basis for new predic- tions. Experience shows that in many applications of machine learning to data mining, the explicit knowledge structures that are acquired, the structural descrip- tions, are at least as important as the ability to perform well on new examples. People frequently use data mining to gain knowledge, not just predictions. Gaining knowl- edge from data certainly sounds like a good idea if you can do it. To find out how, read on! 1.2 SIMPLE EXAMPLES: THE WEATHER AND OTHER PROBLEMS We will be using a lot of examples in this book, which seems particularly appropriate considering that the book is all about learning from examples! There are several standard datasets that we will come back to repeatedly. Different datasets tend to expose new issues and challenges, and it is interesting and instructive to have in mind a variety of problems when considering learning methods. In fact, the need to work with different datasets is so important that a corpus containing around 100 example problems has been gathered together so that different algorithms can be tested and compared on the same set of problems. The set of problems in this section are all unrealistically simple. Serious appli- cation of data mining involves thousands, hundreds of thousands, or even millions of individual cases. But when explaining what algorithms do and how they work, we need simple examples that capture the essence of the problem but are small enough to be comprehensible in every detail. We will be working with the datasets in this section throughout the book, and they are intended to be “academic” in the sense that they will help us to understand what is going on. Some actual fielded applications of learning techniques are discussed in Section 1.3, and many more are covered in the books mentioned in Section 1.7, Further reading, at the end of the chapter. Another problem with actual real-life datasets is that they are often proprietary. No one is going to share their customer and product choice database with you so that you can understand the details of their data mining application and how it works. Corporate data is a valuable asset, the value of which has increased enormously with the development of data mining techniques such as those described in this book. Yet, we are concerned here with understanding how the methods used for data mining work, and understanding the details of these methods so that we can trace their operation on actual data. That is why our illustrative datasets are simple ones. But they are not simplistic: They exhibit the features of real datasets. The Weather Problem The weather problem is a tiny dataset that we will use repeatedly to illustrate machine learning methods. Entirely fictitious, it supposedly concerns the conditions 10 CHAPTER 1 What’s It All About? Table 1.2 Weather Data Outlook Temperature Humidity Windy Play Sunny hot high false no Sunny hot high true no Overcast hot high false yes Rainy mild high false yes Rainy cool normal false yes Rainy cool normal true no Overcast cool normal true yes Sunny mild high false no Sunny cool normal false yes Rainy mild normal false yes Sunny mild normal true yes Overcast mild high true yes Overcast hot normal false yes Rainy mild high true no that are suitable for playing some unspecified game. In general, instances in a dataset are characterized by the values of features, or attributes, that measure different aspects of the instance. In this case there are four attributes: outlook, temperature, humidity, and windy. The outcome is whether to play or not. In its simplest form, shown in Table 1.2, all four attributes have values that are symbolic categories rather than numbers. Outlook can be sunny, overcast, or rainy; temperature can be hot, mild, or cool; humidity can be high or normal; and windy can be true or false. This creates 36 possible combinations (3 × 3 × 2 × 2 = 36), of which 14 are present in the set of input examples. A set of rules learned from this information—not necessarily a very good one— might look like this: If outlook = sunny and humidity = high then play = no If outlook = rainy and windy = true then play = no If outlook = overcast then play = yes If humidity = normal then play = yes If none of the above then play = yes These rules are meant to be interpreted in order: The first one; then, if it doesn’t apply, the second; and so on. A set of rules that are intended to be interpreted in sequence is called a decision list. Interpreted as a decision list, the rules correctly classify all of the examples in the table, whereas taken individually, out of context, some of the rules are incorrect. For example, the rule if humidity = normal then play = yes gets one of the examples wrong (check which one). The meaning of a set of rules depends on how it is interpreted—not surprisingly! In the slightly more complex form shown in Table 1.3, two of the attributes— temperature and humidity—have numeric values. This means that any learning Table 1.3 Weather Data with Some Numeric Attributes Outlook Temperature Humidity Windy Play Sunny 85 85 false no Sunny 80 90 true no Overcast 83 86 false yes Rainy 70 96 false yes Rainy 68 80 false yes Rainy 65 70 true no Overcast 64 65 true yes Sunny 72 95 false no Sunny 69 70 false yes Rainy 75 80 false yes Sunny 75 70 true yes Overcast 72 90 true yes Overcast 81 75 false yes Rainy 71 91 true no scheme must create inequalities involving these attributes rather than simple equality tests as in the former case. This is called a numeric-attribute problem—in this case, a mixed-attribute problem because not all attributes are numeric. Now the first rule given earlier might take the form If outlook = sunny and humidity > 83 then play = no A slightly more complex process is required to come up with rules that involve numeric tests. The rules we have seen so far are classification rules: They predict the classifica- tion of the example in terms of whether to play or not. It is equally possible to disregard the classification and just look for any rules that strongly associate different attribute values. These are called association rules. Many association rules can be derived from the weather data in Table 1.2. Some good ones are If temperature = cool then humidity = normal If humidity = normal and windy = false then play = yes If outlook = sunny and play = no then humidity = high If windy = false and play = no then outlook = sunny and humidity = high All these rules are 100% correct on the given data; they make no false predic- tions. The first two apply to four examples in the dataset, the third to three examples, and the fourth to two examples. And there are many other rules. In fact, nearly 60 association rules can be found that apply to two or more examples of the weather data and are completely correct on this data. And if you look for rules that are less than 100% correct, then you will find many more. There are so many because, unlike 1.2 Simple Examples: The Weather and Other Problems 11 12 CHAPTER 1 What’s It All About? FIGURE 1.1 Rules for the contact lens data. If tear production rate = reduced then recommendation = none. If age = young and astigmatic = no and tear production rate = normal then recommendation = soft If age = pre-presbyopic and astigmatic = no and tear production rate = normal then recommendation = soft If age = presbyopic and spectacle prescription = myope and astigmatic = no then recommendation = none If spectacle prescription = hypermetrope and astigmatic = no and tear production rate = normal then recommendation = soft If spectacle prescription = myope and astigmatic = yes and tear production rate = normal then recommendation = hard If age = young and astigmatic = yes and tear production rate = normal then recommendation = hard If age = pre-presbyopic and spectacle prescription = hypermetrope and astigmatic = yes then recommendation = none If age = presbyopic and spectacle prescription = hypermetrope and astigmatic = yes then recommendation = none classification rules, association rules can “predict” any of the attributes, not just a specified class, and can even predict more than one thing. For example, the fourth rule predicts both that outlook will be sunny and that humidity will be high. Contact Lenses: An Idealized Problem The contact lens data introduced earlier tells you the kind of contact lens to prescribe, given certain information about a patient. Note that this example is intended for illustration only: It grossly oversimplifies the problem and should certainly not be used for diagnostic purposes! The first column of Table 1.1 gives the age of the patient. In case you’re wonder- ing, presbyopia is a form of longsightedness that accompanies the onset of middle age. The second gives the spectacle prescription: Myope means shortsighted and hypermetrope means longsighted. The third shows whether the patient is astigmatic, while the fourth relates to the rate of tear production, which is important in this context because tears lubricate contact lenses. The final column shows which kind of lenses to prescribe, whether hard, soft, or none. All possible combinations of the attribute values are represented in the table. A sample set of rules learned from this information is shown in Figure 1.1. This is a rather large set of rules, but they do correctly classify all the examples. These rules are complete and deterministic: They give a unique prescription for every conceivable example. Generally this is not the case. Sometimes there are situations in which no rule applies; other times more than one rule may apply, resulting in FIGURE 1.2 Decision tree for the contact lens data. tear production rate none reduced astigmatism normal soft no spectacle prescription yes hard myope none hypermetrope conflicting recommendations. Sometimes probabilities or weights may be associated with the rules themselves to indicate that some are more important, or more reliable, than others. You might be wondering whether there is a smaller rule set that performs as well. If so, would you be better off using the smaller rule set, and, if so, why? These are exactly the kinds of questions that will occupy us in this book. Because the examples form a complete set for the problem space, the rules do no more than summarize all the information that is given, expressing it in a different and more concise way. Even though it involves no generalization, this is often a very useful thing to do! People frequently use machine learning techniques to gain insight into the structure of their data rather than to make predictions for new cases. In fact, a prominent and success- ful line of research in machine learning began as an attempt to compress a huge database of possible chess endgames and their outcomes into a data structure of reasonable size. The data structure chosen for this enterprise was not a set of rules but a decision tree. Figure 1.2 shows a structural description for the contact lens data in the form of a decision tree, which for many purposes is a more concise and perspicuous repre- sentation of the rules and has the advantage that it can be visualized more easily. (However, this decision tree, in contrast to the rule set given in Figure 1.1, classifies two examples incorrectly.) The tree calls first for a test on the tear production rate, and the first two branches correspond to the two possible outcomes. If the tear production rate is reduced (the left branch), the outcome is none. If it is normal (the right branch), a second test is made, this time on astigmatism. Eventually, whatever the outcome of the tests, a leaf of the tree is reached that dictates the contact lens recommendation for that case. The question of what is the most natural and easily understood format for the output from a machine learning scheme is one that we will return to in Chapter 3. Irises: A Classic Numeric Dataset The iris dataset, which dates back to seminal work by the eminent statistician R. A. Fisher in the mid- 1930s and is arguably the most famous dataset used in data mining, contains 50 examples of each of three types of plant: Iris setosa, Iris versicolor, and Iris virginica. It is excerpted in Table 1.4. There are four attributes: sepal length, 1.2 Simple Examples: The Weather and Other Problems 13 14 CHAPTER 1 What’s It All About? Table 1.4 Iris Data Sepal Length (cm) Sepal Width (cm) Petal Length (cm) Petal Width (cm) Type 1 5.1 3.5 1.4 0.2 Iris setosa 2 4.9 3.0 1.4 0.2 Iris setosa 3 4.7 3.2 1.3 0.2 Iris setosa 4 4.6 3.1 1.5 0.2 Iris setosa 5 5.0 3.6 1.4 0.2 Iris setosa … 51 7.0 3.2 4.7 1.4 Iris versicolor 52 6.4 3.2 4.5 1.5 Iris versicolor 53 6.9 3.1 4.9 1.5 Iris versicolor 54 5.5 2.3 4.0 1.3 Iris versicolor 55 6.5 2.8 4.6 1.5 Iris versicolor … 101 6.3 3.3 6.0 2.5 Iris virginica 102 5.8 2.7 5.1 1.9 Iris virginica 103 7.1 3.0 5.9 2.1 Iris virginica 104 6.3 2.9 5.6 1.8 Iris virginica 105 6.5 3.0 5.8 2.2 Iris virginica … sepal width, petal length, and petal width (all measured in centimeters). Unlike previous datasets, all attributes have values that are numeric. The following set of rules might be learned from this dataset: If petal-length < 2.45 then Iris-setosa If sepal-width < 2.10 then Iris-versicolor If sepal-width < 2.45 and petal-length < 4.55 then Iris-versicolor If sepal-width < 2.95 and petal-width < 1.35 then Iris-versicolor If petal-length ≥ 2.45 and petal-length < 4.45 then Iris-versicolor If sepal-length ≥ 5.85 and petal-length < 4.75 then Iris-versicolor If sepal-width < 2.55 and petal-length < 4.95 and petal-width < 1.55 then Iris-versicolor If petal-length ≥ 2.45 and petal-length < 4.95 and petal-width < 1.55 then Iris-versicolor If sepal-length ≥ 6.55 and petal-length < 5.05 then Iris-versicolor If sepal-width < 2.75 and petal-width < 1.65 and sepal-length < 6.05 then Iris-versicolor If sepal-length ≥ 5.85 and sepal-length < 5.95 and petal-length < 4.85 then Iris-versicolor If petal-length ≥ 5.15 then Iris-virginica If petal-width ≥ 1.85 then Iris-virginica If petal-width ≥ 1.75 and sepal-width < 3.05 then Iris-virginica If petal-length ≥ 4.95 and petal-width < 1.55 then Iris-virginica These rules are very cumbersome, and we will see in Chapter 3 how more compact rules can be expressed that convey the same information. CPU Performance: Introducing Numeric Prediction Although the iris dataset involves numeric attributes, the outcome—the type of iris—is a category, not a numeric value. Table 1.5 shows some data for which both the outcome and the attributes are numeric. It concerns the relative performance of computer processing power on the basis of a number of relevant attributes; each row represents one of 209 different computer configurations. The classic way of dealing with continuous prediction is to write the outcome as a linear sum of the attribute values with appropriate weights, for example, PRPMYCTMMINMMAX CACH = − + + + + − 55 9 0 0489 0 0153 0 0056 0 6410 ... . . 002700 1 480. . CHMIN CHMAX+ (The abbreviated variable names are given in the second row of the table.) This is called a regression equation, and the process of determining the weights is called regression, a well-known procedure in statistics that we will review in Chapter 4. However, the basic regression method is incapable of discovering nonlinear relation- ships (although variants do exist—indeed, one will be described in Section 6.4), and in Chapter 3 we will examine different representations that can be used for predicting numeric quantities. In the iris and central processing unit (CPU) performance data, all the attributes have numeric values. Practical situations frequently present a mixture of numeric and nonnumeric attributes. Labor Negotiations: A More Realistic Example The labor negotiations dataset in Table 1.6 summarizes the outcome of Canadian contract negotiations in 1987 and 1988. It includes all collective agreements reached in the business and personal services sector for organizations with at least 500 members (teachers, nurses, university staff, police, etc.). Each case concerns one contract, and the outcome is whether the contract is deemed acceptable or unaccept- able. The acceptable contracts are ones in which agreements were accepted by both labor and management. The unacceptable ones are either known offers that fell through because one party would not accept them or acceptable contracts that had been significantly perturbed to the extent that, in the view of experts, they would not have been accepted. There are 40 examples in the dataset (plus another 17 that are normally reserved for test purposes). Unlike the other tables here, Table 1.6 presents the examples as columns rather than as rows; otherwise, it would have to be stretched over several pages. Many of the values are unknown or missing, as indicated by ques- tion marks. This is a much more realistic dataset than the others we have seen. 1.2 Simple Examples: The Weather and Other Problems 15 16 Table 1.5 CPU Performance Data Main Memory (Kb) Channels Performance Cycle Time (ns) Min Max Cache (KB) Min Max MYCT MMIN MMAX CACH CHMIN CHMAX PRP 1 125 256 6000 256 16 128 198 2 29 8000 32,000 32 8 32 269 3 29 8000 32,000 32 8 32 220 4 29 8000 32,000 32 8 32 172 5 29 8000 16,000 32 8 16 132 … 207 125 2000 8000 0 2 14 52 208 480 512 8000 32 0 0 67 209 480 1000 4000 0 0 0 45 Table 1.6 Labor Negotiations Data Attribute Type 1 2 3 … 40 duration (number of years) 1 2 3 2 wage increase 1st year percentage 2% 4% 4.3% 4.5 wage increase 2nd year percentage ? 5% 4.4% 4.0 wage increase 3rd year percentage ? ? ? ? cost-of-living adjustment {none, tcf, tc} none tcf ? none working hours per week (number of hours) 28 35 38 40 pension {none, ret-allw, empl-cntr} none ? ? ? standby pay percentage ? 13% ? ? shift-work supplement percentage ? 5% 4% 4 education allowance {yes, no} yes ? ? ? statutory holidays (number of days) 11 15 12 12 vacation {below-avg, avg, gen} avg gen gen avg long-term disability assistance {yes, no} no ? ? yes dental plan contribution {none, half, full} none ? full full bereavement assistance {yes, no} no ? ? yes health plan contribution {none, half, full} none ? full half acceptability of contract {good, bad} bad good good good 17 18 CHAPTER 1 What’s It All About? FIGURE 1.3 Decision trees for the labor negotiations data. wage increase 1st year working hours per week ≤2.5 statutory holidays >2.5 bad ≤36 health plan contribution >36 good >10 wage increase 1st year ≤10 bad none good half bad full bad ≤4 good >4 wage increase 1st year bad ≤2.5 statutory holidays >2.5 good >10 wage increase 1st year ≤10 bad ≤4 good >4 (a) (b) It contains many missing values, and it seems unlikely that an exact classification can be obtained. Figure 1.3 shows two decision trees that represent the dataset. Figure 1.3(a) is simple and approximate—it doesn’t represent the data exactly. For example, it will predict bad for some contracts that are actually marked good. However, it does make intuitive sense: A contract is bad (for the employee!) if the wage increase in the first year is too small (less than 2.5%). If the first-year wage increase is larger than this, it is good if there are lots of statutory holidays (more than 10 days). Even if there are fewer statutory holidays, it is good if the first-year wage increase is large enough (more than 4%). Figure 1.3(b) is a more complex decision tree that represents the same dataset. Take a detailed look down the left branch. At first sight it doesn’t seem to make sense intuitively that, if the working hours exceed 36, a contract is bad if there is no health-plan contribution or a full health-plan contribution, but is good if there is a half health-plan contribution. It is certainly reasonable that the health-plan contri- bution plays a role in the decision, but it seems anomalous that half is good and both full and none are bad. However, on reflection this could make sense after all, because “good” contracts are ones that have been accepted by both parties: labor and man- agement. Perhaps this structure reflects compromises that had to be made to reach agreement. This kind of detailed reasoning about what parts of decision trees mean is a good way of getting to know your data and thinking about the underlying problem. In fact, Figure 1.3(b) is a more accurate representation of the training dataset than Figure 1.3(a). But it is not necessarily a more accurate representation of the underlying concept of good versus bad contracts. Although it is more accurate on the data that was used to train the classifier, it may perform less well on an inde- pendent set of test data. It may be “overfitted” to the training data—following it too slavishly. The tree in Figure 1.3(a) is obtained from the one in Figure 1.3(b) by a process of pruning, which we will learn more about in Chapter 6. Soybean Classification: A Classic Machine Learning Success An often quoted early success story in the application of machine learning to practi- cal problems is the identification of rules for diagnosing soybean diseases. The data is taken from questionnaires describing plant diseases. There are about 680 exam- ples, each representing a diseased plant. Plants were measured on 35 attributes, each one having a small set of possible values. Examples are labeled with the diagnosis of an expert in plant biology: There are 19 disease categories altogether—horrible- sounding diseases such as diaporthe stem canker, rhizoctonia root rot, and bacterial blight, to mention just a few. Table 1.7 gives the attributes, the number of different values that each can have, and a sample record for one particular plant. The attributes are placed in different categories just to make them easier to read. Here are two example rules, learned from this data: If leaf condition = normal and stem condition = abnormal and stem cankers = below soil line and canker lesion color = brown then diagnosis is rhizoctonia root rot If leaf malformation = absent and stem condition = abnormal and stem cankers = below soil line and canker lesion color = brown then diagnosis is rhizoctonia root rot These rules nicely illustrate the potential role of prior knowledge—often called domain knowledge—in machine learning, for in fact the only difference between the two descriptions is leaf condition is normal versus leaf malformation is absent. Now, in this domain, if the leaf condition is normal then leaf malformation is necessarily absent, so one of these conditions happens to be a special case of the other. Thus, if the first rule is true, the second is necessarily true as well. The only time the second rule comes into play is when leaf malformation is absent but leaf condition is not normal—that is, when something other than malformation is wrong with the leaf. This is certainly not apparent from a casual reading of the rules. Research on this problem in the late 1970s found that these diagnostic rules could be generated by a machine learning algorithm, along with rules for every other disease category, from about 300 training examples. These training examples were carefully selected from the corpus of cases as being quite different from one another—“far apart” in the example space. At the same time, the plant pathologist who had produced the diagnoses was interviewed, and his expertise was translated 1.2 Simple Examples: The Weather and Other Problems 19 20 CHAPTER 1 What’s It All About? Table 1.7 Soybean Data Attribute Number of Values Sample Value environment time of occurrence 7 July precipitation 3 above normal temperature 3 normal cropping history 4 same as last year hail damage 2 yes damaged area 4 scattered severity 3 severe plant height 2 normal plant growth 2 abnormal seed treatment 3 fungicide germination 3 less than 80% seed condition 2 normal mold growth 2 absent discoloration 2 absent size 2 normal shriveling 2 absent fruit condition of fruit pods 3 normal fruit spots 5 — leaves condition 2 abnormal leaf spot size 3 — yellow leaf spot halo 3 absent leaf spot margins 3 — shredding 2 absent leaf malformation 2 absent leaf mildew growth 3 absent stem condition 2 abnormal stem lodging 2 yes stem cankers 4 above soil line canker lesion color 3 — fruiting bodies on stems 2 present external decay of stem 3 firm and dry mycelium on stem 2 absent internal discoloration 3 none sclerotia 2 absent roots condition 3 normal diagnosis 19 diaporthe stem canker 1.3 Fielded Applications 21 into diagnostic rules. Surprisingly, the computer-generated rules outperformed the expert-derived rules on the remaining test examples. The correct disease was ranked at the top 97.5% of the time compared with only 72% for the expert-derived rules. Furthermore, not only did the learning algorithm find rules that outperformed those of the expert collaborator, but the same expert was so impressed that he allegedly adopted the discovered rules in place of his own! 1.3 FIELDED APPLICATIONS The examples that we opened with are speculative research projects, not production systems. And the previous figures are toy problems: They are deliberately chosen to be small so that we can use them to work through algorithms later in the book. Where’s the beef? Here are some applications of machine learning that have actually been put into use. Being fielded applications, the examples that follow tend to stress the use of learning in performance situations, in which the emphasis is on the ability to perform well on new examples. This book also describes the use of learning systems to gain knowledge from decision structures that are inferred from the data. We believe that this is as important—probably even more important in the long run—a use of the technology as making high-performance predictions. Still, it will tend to be under- represented in fielded applications because when learning techniques are used to gain insight, the result is not normally a system that is put to work as an application in its own right. Nevertheless, in three of the following examples, the fact that the decision structure is comprehensible is a key feature in the successful adoption of the application. Web Mining Mining information on the World Wide Web is an exploding growth area. Search engine companies examine the hyperlinks in web pages to come up with a measure of “prestige” for each web page and web site. Dictionaries define prestige as “high standing achieved through success or influence.” A metric called PageRank, intro- duced by Google’s founders and also used in various guises by other search engine developers, attempts to measure the standing of a web page. The more pages that link to your web site, the higher its prestige, especially if the pages that link in have high prestige themselves. The definition sounds circular, but it can be made to work. Search engines use PageRank (among other things) to sort web pages into order before displaying the results of your search. Another way in which search engines tackle the problem of how to rank web pages is to use machine learning based on a training set of example queries— documents that contain the terms in the query and human judgments about how relevant the documents are to that query. Then a learning algorithm analyzes this training data and comes up with a way to predict the relevance judgment for any 22 CHAPTER 1 What’s It All About? document and query. For each document, a set of feature values is calculated that depends on the query term—for example, whether it occurs in the title tag, whether it occurs in the document’s URL, how often it occurs in the document itself, and how often it appears in the anchor text of hyperlinks that point to the document. For multiterm queries, features include how often two different terms appear close together in the document, and so on. There are many possible features—typical algorithms for learning ranks use hundreds or thousands of them. Search engines mine the content of the Web. They also mine the content of your queries—the terms you search for—to select advertisements that you might be interested in. They have a strong incentive to do this accurately because they get paid by advertisers only when users click on their links. Search engine companies mine your clicks because knowledge of which results you click on can be used to improve the search next time. Online booksellers mine the purchasing database to come up with recommendations such as “users who bought this book also bought these ones”; again, they have a strong incentive to present you with compelling, personalized choices. Movie sites recommend movies based on your previous choices and other people’s choices—they win if they make recommendations that keep customers coming back to their web site. And then there are social networks and other personal data. We live in the age of self-revelation: People share their innermost thoughts in blogs and tweets; their photographs, their music and movie tastes, their opinions of books, software, gadgets, and hotels; their social life. They may believe they are doing this anony- mously, or pseudonymously, but often they are incorrect (see Section 1.6). There is huge commercial interest in making money by mining the Web. Decisions Involving Judgment When you apply for a loan, you have to fill out a questionnaire asking for relevant financial and personal information. This information is used by the loan company as the basis for its decision as to whether to lend you money. Such decisions are typically made in two stages. First, statistical methods are used to determine clear “accept” and “reject” cases. The remaining borderline cases are more difficult and call for human judgment. For example, one loan company uses a statistical decision procedure to calculate a numeric parameter based on the information supplied in their questionnaire. Appli- cants are accepted if this parameter exceeds a preset threshold and rejected if it falls below a second threshold. This accounts for 90% of cases, and the remaining 10% are referred to loan officers for a decision. On examining historical data on whether applicants did indeed repay their loans, however, it turned out that half of the bor- derline applicants who were granted loans actually defaulted. Although it would be tempting simply to deny credit to borderline customers, credit industry professionals point out that if only their repayment future could be reliably determined, it is pre- cisely these customers whose business should be wooed; they tend to be active customers of a credit institution because their finances remain in a chronically 1.3 Fielded Applications 23 volatile condition. A suitable compromise must be reached between the viewpoint of a company accountant, who dislikes bad debt, and that of a sales executive, who dislikes turning business away. Enter machine learning. The input was 1000 training examples of borderline cases for which a loan had been made that specified whether the borrower had finally paid off or defaulted. For each training example, about 20 attributes were extracted from the questionnaire, such as age, years with current employer, years at current address, years with the bank, and other credit cards possessed. A machine learning procedure was used to produce a small set of classification rules that made correct predictions on two-thirds of the borderline cases in an independently chosen test set. Not only did these rules improve the success rate of the loan decisions, but the company also found them attractive because they could be used to explain to appli- cants the reasons behind the decision. Although the project was an exploratory one that took only a small development effort, the loan company was apparently so pleased with the result that the rules were put into use immediately. Screening Images Since the early days of satellite technology, environmental scientists have been trying to detect oil slicks from satellite images to give early warning of ecological disasters and deter illegal dumping. Radar satellites provide an opportunity for monitoring coastal waters day and night, regardless of weather conditions. Oil slicks appear as dark regions in the image, the size and shape of which evolve depending on weather and sea conditions. However, other look-alike dark regions can be caused by local weather conditions such as high wind. Detecting oil slicks is an expensive manual process requiring highly trained personnel who assess each region in the image. A hazard detection system has been developed to screen images for subsequent manual processing. Intended to be marketed worldwide to a wide variety of users— government agencies and companies—with different objectives, applications, and geographical areas, this system needs to be highly customizable to individual cir- cumstances. Machine learning allows the system to be trained on examples of spills and nonspills supplied by the user and lets the user control the tradeoff between undetected spills and false alarms. Unlike other machine learning applications, which generate a classifier that is then deployed in the field, here it is the learning scheme itself that will be deployed. The input is a set of raw pixel images from a radar satellite, and the output is a much smaller set of images with putative oil slicks marked by a colored border. First, standard image-processing operations are applied to normalize the image. Then suspicious dark regions are identified. Several dozen attributes are extracted from each region, characterizing its size, shape, area, intensity, sharpness and jag- gedness of the boundaries, proximity to other regions, and information about the background in the vicinity of the region. Finally, standard learning techniques are applied to the resulting attribute vectors. 24 CHAPTER 1 What’s It All About? Several interesting problems were encountered. One was the scarcity of training data. Oil slicks are (fortunately) very rare, and manual classification is extremely costly. Another was the unbalanced nature of the problem: Of the many dark regions in the training data, only a very small fraction were actual oil slicks. A third is that the examples grouped naturally into batches, with regions drawn from each image forming a single batch, and background characteristics varied from one batch to another. Finally, the performance task was to serve as a filter, and the user had to be provided with a convenient means of varying the false-alarm rate. Load Forecasting In the electricity supply industry, it is important to determine future demand for power as far in advance as possible. If accurate estimates can be made for the maximum and minimum load for each hour, day, month, season, and year, utility companies can make significant economies in areas such as setting the operating reserve, maintenance scheduling, and fuel inventory management. An automated load forecasting assistant has been operating at a major utility supplier for more than a decade to generate hourly forecasts two days in advance. The first step was to use data collected over the previous 15 years to create a sophis- ticated load model manually. This model had three components: base load for the year, load periodicity over the year, and the effect of holidays. To normalize for the base load, the data for each previous year was standardized by subtracting the average load for that year from each hourly reading and dividing by the standard deviation over the year. Electric load shows periodicity at three fundamental frequencies: diurnal, where usage has an early morning minimum and midday and afternoon maxima; weekly, where demand is lower at weekends; and seasonal, where increased demand during winter and summer for heating and cooling, respectively, creates a yearly cycle. Major holidays, such as Thanksgiving, Christmas, and New Year’s Day, show sig- nificant variation from the normal load and are each modeled separately by averag- ing hourly loads for that day over the past 15 years. Minor official holidays, such as Columbus Day, are lumped together as school holidays and treated as an offset to the normal diurnal pattern. All of these effects are incorporated by reconstructing a year’s load as a sequence of typical days, fitting the holidays in their correct position, and denormalizing the load to account for overall growth. Thus far, the load model is a static one, constructed manually from historical data, and it implicitly assumes “normal” climatic conditions over the year. The final step was to take weather conditions into account by locating the previous day most similar to the current circumstances and using the historical information from that day as a predictor. The prediction is treated as an additive correction to the static load model. To guard against outliers, the eight most similar days are located and their additive corrections averaged. A database was constructed of temperature, humidity, wind speed, and cloud cover at three local weather centers for each hour of the 15-year historical record, along with the difference between the actual load 1.3 Fielded Applications 25 and that predicted by the static model. A linear regression analysis was performed to determine the relative effects of these parameters on load, and the coefficients were used to weight the distance function used to locate the most similar days. The resulting system yielded the same performance as that of trained human forecasters but was far quicker—taking seconds rather than hours to generate a daily forecast. Human operators can analyze the forecast’s sensitivity to simulated changes in weather and bring up for examination the “most similar” days that the system used for weather adjustment. Diagnosis Diagnosis is one of the principal application areas of expert systems. Although the handcrafted rules used in expert systems often perform well, machine learning can be useful in situations in which producing rules manually is too labor intensive. Preventative maintenance of electromechanical devices such as motors and gen- erators can forestall failures that disrupt industrial processes. Technicians regularly inspect each device, measuring vibrations at various points to determine whether the device needs servicing. Typical faults include shaft misalignment, mechanical loos- ening, faulty bearings, and unbalanced pumps. A particular chemical plant uses more than 1000 different devices, ranging from small pumps to very large turbo-alternators, which until recently were diagnosed by a human expert with 20 years or more of experience. Faults are identified by measuring vibrations at different places on the device’s mounting and using Fourier analysis to check the energy present in three different directions at each harmonic of the basic rotation speed. This information, which is very noisy because of limitations in the measurement and recording pro- cedure, is studied by the expert to arrive at a diagnosis. Although handcrafted expert system rules had been developed for some situations, the elicitation process would have to be repeated several times for different types of machinery; so a learning approach was investigated. Six hundred faults, each comprising a set of measurements along with the expert’s diagnosis, were available, representing 20 years of experience. About half were unsatisfactory for various reasons and had to be discarded; the remainder were used as training examples. The goal was not to determine whether or not a fault existed but to diagnose the kind of fault, given that one was there. Thus, there was no need to include fault-free cases in the training set. The measured attributes were rather low level and had to be augmented by intermediate concepts—that is, func- tions of basic attributes—which were defined in consultation with the expert and embodied some causal domain knowledge. The derived attributes were run through an induction algorithm to produce a set of diagnostic rules. Initially, the expert was not satisfied with the rules because he could not relate them to his own knowledge and experience. For him, mere statistical evidence was not, by itself, an adequate explanation. Further background knowledge had to be used before satisfactory rules were generated. Although the resulting rules were quite complex, the expert liked them because he could justify them in light of his mechanical knowledge. He was 26 CHAPTER 1 What’s It All About? pleased that a third of the rules coincided with ones he used himself and was delighted to gain new insight from some of the others. Performance tests indicated that the learned rules were slightly superior to the handcrafted ones that had previously been elicited from the expert, and this result was confirmed by subsequent use in the chemical factory. It is interesting to note, however, that the system was put into use not because of its good performance but because the domain expert approved of the rules that had been learned. Marketing and Sales Some of the most active applications of data mining have been in the area of marketing and sales. These are domains in which companies possess massive volumes of precisely recorded data, which, it has only recently been realized, is potentially extremely valuable. In these applications, predictions themselves are the chief interest: The structure of how decisions are made is often completely irrelevant. We have already mentioned the problem of fickle customer loyalty and the chal- lenge of detecting customers who are likely to defect so that they can be wooed back into the fold by giving them special treatment. Banks were early adopters of data mining technology because of their successes in the use of machine learning for credit assessment. Data mining is now being used to reduce customer attrition by detecting changes in individual banking patterns that may herald a change of bank, or even life changes, such as a move to another city, that can result in a different bank being chosen. It may reveal, for example, a group of customers with above- average attrition rate who do most of their banking by phone after hours when telephone response is slow. Data mining may determine groups for whom new ser- vices are appropriate, such as a cluster of profitable, reliable customers who rarely get cash advances from their credit cards except in November and December, when they are prepared to pay exorbitant interest rates to see them through the holiday season. In another domain, cellular phone companies fight churn by detecting patterns of behavior that could benefit from new services, and then advertise such services to retain their customer base. Incentives provided specifically to retain existing customers can be expensive, and successful data mining allows them to be precisely targeted to those customers who are likely to yield maximum benefit. Market basket analysis is the use of association techniques to find groups of items that tend to occur together in transactions, typically supermarket checkout data. For many retailers this is the only source of sales information that is available for data mining. For example, automated analysis of checkout data may uncover the fact that customers who buy beer also buy chips, a discovery that could be significant from the supermarket operator’s point of view (although rather an obvious one that prob- ably does not need a data mining exercise to discover). Or analysis may come up with the fact that on Thursdays customers often purchase diapers and beer together, an initially surprising result that, on reflection, makes some sense as young parents 1.3 Fielded Applications 27 stock up for a weekend at home. Such information could be used for many purposes: planning store layouts, limiting special discounts to just one of a set of items that tend to be purchased together, offering coupons for a matching product when one of them is sold alone, and so on. There is enormous added value in being able to identify individual customer’s sales histories. Discount or “loyalty” cards let retailers identify all the purchases that each individual customer makes. This personal data is far more valuable than the cash value of the discount. Identification of individual customers not only allows historical analysis of purchasing patterns but also permits precisely targeted special offers to be mailed out to prospective customers—or perhaps personalized coupons can be printed in real time at the checkout for use during the next grocery run. Supermarkets want you to feel that although we may live in a world of inexorably rising prices, they don’t increase so much for you because the bargains offered by personalized coupons make it attractive for you to stock up on things that you wouldn’t normally have bought. Direct marketing is another popular domain for data mining. Bulk-mail promo- tional offers are expensive and have a low—but highly profitable—response rate. Anything that helps focus promotions, achieving the same or nearly the same response from a smaller sample, is valuable. Commercially available databases containing demographic information that characterizes neighborhoods based on zip codes can be correlated with information on existing customers to predict what kind of people might buy which items. This model can be trialed on information gained in response to an initial mailout, where people send back a response card or call an 800 number for more information, to predict likely future customers. Unlike shopping-mall retailers, direct-mail companies have complete purchasing histories for each individual customer and can use data mining to determine those likely to respond to special offers. Targeted campaigns save money by directing offers only to those likely to want the product. Other Applications There are countless other applications of machine learning. We briefly mention a few more areas to illustrate the breadth of what has been done. Sophisticated manufacturing processes often involve tweaking control parame- ters. Separating crude oil from natural gas is an essential prerequisite to oil refine- ment, and controlling the separation process is a tricky job. British Petroleum used machine learning to create rules for setting the parameters. This now takes just 10 minutes, whereas previously human experts took more than a day. Westinghouse faced problems in their process for manufacturing nuclear fuel pellets and used machine learning to create rules to control the process. This was reported to have saved them more than $10 million per year (in 1984). The Tennessee printing company R. R. Donnelly applied the same idea to control rotogravure printing presses to reduce artifacts caused by inappropriate parameter settings, reducing the number of artifacts from more than 500 each year to less than 30. 28 CHAPTER 1 What’s It All About? In the realm of customer support and service, we have already described adju- dicating loans and marketing and sales applications. Another example arises when a customer reports a telephone problem and the company must decide what kind of technician to assign to the job. An expert system developed by Bell Atlantic in 1991 to make this decision was replaced in 1999 by a set of rules developed using machine learning, which saved more than $10 million per year by making fewer incorrect decisions. There are many scientific applications. In biology, machine learning is used to help identify the thousands of genes within each new genome. In biomedicine, it is used to predict drug activity by analyzing not just the chemical properties of drugs but also their three-dimensional structure. This accelerates drug discovery and reduces its cost. In astronomy, machine learning has been used to develop a fully automatic cataloging system for celestial objects that are too faint to be seen by visual inspection. In chemistry, it has been used to predict the structure of certain organic compounds from magnetic resonance spectra. In all of these applications, machine learning techniques have attained levels of performance—or should we say skill?—that rival or surpass those of human experts. Automation is especially welcome in situations involving continuous monitoring, a job that is time consuming and exceptionally tedious for humans. Ecological applications include the oil spill monitoring described earlier. Other applications are rather less consequential—for example, machine learning is being used to predict preferences for TV programs based on past choices and to advise viewers about available channels. Still other applications may save lives. Intensive-care patients may be monitored to detect changes in variables that cannot be explained by cir- cadian rhythm, medication, and so on, raising an alarm when appropriate. Finally, in a world that relies on vulnerable networked computer systems and is increasingly concerned about cybersecurity, machine learning is used to detect intrusion by recognizing unusual patterns of operation. 1.4 MACHINE LEARNING AND STATISTICS What is the difference between machine learning and statistics? Cynics, looking wryly at the explosion of commercial interest (and hype) in this area, equate data mining to statistics plus marketing. In truth, you should not look for a dividing line between machine learning and statistics because there is a continuum— and a multidimensional one at that—of data analysis techniques. Some derive from the skills taught in standard statistics courses, and others are more closely associated with the kind of machine learning that has arisen out of computer science. Historically, the two sides have had rather different traditions. If forced to point to a single difference of emphasis, it might be that statistics has been more concerned with testing hypotheses, whereas machine learning has been more concerned with formulating the process of generalization as a search through possible hypotheses. But this is a gross oversimplification: Statistics is far more 1.5 Generalization as Search 29 than just hypothesis testing, and many machine learning techniques do not involve any searching at all. In the past, very similar schemes have developed in parallel in machine learning and statistics. One is decision tree induction. Four statisticians (Breiman et al., 1984) published a book, Classification and regression trees, in the mid-1980s, and throughout the 1970s and early 1980s a prominent machine learning researcher, J. Ross Quinlan, was developing a system for inferring classification trees from examples. These two independent projects produced quite similar schemes for generating trees from examples, and the researchers only became aware of one another’s work much later. A second area where similar methods have arisen involves the use of nearest- neighbor methods for classification. These are standard statistical techniques that have been extensively adapted by machine learning researchers, both to improve classification performance and to make the procedure more efficient computation- ally. We will examine both decision tree induction and nearest-neighbor methods in Chapter 4. But now the two perspectives have converged. The techniques we will examine in this book incorporate a great deal of statistical thinking. Right from the beginning, when constructing and refining the initial example set, standard statistical methods apply: visualization of data, selection of attributes, discarding outliers, and so on. Most learning algorithms use statistical tests when constructing rules or trees and for correcting models that are “overfitted” in that they depend too strongly on the details of the particular examples used to produce them (we have already seen an example of this in the two decision trees in Figure 1.3 for the labor negotiations problem). Statistical tests are used to validate machine learning models and to evalu- ate machine learning algorithms. In our study of practical techniques for data mining, we will learn a great deal about statistics. 1.5 GENERALIZATION AS SEARCH One way of visualizing the problem of learning—and one that distinguishes it from statistical approaches—is to imagine a search through a space of possible concept descriptions for one that fits the data. Although the idea of generalization as search is a powerful conceptual tool for thinking about machine learning, it is not essential for understanding the practical schemes described in this book. That is why this section is set apart (boxed), suggesting that it is optional. Suppose, for definiteness, that concept descriptions—the result of learning—are expressed as rules such as the ones given for the weather problem in Section 1.2 (although other concept description languages would do just as well). Suppose that we list all possible sets of rules and then look for ones that satisfy a given set of examples. A big job? Yes. An infinite job? At first glance it seems so because there is no limit to the number of rules there might be. But actually the number of possible rule sets is finite. Note first that each rule is no greater than a fixed maximum size, with at most one term for each attribute: For the weather data of Table 1.2 this involves four terms in all. 30 CHAPTER 1 What’s It All About? Because the number of possible rules is finite, the number of possible rule sets is finite too, although extremely large. However, we’d hardly be interested in sets that contained a very large number of rules. In fact, we’d hardly be interested in sets that had more rules than there are examples because it is difficult to imagine needing more than one rule for each example. So if we were to restrict consideration to rule sets smaller than that, the problem would be substantially reduced, although still very large. The threat of an infinite number of possible concept descriptions seems more serious for the second version of the weather problem in Table 1.3 because these rules contain numbers. If they are real numbers, you can’t enumerate them, even in principle. However, on reflection the problem again disappears because the numbers really just represent breakpoints in the numeric values that appear in the examples. For instance, consider the temperature attribute in Table 1.3. It involves the numbers 64, 65, 68, 69, 70, 71, 72, 75, 80, 81, 83, and 85—12 different numbers. There are 13 possible places in which we might want to put a breakpoint for a rule involving temperature. The problem isn’t infinite after all. So the process of generalization can be regarded as a search through an enormous, but finite, search space. In principle, the problem can be solved by enumerating descriptions and striking out those that do not fit the examples presented. A positive example eliminates all descriptions that it does not match, and a negative one eliminates those it does match. With each example the set of remaining descriptions shrinks (or stays the same). If only one is left, it is the target description—the target concept. If several descriptions are left, they may still be used to classify unknown objects. An unknown object that matches all remaining descriptions should be classified as matching the target; if it fails to match any description it should be classified as being outside the target concept. Only when it matches some descriptions but not others is there ambiguity. In this case if the classification of the unknown object were revealed, it would cause the set of remaining descriptions to shrink because rule sets that classified the object the wrong way would be rejected. Enumerating the Concept Space Regarding it as search is a good way of looking at the learning process. However, the search space, although finite, is extremely big, and it is generally quite impractical to enumerate all possible descriptions and then see which ones fit. In the weather problem there are 4 × 4 × 3 × 3 × 2 = 288 possibilities for each rule. There are four possibilities for the outlook attribute: sunny, overcast, rainy, or it may not participate in the rule at all. Similarly, there are four for temperature, three each for windy and humidity and two for the class. If we restrict the rule set to contain no more than 14 rules (because there are 14 examples in the training set), there are around 2.7 × 1034 possible different rule sets. That’s a lot to enumerate, especially for such a patently trivial problem. Although there are ways of making the enumeration procedure more feasible, a serious problem remains: In practice, it is rare for the process to converge on a unique acceptable description. Either many descriptions are still in the running after the examples are processed or the descriptors are all eliminated. The first case arises when the examples are not sufficiently comprehensive to eliminate all possible descriptions except for the “correct” one. In practice, people often want a single “best” description, and it is necessary to apply some other criteria to select the best one from the set of remaining descriptions. The second problem arises either because the description language is not expressive enough to capture the actual concept or because of noise in the examples. If an example comes in with the “wrong” classification because of an error in some of the attribute values or in the class that is assigned to it, this will likely eliminate the correct description from the space. The result is that the set of remaining descriptions becomes empty. This situation is very likely to happen if the examples contain any noise at all, which inevitably they do except in artificial situations. Another way of looking at generalization as search is to imagine it not as a process of enumerating descriptions and striking out those that don’t apply but as a kind of hill climbing in description space to find the description that best matches the set of examples according to some prespecified matching criterion. This is the way that most practical machine learning methods work. However, except in the most trivial cases, it is impractical to search the whole space exhaustively; most practical algorithms involve heuristic search and cannot guarantee to find the optimal description. Bias Viewing generalization as a search in a space of possible concepts makes it clear that the most important decisions in a machine learning system are: • The concept description language • The order in which the space is searched • The way that overfitting to the particular training data is avoided These three properties are generally referred to as the bias of the search and are called language bias, search bias, and overfitting-avoidance bias. You bias the learning scheme by choosing a language in which to express concepts, by searching in a particular way for an acceptable description, and by deciding when the concept has become so complex that it needs to be simplified. Language Bias The most important question for language bias is whether the concept description language is universal or whether it imposes constraints on what concepts can be learned. If you consider the set of all possible examples, a concept is really just a division of that set into subsets. In the weather example, if you were to enumerate all possible weather conditions, the play concept is a subset of possible weather conditions. A “universal” language is one that is capable of expressing every possible subset of examples. In practice, the set of possible examples is generally huge, and in this respect our perspective is a theoretical, not a practical, one. If the concept description language permits statements involving logical or—that is, disjunctions—then any subset can be represented. If the description language is rule- based, disjunction can be achieved by using separate rules. For example, one possible concept representation is just to enumerate the examples: If outlook = overcast and temperature = hot and humidity = high and windy = false then play = yes If outlook = rainy and temperature = mild and humidity = high and windy = false then play = yes If outlook = rainy and temperature = cool and humidity = normal and windy = false then play = yes If outlook = overcast and temperature = cool and humidity = normal and windy = true then play = yes … If none of the above then play = no This is not a particularly enlightening concept description: It simply records the positive examples that have been observed and assumes that all the rest are negative. Each positive example is given its own rule, and the concept is the disjunction of the rules. Alternatively, you could imagine having individual rules for each of the negative examples, too—an equally uninteresting concept. In either case, the concept description does not perform any generalization; it simply records the original data. On the other hand, if disjunction is not allowed, some possible concepts—sets of examples—may not be able to be represented at all. In that case, a machine learning scheme may simply be unable to achieve good performance. 1.5 Generalization as Search 31 32 CHAPTER 1 What’s It All About? Another kind of language bias is that obtained from knowledge of the particular domain being used. For example, it may be that some combinations of attribute values can never happen. This would be the case if one attribute implied another. We saw an example of this when considering the rules for the soybean problem described in Section 1.2. Then it would be pointless to even consider concepts that involved redundant or impossible combinations of attribute values. Domain knowledge can be used to cut down the search space. Knowledge is power: A little goes a long way, and even a small hint can reduce the search space dramatically. Search Bias In realistic data mining problems, there are many alternative concept descriptions that fit the data, and the problem is to find the “best” one according to some criterion—usually simplicity. We use the term fit in a statistical sense; we seek the best description that fits the data reasonably well. Moreover, it is often computationally infeasible to search the whole space and guarantee that the description found really is the best. Consequently, the search procedure is heuristic, and no guarantees can be made about the optimality of the final result. This leaves plenty of room for bias: Different search heuristics bias the search in different ways. For example, a learning algorithm might adopt a “greedy” search for rules by trying to find the best rule at each stage and adding it to the rule set. However, it may be that the best pair of rules is not just the two rules that are individually found best. Or when building a decision tree, a commitment to split early on using a particular attribute might turn out later to be ill-considered in light of how the tree develops below that node. To get around these problems, a beam search could be used where irrevocable commitments are not made but instead a set of several active alternatives—the number of which is the beam width—are pursued in parallel. This will complicate the learning algorithm quite considerably but has the potential to avoid the myopia associated with a greedy search. Of course, if the beam width is not large enough, myopia may still occur. There are more complex search strategies that help to overcome this problem. A more general and higher-level kind of search bias concerns whether the search is done by starting with a general description and refining it or by starting with a specific example and generalizing it. The former is called a general-to-specific search bias; the latter, a specific-to-general one. Many learning algorithms adopt the former policy, starting with an empty decision tree, or a very general rule, and specializing it to fit the examples. However, it is perfectly possible to work in the other direction. Instance-based methods start with a particular example and see how it can be generalized to cover other nearby examples in the same class. Overfitting-Avoidance Bias Overfitting-avoidance bias is often just another kind of search bias. However, because it addresses a rather special problem, we treat it separately. Recall the disjunction problem described previously. The problem is that if disjunction is allowed, useless concept descriptions that merely summarize the data become possible, whereas if it is prohibited, some concepts are unlearnable. To get around this problem, it is common to search the concept space starting with the simplest concept descriptions and proceeding to more complex ones: simplest-first ordering. This biases the search in favor of simple concept descriptions. Using a simplest-first search and stopping when a sufficiently complex concept description is found is a good way of avoiding overfitting. It is sometimes called forward pruning or prepruning because complex descriptions are pruned away before they are reached. The alternative, backward pruning or postpruning, is also viable. Here, we first find a description that fits the data well and then prune it back to a simpler description that also fits the data. This is not as redundant as it sounds: Often the best way to arrive 1.6 Data Mining and Ethics 33 at a simple theory is to find a complex one and then simplify it. Forward and backward pruning are both a kind of overfitting-avoidance bias. In summary, although generalization as search is a nice way to think about the learning problem, bias is the only way to make it feasible in practice. Different learning algorithms correspond to different concept description spaces searched with different biases. This is what makes it interesting: Different description languages and biases serve some problems well and other problems badly. There is no universal “best” learning method—as every teacher knows! 1.6 DATA MINING AND ETHICS The use of data—particularly data about people—for data mining has serious ethical implications, and practitioners of data mining techniques must act responsibly by making themselves aware of the ethical issues that surround their particular application. When applied to people, data mining is frequently used to discriminate—who gets the loan, who gets the special offer, and so on. Certain kinds of discrimination— racial, sexual, religious, and so on—are not only unethical but also illegal. However, the situation is complex: Everything depends on the application. Using sexual and racial information for medical diagnosis is certainly ethical, but using the same infor- mation when mining loan payment behavior is not. Even when sensitive information is discarded, there is a risk that models will be built that rely on variables that can be shown to substitute for racial or sexual characteristics. For example, people fre- quently live in areas that are associated with particular ethnic identities, and so using a zip code in a data mining study runs the risk of building models that are based on race—even though racial information has been explicitly excluded from the data. Reidentification Recent work in what are being called reidentification techniques has provided sobering insights into the difficulty of anonymizing data. It turns out, for example, that over 85% of Americans can be identified from publicly available records using just three pieces of information: five-digit zip code, birth date (including year), and sex. Don’t know the zip code?—over half of Americans can be identified from just city, birth date, and sex. When the Commonwealth of Massachusetts released medical records summarizing every state employee’s hospital record in the mid- 1990s, the governor gave a public assurance that it had been anonymized by remov- ing all identifying information such as name, address, and social security number. He was surprised to receive his own health records (which included diagnoses and prescriptions) in the mail. Stories abound of companies releasing allegedly anonymous data in good faith, only to find that many individuals are easily identifiable. In 2006, an Internet services company released to the research community the records of 20 million user searches. 34 CHAPTER 1 What’s It All About? The records were anonymized by removing all personal information—or so the company thought. But pretty soon journalists from The New York Times were able to identify the actual person corresponding to user number 4417749 (they sought her permission before exposing her). They did so by analyzing the search terms she used, which included queries for landscapers in her hometown and for several people with the same last name as hers, which reporters correlated with public databases. Two months later, Netflix, an online movie rental service, released 100 million records of movie ratings (from 1 to 5) with their dates. To their surprise, it turned out to be quite easy to identify people in the database and thus discover all the movies they had rated. For example, if you know approximately when (give or take two weeks) a person in the database rated six movies and you know the ratings, you can identify 99% of the people in the database. By knowing only two movies with their ratings and dates, give or take three days, nearly 70% of people can be identified. From just a little information about your friends (or enemies) you can determine all the movies they have rated on Netflix. The moral is that if you really do remove all possible identification information from a database, you will probably be left with nothing useful. Using Personal Information It is widely accepted that before people make a decision to provide personal infor- mation they need to know how it will be used and what it will be used for, what steps will be taken to protect its confidentiality and integrity, what the consequences of supplying or withholding the information are, and any rights of redress they may have. Whenever such information is collected, individuals should be told these things—not in legalistic small print but straightforwardly in plain language they can understand. The potential use of data mining techniques means that the ways in which a repository of data can be used may stretch far beyond what was conceived when the data was originally collected. This creates a serious problem: It is necessary to determine the conditions under which the data was collected and for what purposes it may be used. Does the ownership of data bestow the right to use it in ways other than those purported when it was originally recorded? Clearly, in the case of explic- itly collected personal data, it does not. But in general the situation is complex. Surprising things emerge from data mining. For example, it has been reported that one of the leading consumer groups in France has found that people with red cars are more likely to default on their car loans. What is the status of such a “dis- covery”? What information is it based on? Under what conditions was that informa- tion collected? In what ways is it ethical to use it? Clearly, insurance companies are in the business of discriminating among people based on stereotypes—young males pay heavily for automobile insurance—but such stereotypes are not based solely on statistical correlations; they also draw on commonsense knowledge about the world as well. Whether the preceding finding says something about the kind of person who chooses a red car, or whether it should be discarded as an irrelevancy, is a matter for human judgment based on knowledge of the world rather than on purely statisti- cal criteria. When presented with data, you need to ask who is permitted to have access to it, for what purpose it was collected, and what kind of conclusions are legitimate to draw from it. The ethical dimension raises tough questions for those involved in practical data mining. It is necessary to consider the norms of the community that is used to dealing with the kind of data involved, standards that may have evolved over decades or centuries but ones that may not be known to the information special- ist. For example, did you know that in the library community it is taken for granted that the privacy of readers is a right that is jealously protected? If you call your university library and ask who has such-and-such a textbook out on loan, they will not tell you. This prevents a student being subjected to pressure from an irate profes- sor to yield access to a book that she desperately needs for her latest grant applica- tion. It also prohibits enquiry into the dubious recreational reading tastes of the university ethics committee chairperson. Those who build, say, digital libraries may not be aware of these sensitivities and might incorporate data mining systems that analyze and compare individuals’ reading habits to recommend new books—perhaps even selling the results to publishers! Wider Issues In addition to various community standards for the use of data, logical and sci- entific standards must be adhered to when drawing conclusions from it. If you do come up with conclusions (e.g., red car owners being greater credit risks), you need to attach caveats to them and back them up with arguments other than purely statistical ones. The point is that data mining is just a tool in the whole process. It is people who take the results, along with other knowledge, and decide what action to apply. Data mining prompts another question, which is really a political one concerning the use to which society’s resources are being put. We mentioned earlier the applica- tion of data mining to basket analysis, where supermarket checkout records are analyzed to detect associations among items that people purchase. What use should be made of the resulting information? Should the supermarket manager place the beer and chips together, to make it easier for shoppers, or farther apart to make it less convenient for them, to maximize their time in the store and therefore their likelihood of being drawn into further purchases? Should the manager move the most expensive, most profitable diapers near the beer, increasing sales to harried fathers of a high-margin item, and add further luxury baby products nearby? Of course, anyone who uses advanced technologies should consider the wisdom of what they are doing. If data is characterized as recorded facts, then information is the set of patterns, or expectations, that underlie the data. You could go on to define knowledge as the accumulation of your set of expectations and wisdom as the value attached to knowledge. Although we will not pursue it further here, this issue is worth pondering. 1.6 Data Mining and Ethics 35 36 CHAPTER 1 What’s It All About? As we saw at the very beginning of this chapter, the techniques described in this book may be called upon to help make some of the most profound and intimate deci- sions that life presents. Data mining is a technology that we need to take seriously. 1.7 FURTHER READING To avoid breaking up the flow of the main text, all references are collected in a section at the end of each chapter. This section describes papers, books, and other resources relevant to the material covered in this chapter. The human in vitro fertil- ization research mentioned in the opening was undertaken by the Oxford University Computing Laboratory, and the research on cow culling was performed in the Com- puter Science Department at Waikato University, New Zealand. The weather problem is from Quinlan (1986) and has been widely used to explain machine learning schemes. The corpus of example problems mentioned in the intro- duction to Section 1.2 is available from Asuncion and Newman (2007). The contact lens example is from Cendrowska (1987), who introduced the PRISM rule-learning algorithm that we will encounter in Chapter 4. The iris dataset was described in a classic early paper on statistical inference (Fisher, 1936). The labor negotiations data is from the Collective Bargaining Review, a publication of Labour Canada issued by the Industrial Relations Information Service (BLI 1988), and the soybean problem was first described by Michalski and Chilausky (1980). Some of the applications in Section 1.3 are covered in an excellent paper that gives plenty of other applications of machine learning and rule induction (Langley and Simon, 1995); another source of fielded applications is a special issue of the Machine Learning Journal (Kohavi and Provost, 1998). Chakrabarti (2003) has written an excellent and comprehensive book on techniques of web mining; another, more recent, book is Liu’s Web data mining (2009). The loan company application is described in more detail by Michie (1989), the oil slick detector is from Kubat et al. (1998), the electric load forecasting work is by Jabbour et al. (1988), and the application to preventative maintenance of electromechanical devices is from Saitta and Neri (1998). Fuller descriptions of some of the other projects mentioned in Section 1.3 (including the figures of dollar amounts saved and related literature refer- ences) appear at the web site of the Alberta Ingenuity Centre for Machine Learning. Luan (2002) describes applications for data mining in higher education. Dasu et al. (2006) have some recommendations for successful data mining. Another special issue of the Machine Learning Journal addresses the lessons that have been learned from data mining applications and collaborative problem solving (Lavrac et al., 2004). The “diapers and beer” story is legendary. According to an article in London’s Financial Times (February 7, 1996), The oft-quoted example of what data mining can achieve is the case of a large US supermarket chain which discovered a strong association for many customers 1.7 Further Reading 37 between a brand of babies’ nappies (diapers) and a brand of beer. Most customers who bought the nappies also bought the beer. The best hypothesisers in the world would find it difficult to propose this combination but data mining showed it existed, and the retail outlet was able to exploit it by moving the products closer together on the shelves. However, it seems that it is just a legend after all; Power (2002) traces its history. The book Classification and regression trees, mentioned in Section 1.4, is by Breiman et al. (1984), and Quinlan’s independently derived but similar scheme was described in a series of papers that eventually led to a book (Quinlan, 1993). The first book on data mining was written by Piatetsky-Shapiro and Frawley (1991)—a collection of papers presented at a workshop on knowledge discovery in databases in the late 1980s. Another book from the same stable has appeared since (Fayyad et al., 1996) from a 1994 workshop. There followed a rash of business- oriented books on data mining, focusing mainly on practical aspects of how it can be put into practice with only rather superficial descriptions of the technology that underlies the methods used. They are valuable sources of applications and inspira- tion. For example, Adriaans and Zantige (1996) from Syllogic, a European systems and database consultancy, is an early introduction to data mining. Berry and Linoff (1997), from a Pennsylvania-based firm specializing in data warehousing and data mining, give an excellent and example-studded review of data mining techniques for marketing, sales, and customer support. Cabena et al. (1998), written by people from five international IBM laboratories, contains an overview of the data mining process with many examples of real-world applications. Dhar and Stein (1997) give a business perspective on data mining and include broad-brush, popularized reviews of many of the technologies involved. Groth (1998), working for a provider of data mining software, gives a brief introduction to data mining and then a fairly extensive review of data mining software products; the book includes a CD-ROM containing a demo version of his company’s product. Weiss and Indurkhya (1998) look at a wide variety of statistical techniques for making predictions from what they call “big data.” Han and Kamber (2006) cover data mining from a database perspective, focusing on the discovery of knowledge in large corpo- rate databases; they also discuss mining complex types of data. Hand et al. (2001) produced an interdisciplinary book on data mining from an international group of authors who are well respected in the field. Finally, Nisbet et al. (2009) have produced a comprehensive handbook of statistical analysis and data mining applications. Books on machine learning, on the other hand, tend to be academic texts suited for use in university courses rather than as practical guides. Mitchell (1997) wrote an excellent book that covers many techniques of machine learning, including some—notably genetic algorithms and reinforcement learning—that are not covered here. Langley (1996) offers another good text. Although the previously mentioned book by Quinlan (1993) concentrates on a particular learning algorithm, C4.5, which we will cover in detail in Chapters 4 and 6, it is a good introduction to some of the problems and techniques of machine learning. An absolutely excellent book on 38 CHAPTER 1 What’s It All About? machine learning from a statistical perspective is Hastie et al. (2009). This is quite a theoretically oriented work, and is beautifully produced with apt and telling figures. Russell and Norvig’s Artificial intelligence: A modern approach (2009) is the third edition of a classic text that includes a great deal of information on machine learning and data mining. Pattern recognition is a topic that is closely related to machine learning, and many of the same techniques apply. Duda et al. (2001) is the second edition of a classic and successful book on pattern recognition (Duda and Hart, 1973). Ripley (1996) and Bishop (1995) describe the use of neural networks for pattern recognition; Bishop has a more recent book, Pattern recognition and machine learning (2006). Data mining with neural networks is the subject of a 1996 book by Bigus of IBM, which features the IBM Neural Network Utility Product that he developed. There is a great deal of current interest in support vector machines. Cristianini and Shawe-Taylor (2000) give a nice introduction, and a follow-up work generalizes this to cover additional algorithms, kernels, and solutions with applications to pattern discovery problems in fields such as bioinformatics, text analysis, and image analysis (Shawe-Taylor and Cristianini, 2004). Schölkopf and Smola (2002) provide a com- prehensive introduction to support vector machines and related kernel methods by two young researchers who did their Ph.D. research in this rapidly developing area. The emerging area of reidentification techniques is explored, along with its implications for anonymization, by Ohm (2009). 39Data Mining: Practical Machine Learning Tools and Techniques Copyright © 2011 Elsevier Inc. All rights of reproduction in any form reserved. CHAPTER 2 Input: Concepts, Instances, and Attributes Before delving into the question of how machine learning schemes operate, we begin by looking at the different forms the input might take and, in Chapter 3, the different kinds of output that might be produced. With any software system, understanding what the inputs and outputs are is far more important than knowing what goes on in between, and machine learning is no exception. The input takes the form of concepts, instances, and attributes. We call the thing that is to be learned a concept description. The idea of a concept, like the very idea of learning in the first place, is hard to pin down precisely, and we won’t spend time philosophizing about just what it is and isn’t. In a sense, what we are trying to find—the result of the learning process—is a description of the concept that is intel- ligible in that it can be understood, discussed, and disputed, and operational in that it can be applied to actual examples. The next section explains some distinctions among different kinds of learning problems—distinctions that are very concrete and very important in practical data mining. The information that the learner is given takes the form of a set of instances. In the examples in Chapter 1, each instance was an individual, independent example of the concept to be learned. Of course, there are many things you might like to learn for which the raw data cannot be expressed as individual, independent instances. Perhaps background knowledge should be taken into account as part of the input. Perhaps the raw data is an agglomerated mass that cannot be fragmented into individual instances. Perhaps it is a single sequence—say a time sequence— that cannot meaningfully be cut into pieces. This book is about simple, practical methods of data mining, and we focus on situations where the information can be supplied in the form of individual examples. However, we do introduce one slightly more complicated scenario where the examples for learning contain multiple instances. Each instance is characterized by the values of attributes that measure different aspects of the instance. There are many different types of attributes, although typical data mining schemes deal only with numeric and nominal, or categorical, ones. Finally, we examine the question of preparing input for data mining and introduce a simple format—the one that is used by the Weka system that accompanies this book—for representing the input information as a text file. 40 CHAPTER 2 Input: Concepts, Instances, and Attributes 2.1 WHAT’S A CONCEPT? Four basically different styles of learning appear in data mining applications. In classification learning, the learning scheme is presented with a set of classified examples from which it is expected to learn a way of classifying unseen examples. In association learning, any association among features is sought, not just ones that predict a particular class value. In clustering, groups of examples that belong together are sought. In numeric prediction, the outcome to be predicted is not a discrete class but a numeric quantity. Regardless of the type of learning involved, we call the thing to be learned the concept and the output produced by a learning scheme the concept description. Most of the examples in Chapter 1 are classification problems. The weather data (Tables 1.2 and 1.3) presents a set of days together with a decision for each as to whether to play the game or not. The problem is to learn how to classify new days as play or don’t play. Given the contact lens data (Table 1.1), the problem is to learn how to determine a lens recommendation for a new patient—or more precisely, since every possible combination of attributes is present in the data, the problem is to learn a way of summarizing the given data. For the irises (Table 1.4), the problem is to learn how to determine whether a new iris flower is setosa, versicolor, or virginica, given its sepal length and width and petal length and width. For the labor negotia- tions data (Table 1.6), the problem is to determine whether a new contract is accept- able or not, on the basis of its duration; wage increase in the first, second, and third years; cost of living adjustment; and so forth. We assume throughout this book that each example belongs to one, and only one, class. However, there exist classification scenarios in which individual exam- ples may belong to multiple classes. In technical jargon, these are called multilabeled instances. One simple way to deal with such situations is to treat them as several different classification problems, one for each possible class, where the problem is to determine whether instances belong to that class or not. Classification learning is sometimes called supervised, because, in a sense, the scheme operates under supervision by being provided with the actual outcome for each of the training examples—the play or don’t play judgment, the lens recom- mendation, the type of iris, the acceptability of the labor contract. This outcome is called the class of the example. The success of classification learning can be judged by trying out the concept description that is learned on an independent set of test data for which the true classifications are known but not made available to the machine. The success rate on test data gives an objective measure of how well the concept has been learned. In many practical data mining applications, success is measured more subjectively in terms of how acceptable the learned description— such as the rules or decision tree—is to a human user. Most of the examples in Chapter 1 can be used equally well for association learning, in which there is no specified class. Here, the problem is to discover any structure in the data that is “interesting.” Some association rules for the weather data were given in Section 1.2. Association rules differ from classification rules in two 2.1 What’s a Concept? 41 Table 2.1 Iris Data as a Clustering Problem Sepal Length Sepal Width Petal Length Petal Width 1 5.1 3.5 1.4 0.2 2 4.9 3.0 1.4 0.2 3 4.7 3.2 1.3 0.2 4 4.6 3.1 1.5 0.2 5 5.0 3.6 1.4 0.2 … 51 7.0 3.2 4.7 1.4 52 6.4 3.2 4.5 1.5 53 6.9 3.1 4.9 1.5 54 5.5 2.3 4.0 1.3 55 6.5 2.8 4.6 1.5 … 101 6.3 3.3 6.0 2.5 102 5.8 2.7 5.1 1.9 103 7.1 3.0 5.9 2.1 104 6.3 2.9 5.6 1.8 105 6.5 3.0 5.8 2.2 … ways: They can “predict” any attribute, not just the class, and they can predict more than one attribute’s value at a time. Because of this there are far more association rules than classification rules, and the challenge is to avoid being swamped by them. For this reason, association rules are often limited to those that apply to a certain minimum number of examples—say 80% of the dataset—and have greater than a certain minimum accuracy level—say 95% accurate. Even then, there are usually lots of them, and they have to be examined manually to determine whether they are meaningful or not. Association rules usually involve only nonnumeric attributes; thus, you wouldn’t normally look for association rules in the iris dataset. When there is no specified class, clustering is used to group items that seem to fall naturally together. Imagine a version of the iris data in which the type of iris is omitted, such as in Table 2.1. Then it is likely that the 150 instances fall into natural clusters corresponding to the three iris types. The challenge is to find these clusters and assign the instances to them—and to be able to assign new instances to the clusters as well. It may be that one or more of the iris types splits naturally into subtypes, in which case the data will exhibit more than three natural clusters. The success of clustering is often measured subjectively in terms of how useful the result appears to be to a human user. It may be followed by a second step of classification learning in which rules are learned that give an intelligible description of how new instances should be placed into the clusters. 42 CHAPTER 2 Input: Concepts, Instances, and Attributes Numeric prediction is a variant of classification learning in which the outcome is a numeric value rather than a category. The CPU performance problem is one example. Another, shown in Table 2.2, is a version of the weather data in which what is to be predicted is not play or don’t play but rather the time (in minutes) to play. With numeric prediction problems, as with other machine learning situations, the predicted value for new instances is often of less interest than the structure of the description that is learned, expressed in terms of what the important attributes are and how they relate to the numeric outcome. 2.2 WHAT’S IN AN EXAMPLE? The input to a machine learning scheme is a set of instances. These instances are the things that are to be classified or associated or clustered. Although until now we have called them examples, henceforth we will generally use the more specific term instances to refer to the input. In the standard scenario, each instance is an individual, independent example of the concept to be learned. Instances are charac- terized by the values of a set of predetermined attributes. This was the case in all of the sample datasets described in Chapter 1 (i.e., the weather, the contact lens, the iris, and the labor negotiations problems). Each dataset is represented as a matrix of instances versus attributes, which in database terms is a single relation, or a flat file. Expressing the input data as a set of independent instances is by far the most common situation for practical data mining. However, it is a rather restrictive way of formulating problems, and it is worth spending some time reviewing why. Problems often involve relationships between objects rather than separate, independent Table 2.2 Weather Data with a Numeric Class Outlook Temperature Humidity Windy Play Time Sunny 85 85 false 5 Sunny 80 90 true 0 Overcast 83 86 false 55 Rainy 70 96 false 40 Rainy 68 80 false 65 Rainy 65 70 true 45 Overcast 64 65 true 60 Sunny 72 95 false 0 Sunny 69 70 false 70 Rainy 75 80 false 45 Sunny 75 70 true 50 Overcast 72 90 true 55 Overcast 81 75 false 75 Rainy 71 91 true 10 2.2 What’s in an Example? 43 FIGURE 2.1 A family tree and two ways of expressing the sister-of relation. Peter M = = = Peggy F Grace F Ray M Steven M Graham M Pam F Anna F Nikki F Ian M Pippa F Brian M First person Second person Sister of? First person Second person Sister of? Steven Grace no all the rest no Peter Peggy no Steven Pam yes Peter Steven no Graham Pam yes … … … Ian Pippa yes Steven Peter no Brian Pippa yes Steven Graham no Anna Nikki yes Steven Pam yes Nikki Anna yes … … … yesPippaIan ……… yesNikkiAnna ……… yesAnnaNikki instances. Suppose, to take a specific situation, a family tree is given and we want to learn the concept of sister. Imagine your own family tree, with your relatives (and their genders) placed at the nodes. This tree is the input to the learning process, along with a list of pairs of people and an indication of whether they are sisters or not. Relations Figure 2.1 shows part of a family tree, below which are two tables that each define sisterhood in a slightly different way. A yes in the third column of the individual tables means that the person in the second column is a sister of the person in the first column (that’s just an arbitrary decision we’ve made in setting up this example). The first thing to notice is that there are a lot of nos in the third column of the table on the left, because there are 12 people and 12 × 12 = 144 pairs of people in all, and most pairs of people aren’t sisters. The table on the right, which gives the same information, records only the positive examples and assumes that all others are negative. The idea of specifying only positive examples and adopting a standing assumption that the rest are negative is called the closed-world assumption. It is frequently assumed in theoretical studies; however, it is not of much practical use in real-life problems because they rarely involve “closed” worlds in which you can be certain that all cases are covered. 44 CHAPTER 2 Input: Concepts, Instances, and Attributes Table 2.3 Family Tree Name Gender Parent 1 Parent 2 Peter male ? ? Peggy female ? ? Steven male Peter Peggy Graham male Peter Peggy Pam female Peter Peggy Ian male Grace Ray … Neither table in Figure 2.1 is of any use without the family tree itself. This tree can also be expressed in the form of a table, part of which is shown in Table 2.3. Now the problem is expressed in terms of two relationships, Parent 1 and Parent 2. But these tables do not contain independent sets of instances because values in the Name, Parent 1, and Parent 2 columns of the sister-of relation refer to rows of the family tree relation. We can make them into a single set of instances by collapsing the two tables into a single one, as shown in Table 2.4. We have at last succeeded in transforming the original relational problem into the form of instances, each of which is an individual, independent example of the concept that is to be learned. Of course, the instances are not really independent— there are plenty of relationships among different rows of the table!—but they are independent as far as the concept of sisterhood is concerned. Most machine learning schemes will still have trouble dealing with this kind of data, as we will see in Section 3.4, but at least the problem has been recast into the right form. A simple rule for the sister-of relation is If second person’s gender = female and first person’s parent 1 = second person’s parent 1 then sister-of = yes This example shows how you can take a relationship between different nodes of a tree and recast it into a set of independent instances. In database terms, you take two relations and join them together to make one, a process of flattening that is techni- cally called denormalization. It is always possible to do this with any (finite) set of (finite) relations. The structure of Table 2.4 can be used to describe any relationship between two people—grandparenthood, second cousins twice removed, and so on. Relationships among more people would require a larger table. Relationships in which the maximum number of people is not specified in advance pose a more serious problem. If we want to learn the concept of nuclear family (parents and their children), the number of people involved depends on the size of the largest nuclear family, and although we could guess at a reasonable maximum (10?, 20?), the actual number can only be found by scanning the tree itself. Nevertheless, given a finite set of finite Table 2.4 Sister-of Relation First Person Second Person Sister of?Name Gender Parent 1 Parent 2 Name Gender Parent 1 Parent 2 Steven male Peter Peggy Pam female Peter Peggy yes Graham male Peter Peggy Pam female Peter Peggy yes Ian male Grace Ray Pippa female Grace Ray yes Brian male Grace Ray Pippa female Grace Ray yes Anna female Pam Ian Nikki female Pam Ian yes Nikki female Pam Ian Anna female Pam Ian yes All the rest no 45 46 CHAPTER 2 Input: Concepts, Instances, and Attributes relations we could, at least in principle, form a new “superrelation” that contains one row for every combination of people, and this would be enough to express any relationship between people no matter how many were involved. The computational and storage costs would, however, be prohibitive. Another problem with denormalization is that it produces apparent regularities in the data that are completely spurious and are in fact merely reflections of the original database structure. For example, imagine a supermarket database with a relation for customers and the products they buy, one for products and their suppli- ers, and one for suppliers and their addresses. Denormalizing this will produce a flat file that contains, for each instance, customer, product, supplier, and supplier address. A data mining tool that seeks structure in the database may come up with the fact that customers who buy beer also buy chips, a discovery that could be significant from the supermarket manager’s point of view. However, it may also come up with the fact that the supplier address can be predicted exactly from the supplier—a “discovery” that will not impress the supermarket manager at all. This fact masquer- ades as a significant discovery from the flat file but is present explicitly in the original database structure. Many abstract computational problems involve relations that are not finite, although clearly any actual set of input examples must be finite. Concepts such as ancestor-of involve arbitrarily long paths through a tree, and although the human race, and hence its family tree, may be finite (although prodigiously large), many artificial problems generate data that truly is infinite. Although it may sound abstruse, this situation is the norm in areas such as list processing and logic programming, and it is addressed in a subdiscipline of machine learning called inductive logic programming. Computer scientists usually use recursion to deal with situations in which the number of possible examples is infinite. For example, If person 1 is a parent of person 2 then person 1 is an ancestor of person 2 If person 1 is a parent of person 2 and person 2 is an ancestor of person 3 then person 1 is an ancestor of person 3 This represents a simple recursive definition of ancestor that works no matter how distantly two people are related. Techniques of inductive logic programming can learn recursive rules such as these from a finite set of instances such as those in Table 2.5. The real drawbacks of such techniques, however, are that they do not cope well with noisy data, and they tend to be so slow as to be unusable on anything but small artificial datasets. They are not covered in this book; see Bergadano and Gunetti (1996) for a comprehensive treatment. Other Example Types As we have seen, general relations present substantial challenges, and this book will deal with them no further. Structured examples such as graphs and trees can be 47 Table 2.5 Another Relation First Person Second Person Ancestor of? Name Gender Parent 1 Parent 2 Name Gender Parent 1 Parent 2 Peter male ? ? Steven male Peter Peggy yes Peter male ? ? Pam female Peter Peggy yes Peter male ? ? Anna female Pam Ian yes Peter male ? ? Nikki female Pam Ian yes Pam female Peter Peggy Nikki female Pam Ian yes Grace female ? ? Ian male Grace Ray yes Grace female ? ? Nikki female Pam Ian yes Other examples here yes All the rest no 48 CHAPTER 2 Input: Concepts, Instances, and Attributes viewed as special cases of relations that are often mapped into independent instances by extracting local or global features based on their structure and representing them as attributes. Similarly, sequences of items may be treated by describing them, or their individual items, in terms of a fixed set of properties represented by attributes. Fortunately, most practical data mining problems can be expressed quite effectively as a set of instances, each one being an example of the concept to be learned. In some situations, instead of the individual instances being examples of the concept, each individual example comprises a set of instances that are described by the same attributes. This multi-instance setting covers some important real-world applications. One concerns the inference of characteristics of active drug molecules, where activity corresponds to how well a drug molecule bonds to a “bonding site” on a target molecule. The problem is that the drug molecule can assume alternative shapes by rotating its bonds. It is classed as positive if just one of these shapes actu- ally binds to the site and has the desired effect—but it is not known which shape it is. On the other hand, a drug molecule is negative if none of the shapes bind suc- cessfully. In this case, a multiple instance is a set of shapes, and the entire set is classified as positive or negative. Multi-instance problems often also arise naturally when relations from a database are joined—that is, when several rows from a secondary relation are associated with the same row in the target relation. For example, we may want to classify computer users as experts or novices based on descriptions of user sessions that are stored in a secondary table. The target relation just has the classification and the user ID. Joining the two tables creates a flat file. However, the rows pertaining to an indi- vidual user are not independent. Classification is performed on a per-user basis, so the set of session instances associated with the same user should be viewed as a single example for learning. The goal of multi-instance learning is still to produce a concept description, but now the task is more difficult because the learning algorithm has to contend with incomplete information about each training example. Rather than seeing each example in terms of a single definitive attribute vector, the learning algorithm sees each example as a set of attribute vectors. Things would be easy if only the algorithm knew which member of the set was responsible for the example’s classification— but it does not. Several special learning algorithms have been developed to tackle the multi- instance problem; we describe some of them in Chapter 6. It is also possible to apply standard machine learning schemes by recasting the problem as a single table comprising independent instances. Chapter 4 gives some ways of achieving this. In summary, the input to a data mining scheme is generally expressed as a table of independent instances of the concept to be learned. Because of this it has been suggested, disparagingly, that we should really talk of file mining rather than data- base mining. Relational data is more complex than a flat file. A finite set of finite relations can always be recast into a single table, although often at enormous cost in space. Moreover, denormalization can generate spurious regularities in the data, 2.3 What’s in an Attribute? 49 and it is essential to check the data for such artifacts before applying a learning scheme. Potentially infinite concepts can be dealt with by learning rules that are recursive, although that is beyond the scope of this book. Finally, some important real-world problems are most naturally expressed in a multi-instance format, where each example is actually a separate set of instances. 2.3 WHAT’S IN AN ATTRIBUTE? Each instance that provides the input to machine learning is characterized by its values on a fixed, predefined set of features or attributes. The instances are the rows of the tables that we have shown for the weather, the contact lens, the iris, and the CPU performance problems, and the attributes are the columns. (The labor negotia- tions data was an exception: We presented this with instances in columns and attri- butes in rows for space reasons.) The use of a fixed set of features imposes another restriction on the kinds of problems generally considered in practical data mining. What if different instances have different features? If the instances were transportation vehicles, then number of wheels is a feature that applies to many vehicles but not to ships, for example, whereas number of masts might be a feature that applies to ships but not to land vehicles. The standard workaround is to make each possible feature an attribute and to use a special “irrelevant value” flag to indicate that a particular attribute is not available for a particular case. A similar situation arises when the existence of one feature (say, spouse’s name) depends on the value of another (married or single). The value of an attribute for a particular instance is a measurement of the quantity to which the attribute refers. There is a broad distinction between quantities that are numeric and ones that are nominal. Numeric attributes, sometimes called continuous attributes, measure numbers—either real or integer valued. Note that the term con- tinuous is routinely abused in this context; integer-valued attributes are certainly not continuous in the mathematical sense. Nominal attributes take on values in a pre- specified, finite set of possibilities and are sometimes called categorical. But there are other possibilities. Statistics texts often introduce “levels of measurement” such as nominal, ordinal, interval, and ratio. Nominal quantities have values that are distinct symbols. The values themselves serve just as labels or names—hence the term nominal, which comes from the Latin word for name. For example, in the weather data the attribute outlook has the values sunny, overcast, and rainy. No relation is implied among these three—no ordering or distance measure. It certainly does not make sense to add the values together, multiply them, or even compare their size. A rule using such an attribute can only test for equality or inequality, as in outlook: sunny → no overcast → yes rainy → yes 50 CHAPTER 2 Input: Concepts, Instances, and Attributes Ordinal quantities are ones that make it possible to rank-order the categories. However, although there is a notion of ordering, there is no notion of distance. For example, in the weather data the attribute temperature has values hot, mild, and cool. These are ordered. Whether you say that hot mild cool hot mild cool> > < 7.5 MMAX ≤8.5 LM4 (50/22.1%) >8.5 LM1 (65/7.32%) ≤4250 CACH >4250 LM2 (26/6.37%) ≤0.5 LM3 (24/14.5%) (0.5,8.5] LM5 (21/45.5%) ≤28,000 LM6 (23/63.5%) >28,000 (b) (a) CHMIN CACH ≤7.5 MMAX >7.5 MMAX ≤8.5 64.6 (24/19.2%) (8.5,28] MMAX >28 19.3 (28/8.7%) ≤2500 29.8 (37/8.18%) (2500,4250] CACH >4250 MYCT ≤0.5 59.3 (24/16.9%) (0.5,8.5] 37.3 (19/11.3%) ≤550 18.3 (7/3.83%) >550 75.7 (10/24.6%) ≤10,000 133 (16/28.8%) >10,000 157 (21/73.7%) ≤28,000 CHMAX >28,000 MMIN ≤58 783 (5/359%) >58 281 (11/56%) ≤12,000 492 (7/53.9%) >12,000 PRP = – 56.1 + 0.049 MYCT + 0.015 MMIN + 0.006 MMAX + 0.630 CACH – 0.270 CHMIN + 1.46 CHMAX LM1 PRP = 8.29 + 0.004 MMAX + 2.77 CHMIN LM2 PRP = 20.3 + 0.004 MMIN - 3.99 CHMIN + 0.946 CHMAX LM3 PRP = 38.1 + 0.012 MMIN LM4 PRP = 19.5 + 0.002 MMAX + 0.698 CACH + 0.969 CHMAX LM5 PRP = 285 - 1.46 MYCT + 1.02 CACH - 9.39 CHMIN LM6 PRP = -65.8 + 0.03 MMIN - 2.94 CHMIN + 4.98 CHMAX 3.4 Rules 69 FIGURE 3.5 Decision tree for a simple disjunction. a b yes c no x yes c no d yes no x yes no d yes no x yes no Classification Rules It is easy to read a set of classification rules directly off a decision tree. One rule is generated for each leaf. The antecedent of the rule includes a condition for every node on the path from the root to that leaf, and the consequent of the rule is the class assigned by the leaf. This procedure produces rules that are unambiguous in that the order in which they are executed is irrelevant. However, in general, rules that are read directly off a decision tree are far more complex than necessary, and rules derived from trees are usually pruned to remove redundant tests. Because decision trees cannot easily express the disjunction implied among the different rules in a set, transforming a general set of rules into a tree is not quite so straightforward. A good illustration of this occurs when the rules have the same structure but different attributes, like If a and b then x If c and d then x Then it is necessary to break the symmetry and choose a single test for the root node. If, for example, a is chosen, the second rule must, in effect, be repeated twice in the tree, as shown in Figure 3.5. This is known as the replicated subtree problem. 70 CHAPTER 3 Output: Knowledge Representation FIGURE 3.6 The exclusive-or problem. (a) (b) 1 a0 b a b 0 1 x = 1? y = 1? no y = 1? yes b no a yes a no b yes If x = 1 and y = 0 then class = a If x = 0 and y = 1 then class = a If x = 0 and y = 0 then class = b If x = 1 and y = 1 then class = b The replicated subtree problem is sufficiently important that it is worth looking at a couple more examples. The left diagram of Figure 3.6 shows an exclusive-or function for which the output is a if x = 1 or y = 1 but not both. To make this into a tree, you have to split on one attribute first, leading to a structure like the one shown in the center. In contrast, rules can faithfully reflect the true symmetry of the problem with respect to the attributes, as shown on the right. In this example the rules are not notably more compact than the tree. In fact, they are just what you would get by reading rules off the tree in the obvious way. But in other situations, rules are much more compact than trees, particularly if it is possible to have a “default” rule that covers cases not specified by the other rules. For example, to capture the effect of the rules in Figure 3.7—in which there are four attributes, x, y, z, and w, which can each be 1, 2, or 3—requires the tree shown on the right. Each of the three small gray triangles to the upper right should actually contain the whole three-level subtree that is displayed in gray, a rather extreme example of the replicated subtree problem. This is a distressingly complex description of a rather simple concept. One reason why rules are popular is that each rule seems to represent an inde- pendent “nugget” of knowledge. New rules can be added to an existing rule set without disturbing ones already there, whereas to add to a tree structure may require reshaping the whole tree. However, this independence is something of an illusion because it ignores the question of how the rule set is executed. We explained previ- ously the fact that if rules are meant to be interpreted in order as a “decision list,” some of them, taken individually and out of context, may be incorrect. On the other hand, if the order of interpretation is supposed to be immaterial, then it is not clear what to do when different rules lead to different conclusions for the same instance. This situation cannot arise for rules that are read directly off a decision tree because 3.4 Rules 71 FIGURE 3.7 Decision tree with a replicated subtree. x y 1 2 3 a 1 z 2 3 w 1 b 2 b 3 a 1 b 2 b 3 If x = 1 and y = 1 then class = a If z = 1 and w = 1 then class = a Otherwise class = b the redundancy included in the structure of the rules prevents any ambiguity in interpretation. But it does arise when rules are generated in other ways. If a rule set gives multiple classifications for a particular example, one solution is to give no conclusion at all. Another is to count how often each rule fires on the training data and go with the most popular one. These strategies can lead to radically different results. A different problem occurs when an instance is encountered that the rules fail to classify at all. Again, this cannot occur with decision trees, or with rules read directly off them, but it can easily happen with general rule sets. One way of dealing with this situation is to decide not to classify such an example; another is to choose the most frequently occurring class as a default. Again, radically differ- ent results may be obtained for these strategies. Individual rules are simple, and sets of rules seem deceptively simple—but given just a set of rules with no additional information, it is not clear how it should be interpreted. A particularly straightforward situation occurs when rules lead to a class that is Boolean (say, yes and no), and when only rules leading to one outcome (say, yes) are expressed. The assumption is that if a particular instance is not in class yes, then 72 CHAPTER 3 Output: Knowledge Representation it must be in class no—a form of closed-world assumption. If this is the case, rules cannot conflict and there is no ambiguity in rule interpretation: Any interpretation strategy will give the same result. Such a set of rules can be written as a logic expression in what is called disjunctive normal form: that is, as a disjunction (OR) of conjunctive (AND) conditions. It is this simple special case that seduces people into assuming that rules are very easy to deal with, for here each rule really does operate as a new, independent piece of information that contributes in a straightforward way to the disjunction. Unfor- tunately, it only applies to Boolean outcomes and requires the closed-world assump- tion, and both these constraints are unrealistic in most practical situations. Machine learning algorithms that generate rules invariably produce ordered rule sets in multi class situations, and this sacrifices any possibility of modularity because the order of execution is critical. Association Rules Association rules are no different from classification rules except that they can predict any attribute, not just the class, and this gives them the freedom to predict combinations of attributes too. Also, association rules are not intended to be used together as a set, as classification rules are. Different association rules express dif- ferent regularities that underlie the dataset, and they generally predict different things. Because so many different association rules can be derived from even a very small dataset, interest is restricted to those that apply to a reasonably large number of instances and have a reasonably high accuracy on the instances to which they apply. The coverage of an association rule is the number of instances for which it predicts correctly—this is often called its support. Its accuracy—often called confidence—is the number of instances that it predicts correctly, expressed as a proportion of all instances to which it applies. For example, with the rule If temperature = cool then humidity = normal the coverage is the number of days that are both cool and have normal humidity (4 in the data of Table 1.2), and the accuracy is the proportion of cool days that have normal humidity (100% in this case). It is usual to specify minimum coverage and accuracy values, and to seek only those rules for which coverage and accuracy are both at least these specified minima. In the weather data, for example, there are 58 rules with coverage and accuracy that are at least 2 and 95%, respectively. (It may also be convenient to specify coverage as a percentage of the total number of instances instead.) Association rules that predict multiple consequences must be interpreted rather carefully. For example, with the weather data in Table 1.2 we saw this rule: If windy = false and play = no then outlook = sunny and humidity = high 3.4 Rules 73 Table 3.1 New Iris Flower Sepal Length Sepal Width Petal Length Petal Width Type 5.1 3.5 2.6 0.2 ? This is not just a shorthand expression for the two separate rules If windy = false and play = no then outlook = sunny If windy = false and play = no then humidity = high It does indeed imply that these two rules exceed the minimum coverage and accuracy figures—but it also implies more. The original rule means that the number of examples that are nonwindy, nonplaying, with sunny outlook and high humidity, is at least as great as the specified minimum coverage figure. It also means that the number of such days, expressed as a proportion of nonwindy, nonplaying days, is at least the specified minimum accuracy figure. This implies that the rule If humidity = high and windy = false and play = no then outlook = sunny also holds, because it has the same coverage as the original rule, and its accuracy must be at least as high as the original rule’s because the number of high-humidity, nonwindy, nonplaying days is necessarily less than that of nonwindy, nonplaying days—which makes the accuracy greater. As we have seen, there are relationships between particular association rules: Some rules imply others. To reduce the number of rules that are produced, in cases where several rules are related it makes sense to present only the strongest one to the user. In the previous example, only the first rule should be printed. Rules with Exceptions Returning to classification rules, a natural extension is to allow them to have excep- tions. Then incremental modifications can be made to a rule set by expressing exceptions to existing rules rather than reengineering the entire set. For example, consider the iris problem described earlier. Suppose a new flower was found with the dimensions given in Table 3.1, and an expert declared it to be an instance of Iris setosa. If this flower was classified by the rules given in Chapter 1 (see page 14) for this problem, it would be misclassified by two of them: If petal-length ≥ 2.45 and petal-length < 4.45 then Iris-versicolor If petal-length ≥ 2.45 and petal-length < 4.95 and petal-width < 1.55 then Iris-versicolor These rules require modification so that the new instance can be treated correctly. However, simply changing the bounds for the attribute–value tests in these rules may not suffice because the instances used to create the rule set may then be mis- classified. Fixing up a rule set is not as simple as it sounds. 74 CHAPTER 3 Output: Knowledge Representation FIGURE 3.8 Rules for the iris data. Default: Iris-setosa 1 except if petal-length >= 2 553.5 < htgnel-latep dna 54.2 3 57.1 < htdiw-latep dna 4 rolocisrev-sirI neht except if petal-length >= 4.95 and petal-width < 1.55 5 6 acinigriv-sirI neht else if sepal-length < 4.95 and sepal-width >= 2.45 7 8 acinigriv-sirI neht else if petal-length >= 9 53.3 01 acinigriv-sirI neht except if petal-length < 4.85 and sepal-length < 5.95 11 21 rolocisrev-sirI neht Instead of changing the tests in the existing rules, an expert might be consulted to explain why the new flower violates them, giving explanations that could be used to extend the relevant rules only. For example, the first of these two rules misclas- sifies the new Iris setosa as an instance of the genus Iris versicolor. Instead of altering the bounds on any of the inequalities in the rule, an exception can be made based on some other attribute: If petal-length ≥ 2.45 and petal-length < 4.45 then Iris-versicolor EXCEPT if petal-width < 1.0 then Iris-setosa This rule says that a flower is Iris versicolor if its petal length is between 2.45 cm and 4.45 cm except when its petal width is less than 1.0 cm, in which case it is Iris setosa. Of course, we might have exceptions to the exceptions, exceptions to these, and so on, giving the rule set something of the character of a tree. As well as being used to make incremental changes to existing rule sets, rules with exceptions can be used to represent the entire concept description in the first place. Figure 3.8 shows a set of rules that correctly classify all examples in the iris dataset given in Chapter 1. These rules are quite difficult to comprehend at first. Let’s follow them through. A default outcome has been chosen, Iris setosa, and is shown in the first line. For this dataset, the choice of default is rather arbitrary because there are 50 examples of each type. Normally, the most frequent outcome is chosen as the default. Subsequent rules give exceptions to this default. The first if … then, on lines 2 through 4, gives a condition that leads to the classification Iris versicolor. However, there are two exceptions to this rule (lines 5–8), which we will deal with in a moment. If the conditions on lines 2 and 3 fail, the else clause on line 9 is reached, which essentially specifies a second exception to the original default. If the condition 3.4 Rules 75 on line 9 holds, the classification is Iris virginica (line 10). Again, there is an excep- tion to this rule (on lines 11 and 12). Now return to the exception on lines 5 through 8. This overrides the Iris versi- color conclusion on line 4 if either of the tests on lines 5 and 7 holds. As it happens, these two exceptions both lead to the same conclusion, Iris virginica (lines 6 and 8). The final exception is the one on lines 11 and 12, which overrides the Iris virgi- nica conclusion on line 10 when the condition on line 11 is met, and leads to the classification Iris versicolor. You will probably need to ponder these rules for some minutes before it becomes clear how they are intended to be read. Although it takes some time to get used to reading them, sorting out the excepts and if … then … elses becomes easier with familiarity. People often think of real problems in terms of rules, exceptions, and exceptions to the exceptions, so it is often a good way to express a complex rule set. But the main point in favor of this way of representing rules is that it scales up well. Although the whole rule set is a little hard to comprehend, each individual conclusion, each individual then statement, can be considered just in the context of the rules and exceptions that lead to it, whereas with decision lists, all prior rules need to be reviewed to determine the precise effect of an individual rule. This locality property is crucial when trying to understand large rule sets. Psychologically, people familiar with the data think of a particular set of cases, or kind of case, when looking at any one conclusion in the exception structure, and when one of these cases turns out to be an exception to the conclusion, it is easy to add an except clause to cater for it. It is worth pointing out that the default … except if … then structure is logically equivalent to an if … then … else, where the else is unconditional and specifies exactly what the default did. An unconditional else is, of course, a default. (Note that there are no unconditional elses in the preceding rules.) Logically, the exception-based rules can be very simply rewritten in terms of regular if … then … else clauses. What is gained by the formulation in terms of exceptions is not logical but psychological. We assume that the defaults and the tests that occur early on apply more widely than the exceptions further down. If this is indeed true for the domain, and the user can see that it is plausible, the expression in terms of (common) rules and (rare) excep- tions will be easier to grasp than a different, but logically equivalent, structure. More Expressive Rules We have assumed implicitly that the conditions in rules involve testing an attribute value against a constant. But this may not be ideal. Suppose, to take a concrete example, we have the set of eight building blocks of the various shapes and sizes illustrated in Figure 3.9, and we wish to learn the concept of standing up. This is a classic two-class problem with classes standing and lying. The four shaded blocks are positive (standing) examples of the concept, and the unshaded blocks are nega- tive (lying) examples. The only information the learning algorithm will be given is the width, height, and number of sides of each block. The training data is shown in Table 3.2. 76 CHAPTER 3 Output: Knowledge Representation FIGURE 3.9 The shapes problem: shaded = standing; unshaded = lying. Table 3.2 Training Data for the Shapes Problem Width Height Sides Class 2 4 4 standing 3 6 4 standing 4 3 4 lying 7 8 3 standing 7 6 3 lying 2 9 4 standing 9 1 4 lying 10 2 3 lying A conventional rule set that might be produced for this data is if width ≥ 3.5 and height < 7.0 then lying if height ≥ 3.5 then standing In case you’re wondering, 3.5 is chosen as the breakpoint for width because it is halfway between the width of the thinnest lying block, namely 4, and the width of the fattest standing block whose height is less than 7, namely 3. Also, 7.0 is chosen as the breakpoint for height because it is halfway between the height of the tallest lying block, namely 6, and the shortest standing block whose width is greater than 3.5, namely 8. It is common to place numeric thresholds halfway between the values that delimit the boundaries of a concept. 3.4 Rules 77 Although these two rules work well on the examples given, they are not very good. Many new blocks would not be classified by either rule (e.g., one with width 1 and height 2), and it is easy to devise many legitimate blocks that the rules would not fit. A person classifying the eight blocks would probably notice that “standing blocks are those that are taller than they are wide.” This rule does not compare attribute values with constants; it compares attributes with one another: if width > height then lying if height > width then standing The actual values of the height and width attributes are not important, just the result of comparing the two. Many machine learning schemes do not consider relations between attributes because there is a considerable cost in doing so. One way of rectifying this is to add extra, secondary attributes that say whether two primary attributes are equal or not, or give the difference between them if they are numeric. For example, we might add a binary attribute is width < height? to Table 3.2. Such attributes are often added as part of the data engineering process. With a seemingly rather small further enhancement, the expressive power of the knowledge representation can be extended greatly. The trick is to express rules in a way that makes the role of the instance explicit: if width(block) > height(block) then lying(block) if height(block) > width(block) then standing(block) Although this may not seem like much of an extension, it is if instances can be decomposed into parts. For example, if a tower is a pile of blocks, one on top of the other, the fact that the topmost block of the tower is standing can be expressed by if height(tower.top) > width(tower.top) then standing(tower.top) Here, tower.top is used to refer to the topmost block. So far, nothing has been gained. But if tower.rest refers to the rest of the tower, then the fact that the tower is composed entirely of standing blocks can be expressed by the rules if height(tower.top) > width(tower.top) and standing(tower.rest) then standing(tower) The apparently minor addition of the condition standing(tower.rest) is a recursive expression that will turn out to be true only if the rest of the tower is composed of standing blocks. That will be tested by a recursive application of the same rule. Of course, it is necessary to ensure that the recursion “bottoms out” properly by adding a further rule, such as if tower=empty then standing(tower.top) Sets of rules like this are called logic programs, and this area of machine learning is called inductive logic programming. We will not be treating it further in this book. 78 CHAPTER 3 Output: Knowledge Representation 3.5 INSTANCE-BASED REPRESENTATION The simplest form of learning is plain memorization, or rote learning. Once a set of training instances has been memorized, on encountering a new instance the memory is searched for the training instance that most strongly resembles the new one. The only problem is how to interpret “resembles”—we will explain that shortly. First, however, note that this is a completely different way of representing the “knowl- edge” extracted from a set of instances: Just store the instances themselves and operate by relating new instances whose class is unknown to existing ones whose class is known. Instead of trying to create rules, work directly from the examples themselves. This is known as instance-based learning. In a sense, all the other learn- ing methods are instance-based too, because we always start with a set of instances as the initial training information. But the instance-based knowledge representation uses the instances themselves to represent what is learned, rather than inferring a rule set or decision tree and storing it instead. In instance-based learning, all the real work is done when the time comes to classify a new instance rather than when the training set is processed. In a sense, then, the difference between this method and the others that we have seen is the time at which the “learning” takes place. Instance-based learning is lazy, deferring the real work as long as possible, whereas other methods are eager, producing a generalization as soon as the data has been seen. In instance-based classification, each new instance is compared with existing ones using a distance metric, and the closest existing instance is used to assign the class to the new one. This is called the nearest-neighbor classification method. Sometimes more than one nearest neigh- bor is used, and the majority class of the closest k neighbors (or the distance- weighted average if the class is numeric) is assigned to the new instance. This is termed the k-nearest-neighbor method. Computing the distance between two examples is trivial when examples have just one numeric attribute: It is just the difference between the two attribute values. It is almost as straightforward when there are several numeric attributes: Generally, the standard Euclidean distance is used. However, this assumes that the attributes are normalized and are of equal importance, and one of the main problems in learn- ing is to determine which are the important features. When nominal attributes are present, it is necessary to come up with a “distance” between different values of that attribute. What are the distances between, say, the values red, green, and blue? Usually, a distance of zero is assigned if the values are identical; otherwise, the distance is one. Thus, the distance between red and red is zero but the distance between red and green is one. However, it may be desirable to use a more sophisticated representation of the attributes. For example, with more colors one could use a numeric measure of hue in color space, making yellow closer to orange than it is to green and ocher closer still. Some attributes will be more important than others, and this is usually reflected in the distance metric by some kind of attribute weighting. Deriving suitable attri- bute weights from the training set is a key problem in instance-based learning. 3.5 Instance-Based Representation 79 It may not be necessary, or desirable, to store all the training instances. For one thing, this may make the nearest-neighbor calculation unbearably slow. For another, it may consume unrealistic amounts of storage. Generally, some regions of attribute space are more stable than others with regard to class, and just a few exemplars are needed inside stable regions. For example, you might expect the required density of exemplars that lie well inside class boundaries to be much less than the density that is needed near class boundaries. Deciding which instances to save and which to discard is another key problem in instance-based learning. An apparent drawback to instance-based representations is that they do not make explicit the structures that are learned. In a sense, this violates the notion of learning that we presented at the beginning of this book; instances do not really “describe” the patterns in data. However, the instances combine with the distance metric to carve out boundaries in instance space that distinguish one class from another, and this is a kind of explicit representation of knowledge. For example, given a single instance of each of two classes, the nearest-neighbor rule effectively splits the instance space along the perpendicular bisector of the line joining the instances. Given several instances of each class, the space is divided by a set of lines that represent the perpendicular bisectors of selected lines joining an instance of one class to one of another class. Figure 3.10(a) illustrates a nine-sided polygon that separates the filled-circle class from the open-circle class. This polygon is implicit in the operation of the nearest-neighbor rule. When training instances are discarded, the result is to save just a few critical examples of each class. Figure 3.10(b) shows only the examples that actually get used in nearest-neighbor decisions: The others (the light-gray ones) can be discarded without affecting the result. These examples serve as a kind of explicit knowledge representation. Some instance-based representations go further and explicitly generalize the instances. Typically, this is accomplished by creating rectangular regions that enclose examples of the same class. Figure 3.10(c) shows the rectangular regions that might be produced. Unknown examples that fall within one of the rectangles will be assigned the corresponding class; ones that fall outside all rectangles will be subject to the usual nearest-neighbor rule. Of course, this produces different decision bound- aries from the straightforward nearest-neighbor rule, as can be seen by superimposing the polygon in Figure 3.10(a) onto the rectangles. Any part of the polygon that lies within a rectangle will be chopped off and replaced by the rectangle’s boundary. Rectangular generalizations in instance space are just like rules with a special form of condition, one that tests a numeric variable against an upper and lower bound and selects the region in between. Different dimensions of the rectangle correspond to tests on different attributes being ANDed together. Choosing snug-fitting rectan- gular regions as tests leads to more conservative rules than those generally produced by rule-based machine learning schemes, because for each boundary of the region, there is an actual instance that lies on (or just inside) that boundary. Tests such as x < a (where x is an attribute value and a is a constant) encompass an entire half-space—they apply no matter how small x is as long as it is less than a. 80 CHAPTER 3 Output: Knowledge Representation FIGURE 3.10 Different ways of partitioning the instance space. (a) (b) (c) (d) When doing rectangular generalization in instance space you can afford to be conservative, because if a new example is encountered that lies outside all regions, you can fall back on the nearest-neighbor metric. With rule-based methods the example cannot be classified, or receives just a default classification, if no rules apply to it. The advantage of more conservative rules is that, although incomplete, they may be more perspicuous than a complete set of rules that covers all cases. Finally, ensuring that the regions do not overlap is tantamount to ensuring that at most one rule can apply to an example, eliminating another of the difficulties of rule-based systems—what to do when several rules apply. A more complex kind of generalization is to permit rectangular regions to nest one within another. Then a region that is basically all one class can contain an inner region with a different class, as illustrated in Figure 3.10(d). It is possible 3.6 Clusters 81 to allow nesting within nesting so that the inner region can itself contain its own inner region of a different class—perhaps the original class of the outer region. This is analogous to allowing rules to have exceptions and exceptions to the exceptions, as in Section 3.4. It is worth pointing out a slight danger to the technique of visualizing instance- based learning in terms of boundaries in example space: It makes the implicit assumption that attributes are numeric rather than nominal. If the various values that a nominal attribute can take on were laid out along a line, generalizations involving a segment of that line would make no sense: Each test involves either one value for the attribute or all values for it (or perhaps an arbitrary subset of values). Although you can more or less easily imagine extending the examples in Figure 3.10 to several dimensions, it is much harder to imagine how rules involving nominal attributes will look in multidimensional instance space. Many machine learning situations involve numerous attributes, and our intuitions tend to lead us astray when extended to high-dimensional spaces. 3.6 CLUSTERS When a cluster rather than a classifier is learned, the output takes the form of a diagram that shows how the instances fall into clusters. In the simplest case this involves associating a cluster number with each instance, which might be depicted by laying the instances out in two dimensions and partitioning the space to show each cluster, as illustrated in Figure 3.11(a). Some clustering algorithms allow one instance to belong to more than one cluster, so the diagram might lay the instances out in two dimensions and draw overlapping subsets representing each cluster—a Venn diagram, as in Figure 3.11(b). Some algorithms associate instances with clusters probabilistically rather than cat- egorically. In this case, for every instance there is a probability or degree of mem- bership with which it belongs to each of the clusters. This is shown in Figure 3.11(c). This particular association is meant to be a probabilistic one, so the numbers for each example sum to 1—although that is not always the case. Other algorithms produce a hierarchical structure of clusters so that at the top level the instance space divides into just a few clusters, each of which divides into its own subcluster at the next level down, and so on. In this case a diagram such as the one in Figure 3.11(d) is used, in which elements joined together at lower levels are more tightly clustered than ones joined together at higher levels. Such diagrams are called dendrograms. This term means just the same thing as tree diagrams (the Greek word dendron means “tree”), but in clustering the more exotic version seems to be preferred—perhaps because biological species are a prime application area for clustering techniques, and ancient languages are often used for naming in biology. Clustering is often followed by a stage in which a decision tree or rule set is inferred that allocates each instance to the cluster in which it belongs. Then, the clustering operation is just one step on the way to a structural description. 82 CHAPTER 3 Output: Knowledge Representation FIGURE 3.11 Different ways of representing clusters. (a) d e c h j a k g i f b d e c h j a k g i f b (c) gaciedkbjfh (d) (b) 1 2 3 a 0.4 0.1 0.5 b 0.1 0.8 0.1 c 0.3 0.3 0.4 d 0.1 0.1 0.8 e 0.4 0.2 0.4 f 0.1 0.4 0.5 g 0.7 0.2 0.1 h 0.5 0.4 0.1 … 3.7 Further Reading 83 3.7 FURTHER READING Knowledge representation is a key topic in classical artificial intelligence and early work on it is well represented by a comprehensive series of papers edited by Brachman and Levesque (1985). The area of inductive logic programming and associated topics are covered in detail by de Raedt’s book, Logical and relational learning (2008). We mentioned the problem of dealing with conflict among different rules. Various ways of doing this, called conflict resolution strategies, have been developed for use with rule-based programming systems. These are described in books on rule-based programming such as Brownstown et al. (1985). Again, however, they are designed for use with handcrafted rule sets rather than ones that have been learned. The use of handcrafted rules with exceptions for a large dataset has been studied by Gaines and Compton (1995), and Richards and Compton (1998) describe their role as an alternative to classic knowledge engineering. Further information on the various styles of concept representation can be found in the papers that describe machine learning methods for inferring concepts from examples, and these are covered in Section 4.10, Further Reading, and the Discussion sections of Chapter 6. This page intentionally left blank 85Data Mining: Practical Machine Learning Tools and Techniques Copyright © 2011 Elsevier Inc. All rights of reproduction in any form reserved. Algorithms: The Basic Methods CHAPTER 4 Now that we’ve seen how the inputs and outputs can be represented, it’s time to look at the learning algorithms themselves. This chapter explains the basic ideas behind the techniques that are used in practical data mining. We will not delve too deeply into the trickier issues—advanced versions of the algorithms, optimizations that are possible, complications that arise in practice. These topics are deferred to Chapter 6, where we come to grips with real implementations of machine learning schemes such as the ones included in data mining toolkits and used for real-world applications. It is important to understand these more advanced issues so that you know what is really going on when you analyze a particular dataset. In this chapter we look at the basic ideas. One of the most instructive lessons is that simple ideas often work very well, and we strongly recommend the adoption of a “simplicity-first” methodology when analyzing practical datasets. There are many different kinds of simple structure that datasets can exhibit. In one dataset, there might be a single attribute that does all the work and the others are irrelevant or redundant. In another dataset, the attributes might contribute independently and equally to the final outcome. A third might have a simple logical structure, involving just a few attributes, which can be captured by a decision tree. In a fourth, there may be a few independent rules that govern the assignment of instances to different classes. A fifth might exhibit dependencies among different subsets of attributes. A sixth might involve linear dependence among numeric attributes, where what matters is a weighted sum of attribute values with appropriately chosen weights. In a seventh, classifications appropriate to particular regions of instance space might be governed by the distances between the instances themselves. And in an eighth, it might be that no class values are provided: The learning is unsupervised. In the infinite variety of possible datasets there are many different kinds of structures that can occur, and a data mining tool—no matter how capable—that is looking for one class of structure may completely miss regularities of a different kind, regardless of how rudimentary those may be. The result is a baroque and opaque classification structure of one kind instead of a simple, elegant, immediately comprehensible structure of another. Each of the eight examples of different kinds of datasets just sketched leads to a different machine learning scheme that is well suited to discovering the underlying concept. The sections of this chapter look at each of these structures in turn. A final 86 CHAPTER 4 Algorithms: The Basic Methods section introduces simple ways of dealing with multi-instance problems, where each example comprises several different instances. 4.1 INFERRING RUDIMENTARY RULES Here’s an easy way to find very simple classification rules from a set of instances. Called 1R for 1-rule, it generates a one-level decision tree expressed in the form of a set of rules that all test one particular attribute. 1R is a simple, cheap method that often comes up with quite good rules for characterizing the structure in data. It turns out that simple rules frequently achieve surprisingly high accu- racy. Perhaps this is because the structure underlying many real-world datasets is quite rudimentary, and just one attribute is sufficient to determine the class of an instance quite accurately. In any event, it is always a good plan to try the simplest things first. The idea is this: We make rules that test a single attribute and branch accord- ingly. Each branch corresponds to a different value of the attribute. It is obvious what is the best classification to give each branch: Use the class that occurs most often in the training data. Then the error rate of the rules can easily be determined. Just count the errors that occur on the training data—that is, the number of instances that do not have the majority class. Each attribute generates a different set of rules, one rule for every value of the attribute. Evaluate the error rate for each attribute’s rule set and choose the best. It’s that simple! Figure 4.1 shows the algorithm in the form of pseudocode. To see the 1R method at work, consider the weather data of Table 1.2 on page 10 (we will encounter it many times again when looking at how learning algorithms work). To classify on the final column, play, 1R considers four sets of rules, one for each attribute. These rules are shown in Table 4.1. An asterisk indicates that a random choice has been made between two equally likely outcomes. The number of errors is given for each rule, along with the total number of errors for the rule set as a whole. 1R chooses the attribute that produces rules with the smallest number of FIGURE 4.1 Pseudocode for 1R. For each attribute, For each value of that attribute, make a rule as follows: count how often each class appears find the most frequent class make the rule assign that class to this attribute value. Calculate the error rate of the rules. Choose the rules with the smallest error rate. 4.1 Inferring Rudimentary Rules 87 64 65 68 69 70 71 72 72 75 75 80 81 83 85 yes no yes yes yes no no yes yes yes no yes yes no Table 4.1 Evaluating Attributes in the Weather Data Attribute Rules Errors Total Errors 1 outlook sunny → no 2/5 4/14 overcast → yes 0/4 rainy → yes 2/5 2 temperature hot → no* 2/4 5/14 mild → yes 2/6 cool → yes 1/4 3 humidity high → no 3/7 4/14 normal → yes 1/7 4 windy false → yes 2/8 5/14 true → no* 3/6 *A random choice has been made between two equally likely outcomes. errors—that is, the first and third rule sets. Arbitrarily breaking the tie between these two rule sets gives outlook: sunny → no overcast → yes rainy → yes We noted at the outset that the game for the weather data is unspecified. Oddly enough, it is apparently played when it is overcast or rainy but not when it is sunny. Perhaps it’s an indoor pursuit. Missing Values and Numeric Attributes Although a very rudimentary learning scheme, 1R does accommodate both missing values and numeric attributes. It deals with these in simple but effective ways. Missing is treated as just another attribute value so that, for example, if the weather data had contained missing values for the outlook attribute, a rule set formed on outlook would specify four possible class values, one for each of sunny, overcast, and rainy, and a fourth for missing. We can convert numeric attributes into nominal ones using a simple discreti- zation method. First, sort the training examples according to the values of the numeric attribute. This produces a sequence of class values. For example, sorting the numeric version of the weather data (Table 1.3, page 11) according to the values of temperature produces the sequence Discretization involves partitioning this sequence by placing breakpoints in it. One possibility is to place breakpoints wherever the class changes, producing the following eight categories: 88 CHAPTER 4 Algorithms: The Basic Methods yes | no | yes yes yes | no no | yes yes yes | no | yes yes | no Choosing breakpoints halfway between the examples on either side places them at 64.5, 66.5, 70.5, 72, 77.5, 80.5, and 84. However, the two instances with value 72 cause a problem because they have the same value of temperature but fall into different classes. The simplest fix is to move the breakpoint at 72 up one example, to 73.5, producing a mixed partition in which no is the majority class. A more serious problem is that this procedure tends to form an excessively large number of categories. The 1R method will naturally gravitate toward choos- ing an attribute that splits into many categories, because this will partition the dataset into many pieces, making it more likely that instances will have the same class as the majority in their partition. In fact, the limiting case is an attribute that has a different value for each instance—that is, an identification code attribute that pinpoints instances uniquely—and this will yield a zero error rate on the training set because each partition contains just one instance. Of course, highly branching attributes do not usually perform well on test examples; indeed, the identification code attribute will never get any examples outside the training set correct. This phenomenon is known as overfitting; we have already described overfitting- avoidance bias in Chapter 1, and we will encounter this problem repeatedly in subsequent chapters. For 1R, overfitting is likely to occur whenever an attribute has a large number of possible values. Consequently, when discretizing a numeric attribute, a minimum limit is imposed on the number of examples of the majority class in each partition. Suppose that minimum is set at 3. This eliminates all but two of the preceding partitions. Instead, the partitioning process begins yes no yes yes | yes … ensuring that there are three occurrences of yes, the majority class, in the first parti- tion. However, because the next example is also yes, we lose nothing by including that in the first partition, too. This leads to a new division of yes no yes yes yes | no no yes yes yes | no yes yes no where each partition contains at least three instances of the majority class, except the last one, which will usually have less. Partition boundaries always fall between examples of different classes. Whenever adjacent partitions have the same majority class, as do the first two partitions above, they can be merged together without affecting the meaning of the rule sets. Thus, the final discretization is yes no yes yes yes no no yes yes yes | no yes yes no which leads to the rule set temperature: ≤ 77.5 → yes > 77.5 → no 4.1 Inferring Rudimentary Rules 89 The second rule involved an arbitrary choice; as it happens, no was chosen. If yes had been chosen instead, there would be no need for any breakpoint at all—and as this example illustrates, it might be better to use the adjacent categories to help break ties. In fact, this rule generates five errors on the training set and so is less effective than the preceding rule for outlook. However, the same procedure leads to this rule for humidity: humidity: ≤ 82.5 → yes > 82.5 and ≤ 95.5 → no > 95.5 → yes This generates only three errors on the training set and is the best 1-rule for the data in Table 1.3. Finally, if a numeric attribute has missing values, an additional category is created for them, and the discretization procedure is applied just to the instances for which the attribute’s value is defined. Discussion In a seminal paper entitled “Very simple classification rules perform well on most commonly used datasets” (Holte, 1993), a comprehensive study of the performance of the 1R procedure was reported on 16 datasets frequently used by machine learning researchers to evaluate their algorithms. Cross-validation, an evaluation technique that we will explain in Chapter 5, was used to ensure that the results were the same as would be obtained on independent test sets. After some experimentation, the minimum number of examples in each partition of a numeric attribute was set at six, not three as used in our illustration. Surprisingly, despite its simplicity 1R did well in comparison with the state- of-the-art learning schemes, and the rules it produced turned out to be just a few percentage points less accurate, on almost all of the datasets, than the decision trees produced by a state-of-the-art decision tree induction scheme. These trees were, in general, considerably larger than 1R’s rules. Rules that test a single attribute are often a viable alternative to more complex structures, and this strongly encourages a simplicity-first methodology in which the baseline performance is established using simple, rudimentary techniques before progressing to more sophis- ticated learning schemes, which inevitably generate output that is harder for people to interpret. The 1R procedure learns a one-level decision tree whose leaves represent the various different classes. A slightly more expressive technique is to use a different rule for each class. Each rule is a conjunction of tests, one for each attribute. For numeric attributes the test checks whether the value lies within a given interval; for nominal ones it checks whether it is in a certain subset of that attribute’s values. These two types of tests—that is, intervals and subsets—are learned from the training data pertaining to each of the classes. For a numeric attribute, the end 90 CHAPTER 4 Algorithms: The Basic Methods points of the interval are the minimum and the maximum values that occur in the training data for that class. For a nominal one, the subset contains just those values that occur for that attribute in the training data for the individual class. Rules representing different classes usually overlap, and at prediction time the one with the most matching tests is predicted. This simple technique often gives a useful first impression of a dataset. It is extremely fast and can be applied to very large quantities of data. 4.2 STATISTICAL MODELING The 1R method uses a single attribute as the basis for its decisions and chooses the one that works best. Another simple technique is to use all attributes and allow them to make contributions to the decision that are equally important and independent of one another, given the class. This is unrealistic, of course: What makes real-life datasets interesting is that the attributes are certainly not equally important or inde- pendent. But it leads to a simple scheme that, again, works surprisingly well in practice. Table 4.2 shows a summary of the weather data obtained by counting how many times each attribute–value pair occurs with each value (yes and no) for play. For example, you can see from Table 1.2 (page 10) that outlook is sunny for five examples, two of which have play = yes and three of which have play = no. The cells in the first row of the new table simply count these occurrences for all pos- sible values of each attribute, and the play figure in the final column counts the total number of occurrences of yes and no. The lower part of the table contains the same information expressed as fractions, or observed probabilities. For example, of the nine days that play is yes, outlook is sunny for two, yielding a fraction of 2/9. For play the fractions are different: They are the proportion of days that play is yes and no, respectively. Now suppose we encounter a new example with the values that are shown in Table 4.3. We treat the five features in Table 4.2—outlook, temperature, humidity, windy, and the overall likelihood that play is yes or no—as equally important, inde- pendent pieces of evidence and multiply the corresponding fractions. Looking at the outcome yes gives Likelihood of yes = × × × × =2 9 3 9 3 9 3 9 9 14 0 0053. The fractions are taken from the yes entries in the table according to the values of the attributes for the new day, and the final 9/14 is the overall fraction rep- resenting the proportion of days on which play is yes. A similar calculation for the outcome no leads to Likelihood of no = × × × × =3 5 1 5 4 5 3 5 5 14 0 0206. 91 Table 4.2 Weather Data with Counts and Probabilities Outlook Temperature Humidity Windy Play yes no yes no yes no yes no yes no sunny 2 3 hot 2 2 high 3 4 false 6 2 9 5 overcast 4 0 mild 4 2 normal 6 1 true 3 3 rainy 3 2 cool 3 1 sunny 2/9 3/5 hot 2/9 2/5 high 3/9 4/5 false 6/9 2/5 9/14 5/14 overcast 4/9 0/5 mild 4/9 2/5 normal 6/9 1/5 true 3/9 3/5 rainy 3/9 2/5 cool 3/9 1/5 92 CHAPTER 4 Algorithms: The Basic Methods Table 4.3 A New Day Outlook Temperature Humidity Windy Play Sunny cool high true ? This indicates that for the new day, no is more likely than yes—four times more likely. The numbers can be turned into probabilities by normalizing them so that they sum to 1: Probability of yes = + =0 0053 0 0053 0 0206 20 5. ...% Probability of no = + =0 0206 0 0053 0 0206 79 5. ...% This simple and intuitive method is based on Bayes’ rule of conditional probability. Bayes’ rule says that if you have a hypothesis H and evidence E that bears on that hypothesis, then Pr[ | ] Pr[ | ]Pr[ ] Pr[ ]HEEHH E= We use the notation that Pr[A] denotes the probability of an event A and Pr[A | B] denotes the probability of A conditional on another event B. The hypothesis H is that play will be, say, yes, and Pr[H | E] is going to turn out to be 20.5%, just as determined previously. The evidence E is the particular combination of attribute values for the new day—outlook = sunny, temperature = cool, humidity = high, and windy = true. Let’s call these four pieces of evidence E1, E2, E3, and E4, respectively. Assuming that these pieces of evidence are independent (given the class), their combined probability is obtained by multiplying the probabilities: Pr[ | ] Pr[ | ]Pr[ | ]Pr[ | ]Pr[ | ]Pr[yes E E yes E yes E yes E yes y= × × × ×1 2 3 4 ees E ] Pr[ ] Don’t worry about the denominator: We will ignore it and eliminate it in the final normalizing step when we make the probabilities for yes and no sum to 1, just as we did previously. The Pr[yes] at the end is the probability of a yes outcome without knowing any of the evidence E—that is, without knowing anything about the particular day in question—and it’s called the prior probability of the hypothesis H. In this case, it’s just 9/14, because 9 of the 14 training examples had a yes 4.2 Statistical Modeling 93 value for play. Substituting the fractions in Table 4.2 for the appropriate evidence probabilities leads to Pr[ | ] Pr[ ]yes E E= × × × ×2 9 3 9 3 9 3 9 9 14 just as we calculated previously. Again, the Pr[E] in the denominator will disappear when we normalize. This method goes by the name of Naïve Bayes because it’s based on Bayes’ rule and “naïvely” assumes independence—it is only valid to multiply probabilities when the events are independent. The assumption that attributes are independent (given the class) in real life certainly is a simplistic one. But despite the disparaging name, Naïve Bayes works very effectively when tested on actual datasets, particularly when combined with some of the attribute selection procedures, which are introduced in Chapter 7, that eliminate redundant, and hence nonindependent, attributes. Things go badly awry in Naïve Bayes if a particular attribute value does not occur in the training set in conjunction with every class value. Suppose that in the training data the attribute value outlook = sunny was always associated with the outcome no. Then the probability of outlook = sunny being given a yes—that is, Pr[outlook = sunny | yes]—would be zero, and because the other probabilities are multiplied by this, the final probability of yes in the previous example would be zero no matter how large they were. Probabilities that are zero hold a veto over the other ones. This is not a good idea. But the bug is easily fixed by minor adjustments to the method of calculating probabilities from frequencies. For example, the upper part of Table 4.2 shows that for play = yes, outlook is sunny for two examples, overcast for four, and rainy for three, and the lower part gives these events probabilities of 2/9, 4/9, and 3/9, respectively. Instead, we could add 1 to each numerator, and compensate by adding 3 to the denominator, giving probabilities of 3/12, 5/12, and 4/12, respectively. This will ensure that an attribute value that occurs zero times receives a probability which is nonzero, albeit small. The strategy of adding 1 to each count is a standard technique called the Laplace estimator after the great eighteenth-century French mathematician Pierre Laplace. Although it works well in practice, there is no particular reason for adding 1 to the counts: We could instead choose a small constant µ and use 2 3 9 4 3 9 3 3 9 + + + + + + µ µ µ µ µ µ , , and The value of µ, which was set to 3 before, effectively provides a weight that determines how influential the a priori values of 1/3, 1/3, and 1/3 are for each of the three possible attribute values. A large µ says that these priors are very important compared with the new evidence coming in from the training set, whereas a small one gives them less influence. Finally, there is no particular reason for dividing µ into three equal parts in the numerators: We could use 94 CHAPTER 4 Algorithms: The Basic Methods 2 9 4 9 3 9 1 2 3+ + + + + + µ µ µ µ µ µ p p p, , and instead, where p1, p2, and p3 sum to 1. Effectively, these three numbers are a priori probabilities of the values of the outlook attribute being sunny, overcast, and rainy, respectively. This is now a fully Bayesian formulation where prior probabilities have been assigned to everything in sight. It has the advantage of being completely rigorous, but the disadvantage that it is not usually clear just how these prior probabilities should be assigned. In practice, the prior probabilities make little difference provided that there are a reasonable number of training instances, and people generally just estimate frequencies using the Laplace estimator by initializing all counts to 1 instead of 0. Missing Values and Numeric Attributes One of the really nice things about Naïve Bayes is that missing values are no problem at all. For example, if the value of outlook were missing in the example of Table 4.3, the calculation would simply omit this attribute, yielding Likelihood of yes = × × × =3 9 3 9 3 9 9 14 0 0238. Likelihood of no = × × × =1 5 4 5 3 5 5 14 0 0343. These two numbers are individually a lot higher than they were before because one of the fractions is missing. But that’s not a problem because a fraction is missing in both cases, and these likelihoods are subject to a further normalization process. This yields probabilities for yes and no of 41% and 59%, respectively. If a value is missing in a training instance, it is simply not included in the fre- quency counts, and the probability ratios are based on the number of values that actually occur rather than on the total number of instances. Numeric values are usually handled by assuming that they have a “normal” or “Gaussian” probability distribution. Table 4.4 gives a summary of the weather data with numeric features from Table 1.3. For nominal attributes, we calculate counts as before, while for numeric ones we simply list the values that occur. Then, instead of normalizing counts into probabilities as we do for nominal attributes, we calculate the mean and the standard deviation for each class and each numeric attribute. The mean value of temperature over the yes instances is 73, and its standard deviation is 6.2. The mean is simply the average of the values—that is, the sum divided by the number of values. The standard deviation is the square root of the sample variance, which we calculate as follows: Subtract the mean from each value, square the result, sum them together, and then divide by one less than the number of values. After we have found this “sample variance,” take its square root to yield the standard deviation. This is the standard way of calculating the mean and the standard deviation of a set of numbers. (The “one less than” has to do with the number of degrees of freedom in the sample, a statistical notion that we don’t want to get into here.) 95 Table 4.4 Numeric Weather Data with Summary Statistics Outlook Temperature Humidity Windy Play yes no yes no yes no yes no yes no sunny 2 3 83 85 86 85 false 6 2 9 5 overcast 4 0 70 80 96 90 true 3 3 rainy 3 2 68 65 80 70 64 72 65 95 69 71 70 91 75 80 75 70 72 90 81 75 sunny 2/9 3/5 mean 73 74.6 mean 79.1 86.2 false 6/9 2/5 9/14 5/14 overcast 4/9 0/5 std. dev. 6.2 7.9 std. dev. 10.2 9.7 true 3/9 3/5 rainy 3/9 2/5 96 CHAPTER 4 Algorithms: The Basic Methods Table 4.5 Another New Day Outlook Temperature Humidity Windy Play Sunny 66 90 true ? The probability density function for a normal distribution with mean µ and standard deviation σ is given by the rather formidable expression f x e x () () = − −1 2 2 22 πσ µ σ But fear not! All this means is that if we are considering a yes outcome when temperature has a value of, say, 66, we just need to plug x = 66, µ = 73, and σ = 6.2 into the formula. So the value of the probability density function is f temperature yes e( | ) . . () .= = × = − − ×66 1 2 6 2 0 0340 66 73 2 6 2 2 2 π And by the same token, the probability density of a yes outcome when humidity has a value of, say, 90, is calculated in the same way: f humidity yes( | ) .= =90 0 0221 The probability density function for an event is very closely related to its prob- ability. However, it is not quite the same thing. If temperature is a continuous scale, the probability of the temperature being exactly 66—or exactly any other value, such as 63.14159262—is zero. The real meaning of the density function f(x) is that the probability that the quantity lies within a small region around x, say between x − ε/2 and x + ε/2, is ε × f(x). You might think we ought to factor in the accuracy figure ε when using these density values, but that’s not necessary. The same ε would appear in both the yes and no likelihoods that follow and cancel out when the probabilities were calculated. Using these probabilities for the new day in Table 4.5 yields Likelihood of yes = × × × × =2 9 0 0340 0 0221 3 9 9 14 0 000036... Likelihood of no = × × × × =3 5 0 0279 0 0381 3 5 5 14 0 000137... which leads to probabilities Probability of yes = + =0 000036 0 000036 0 000137 20 8. ...% 4.2 Statistical Modeling 97 Probability of no = + =0 000137 0 000036 0 000137 79 2. ...% These figures are very close to the probabilities calculated earlier for the new day in Table 4.3 because the temperature and humidity values of 66 and 90 yield similar probabilities to the cool and high values used before. The normal-distribution assumption makes it easy to extend the Naïve Bayes classifier to deal with numeric attributes. If the values of any numeric attributes are missing, the mean and standard deviation calculations are based only on the ones that are present. Naïve Bayes for Document Classification An important domain for machine learning is document classification, in which each instance represents a document and the instance’s class is the document’s topic. Documents might be news items and the classes might be domestic news, overseas news, financial news, and sports. Documents are characterized by the words that appear in them, and one way to apply machine learning to document classification is to treat the presence or absence of each word as a Boolean attribute. Naïve Bayes is a popular technique for this application because it is very fast and quite accurate. However, this does not take into account the number of occurrences of each word, which is potentially useful information when determining the category of a document. Instead, a document can be viewed as a bag of words—a set that contains all the words in the document, with multiple occurrences of a word appearing mul- tiple times (technically, a set includes each of its members just once, whereas a bag can have repeated elements). Word frequencies can be accommodated by applying a modified form of Naïve Bayes called multinominal Naïve Bayes. Suppose n1, n2, …, nk is the number of times word i occurs in the document, and P1, P2, …, Pk is the probability of obtaining word i when sampling from all the documents in category H. Assume that the probability is independent of the word’s context and position in the document. These assumptions lead to a multinomial distribution for document probabilities. For this distribution, the probability of a document E given its class H—in other words, the formula for computing the probability Pr[E | H] in Bayes’ rule—is Pr[ ]EHN P n i n ii k i | ! != × = ∏1 where N = n1 + n2 + … + nk is the number of words in the document. The reason for the factorials is to account for the fact that the ordering of the occurrences of each word is immaterial according to the bag-of-words model. Pi is estimated by computing the relative frequency of word i in the text of all training documents pertaining to category H. In reality, there could be a further term that gives the probability that the model for category H generates a document whose length is the same as the length of E, but it is common to assume that this is the same for all classes and hence can be dropped. 98 CHAPTER 4 Algorithms: The Basic Methods For example, suppose there are only two words, yellow and blue, in the vocabu- lary, and a particular document class H has Pr[yellow | H] = 75% and Pr[blue | H] = 25% (you might call H the class of yellowish green documents). Suppose E is the document blue yellow blue with a length of N = 3 words. There are four possible bags of three words. One is {yellow yellow yellow}, and its probability according to the preceding formula is Pr [{}| ] ! . ! . !yellow yellow yellow H = × × =3 0 75 3 0 25 0 27 64 3 0 The other three, with their probabilities, are Pr[{ blue blue blue H}| ] = 1 64 Pr[{ yellow yellow blue H}| ] = 27 64 Pr[{ yellow blue blue H}| ] = 9 64 E corresponds to the last case (recall that in a bag of words the order is immaterial); thus, its probability of being generated by the yellowish green document model is 9/64, or 14%. Suppose another class, very bluish green documents (call it H′), has Pr[yellow | H′] = 10% and Pr[blue | H′] = 90%. The probability that E is generated by this model is 24%. If these are the only two classes, does that mean that E is in the very bluish green document class? Not necessarily. Bayes’ rule, given earlier, says that you have to take into account the prior probability of each hypothesis. If you know that in fact very bluish green documents are twice as rare as yellowish green ones, this would be just sufficient to outweigh the 14 to 24% disparity and tip the balance in favor of the yellowish green class. The factorials in the probability formula don’t actually need to be computed because, being the same for every class, they drop out in the normalization process anyway. However, the formula still involves multiplying together many small prob- abilities, which soon yields extremely small numbers that cause underflow on large documents. The problem can be avoided by using logarithms of the probabilities instead of the probabilities themselves. In the multinomial Naïve Bayes formulation a document’s class is determined not just by the words that occur in it but also by the number of times they occur. In general, it performs better than the ordinary Naïve Bayes model for document clas- sification, particularly for large dictionary sizes. 4.3 Divide-and-Conquer: Constructing Decision Trees 99 Discussion Naïve Bayes gives a simple approach, with clear semantics, to representing, using, and learning probabilistic knowledge. It can achieve impressive results. People often find that Naïve Bayes rivals, and indeed outperforms, more sophisticated classifiers on many datasets. The moral is, always try the simple things first. Over and over again people have eventually, after an extended struggle, managed to obtain good results using sophisticated learning schemes, only to discover later that simple methods such as 1R and Naïve Bayes do just as well—or even better. There are many datasets for which Naïve Bayes does not do well, however, and it is easy to see why. Because attributes are treated as though they were independent given the class, the addition of redundant ones skews the learning process. As an extreme example, if you were to include a new attribute with the same values as temperature to the weather data, the effect of the temperature attribute would be multiplied: All of its probabilities would be squared, giving it a great deal more influence in the decision. If you were to add 10 such attributes, the decisions would effectively be made on temperature alone. Dependencies between attributes inevi- tably reduce the power of Naïve Bayes to discern what is going on. They can, however, be ameliorated by using a subset of the attributes in the decision procedure, making a careful selection of which ones to use. Chapter 7 shows how. The normal-distribution assumption for numeric attributes is another restriction on Naïve Bayes as we have formulated it here. Many features simply aren’t nor- mally distributed. However, there is nothing to prevent us from using other distributions—there is nothing magic about the normal distribution. If you know that a particular attribute is likely to follow some other distribution, standard esti- mation procedures for that distribution can be used instead. If you suspect it isn’t normal but don’t know the actual distribution, there are procedures for “kernel density estimation” that do not assume any particular distribution for the attribute values. Another possibility is simply to discretize the data first. 4.3 DIVIDE-AND-CONQUER: CONSTRUCTING DECISION TREES The problem of constructing a decision tree can be expressed recursively. First, select an attribute to place at the root node, and make one branch for each possible value. This splits up the example set into subsets, one for every value of the attribute. Now the process can be repeated recursively for each branch, using only those instances that actually reach the branch. If at any time all instances at a node have the same classification, stop developing that part of the tree. The only thing left is how to determine which attribute to split on, given a set of examples with different classes. Consider (again!) the weather data. There are four possibilities for each split, and at the top level they produce the trees in Figure 4.2. 100 CHAPTER 4 Algorithms: The Basic Methods FIGURE 4.2 Tree stumps for the weather data: (a) outlook, (b) temperature, (c) humidity, and (d) windy. outlook yes yes no no no sunny yes yes yes yes overcast yes yes yes no no rainy temperature yes yes no no hot yes yes yes yes no no mild yes yes yes no cool (a) (b) humidity yes yes yes no no no no high yes yes yes yes yes yes no normal windy yes yes yes yes yes yes no no false yes yes yes no no no true (c) (d) Which is the best choice? The number of yes and no classes is shown at the leaves. Any leaf with only one class—yes or no—will not have to be split further, and the recursive process down that branch will terminate. Because we seek small trees, we would like this to happen as soon as possible. If we had a measure of the purity of each node, we could choose the attribute that produces the purest daughter nodes. Take a moment to look at Figure 4.2 and ponder which attribute you think is the best choice. The measure of purity that we will use is called the information and is measured in units called bits. Associated with each node of the tree, it represents the expected amount of information that would be needed to specify whether a new instance should be classified yes or no, given that the example reached that node. Unlike the bits in computer memory, the expected amount of information usually involves frac- tions of a bit—and is often less than 1! It is calculated based on the number of yes and no classes at the node. We will look at the details of the calculation shortly, but first let’s see how it’s used. When evaluating the first tree in Figure 4.2, the number of yes and no classes at the leaf nodes are [2, 3], [4, 0], and [3, 2], respectively, and the information values of these nodes are info bits([ , ]) .23 0 971= info bits([ , ]) .40 0 0= info bits([ , ]) .32 0 971= We calculate the average information value of these, taking into account the number of instances that go down each branch—five down the first and third and four down the second: info([ , ], [ , ], [ , ]) ( ) . ( ) ( ) . . 2 3 4 0 3 2 5 14 0 971 4 14 0 5 14 0 971 0 = × + × + × = 6693 bits This average represents the amount of information that we expect would be nec essary to specify the class of a new instance, given the tree structure in Figure 4.2(a). Before any of the nascent tree structures in Figure 4.2 were created, the training examples at the root comprised nine yes and five no nodes, corresponding to an information value of info bits([ , ]) .95 0 940= Thus, the tree in Figure 4.2(a) is responsible for an information gain of gain( ) info infooutlook = − = −([,]) ([,],[,],[,]) . .9 5 2 3 4 0 3 2 0 940 0 6693 0 247= . bits which can be interpreted as the informational value of creating a branch on the outlook attribute. The way forward is clear. We calculate the information gain for each attribute and split on the one that gains the most information. In the situation that is shown in Figure 4.2: • gain(outlook) = 0.247 bits • gain(temperature) = 0.029 bits • gain(humidity) = 0.152 bits • gain(windy) = 0.048 bits Therefore, we select outlook as the splitting attribute at the root of the tree. Hope- fully this accords with your intuition as the best one to select. It is the only choice for which one daughter node is completely pure, and this gives it a considerable advantage over the other attributes. Humidity is the next best choice because it produces a larger daughter node that is almost completely pure. Then we continue, recursively. Figure 4.3 shows the possibilities for a further branch at the node reached when outlook is sunny. Clearly, a further split on 4.3 Divide-and-Conquer: Constructing Decision Trees 101 102 CHAPTER 4 Algorithms: The Basic Methods FIGURE 4.3 Expanded tree stumps for the weather data: (a) temperature, (b) humidity, and (c) windy. outlook windy sunny ... ... yes no no false yes no true (c) outlook temperature sunny ... ... no no hot yes no mild yes cool outlook humidity sunny ... ... no no no high yes yes normal (a) (b) outlook will produce nothing new, so we only consider the other three attributes. The information gain for each turns out to be • gain(temperature) = 0.571 bits • gain(humidity) = 0.971 bits • gain(windy) = 0.020 bits Therefore, we select humidity as the splitting attribute at this point. There is no need to split these nodes any further, so this branch is finished. Continued application of the same idea leads to the decision tree of Figure 4.4 for the weather data. Ideally, the process terminates when all leaf nodes are pure—that is, when they contain instances that all have the same classification. However, it might not be possible to reach this happy situation because there is nothing to stop the train- ing set containing two examples with identical sets of attributes but different classes. Consequently, we stop when the data cannot be split any further. Alternatively, one could stop if the information gain is zero. This is slightly more conservative because FIGURE 4.4 Decision tree for the weather data. outlook humidity sunny yes overcast windy rainy no high yes normal yes false no true it is possible to encounter cases where the data can be split into subsets exhibiting identical class distributions, which would make the information gain zero. Calculating Information Now it is time to explain how to calculate the information measure that is used as the basis for evaluating different splits. We describe the basic idea in this section, then in the next we examine a correction that is usually made to counter a bias toward selecting splits on attributes with large numbers of possible values. Before examining the detailed formula for calculating the amount of information required to specify the class of an example given that it reaches a tree node with a certain number of yes’s and no’s, consider first the kind of properties we would expect this quantity to have 1. When the number of either yes’s or no’s is zero, the information is zero. 2. When the number of yes’s and no’s is equal, the information reaches a maximum. Moreover, the measure should be applicable to multiclass situations, not just to two- class ones. The information measure relates to the amount of information obtained by making a decision, and a more subtle property of information can be derived by considering the nature of decisions. Decisions can be made in a single stage, or they can be made in several stages, and the amount of information involved is the same in both cases. For example, the decision involved in info([ , , ])23 4 can be made in two stages. First decide whether it’s the first case or one of the other two cases: info([ , ])27 4.3 Divide-and-Conquer: Constructing Decision Trees 103 104 CHAPTER 4 Algorithms: The Basic Methods and then decide which of the other two cases it is: info([ , ])34 In some cases the second decision will not need to be made, namely, when the deci- sion turns out to be the first one. Taking this into account leads to the equation info info info([ , , ]) ([ , ]) ( ) ([ , ])2 3 4 2 7 7 9 3 4= + × Of course, there is nothing special about these particular numbers, and a similar relationship should hold regardless of the actual values. Thus, we could add a further criterion to the list above: 3. The information should obey the multistage property that we have illustrated. Remarkably, it turns out that there is only one function that satisfies all these properties, and it is known as the information value or entropy: entropy( , , , ) log log logp p p p p p p p pn n n1 2 1 1 2 2… …= − − − The reason for the minus signs is that logarithms of the fractions p1, p2, … , pn are negative, so the entropy is actually positive. Usually the logarithms are expressed in base 2, and then the entropy is in units called bits—just the usual kind of bits used with computers. The arguments p1, p2, … of the entropy formula are expressed as fractions that add up to 1, so that, for example, info entropy([ , , ]) ( , , )2 3 4 2 9 3 9 4 9= Thus, the multistage decision property can be written in general as entropy( , , ) entropy( , ) ( ) entropy ,p q r p q r q r q q r r q r= + + + × + + where p + q + r = 1. Because of the way the log function works, you can calculate the information measure without having to work out the individual fractions: info([ , , ]) log log log [ log log l 234 29 2939 3949 49 2 2 3 3 4 = − × − × − × = − − − oog log ]49 9 9+ This is the way that the information measure is usually calculated in practice. So the information value for the first node of Figure 4.2(a) is info bits([ , ]) log log .2 3 2 5 2 5 3 5 3 5 0 971= − × − × = Highly Branching Attributes When some attributes have a large number of possible values, giving rise to a mul- tiway branch with many child nodes, a problem arises with the information gain calculation. The problem can best be appreciated in the extreme case when an attri- bute has a different value for each instance in the dataset—as, for example, an identification code attribute might. Table 4.6 gives the weather data with this extra attribute. Branching on ID code produces the tree stump in Figure 4.5. The information required to specify the class given the value of this attribute is info info info info info([ , ]) ([ , ]) ([ , ]) ([ , ]) ([ ,0 1 0 1 1 0 1 0 0+ + + + +… 11]) which is 0 because each of the 14 terms is 0. This is not surprising: The ID code attribute identifies the instance, which determines the class without any ambiguity— just as Table 4.6 shows. Consequently, the information gain of this attribute is just the information at the root, info([9,5]) = 0.940 bits. This is greater than the informa- tion gain of any other attribute, and so ID code will inevitably be chosen as the splitting attribute. But branching on the identification code is no good for predicting the class of unknown instances and tells nothing about the structure of the decision, which after all are the twin goals of machine learning. The overall effect is that the information gain measure tends to prefer attributes with large numbers of possible values. To compensate for this, a modification of the measure called the gain ratio is widely used. The gain ratio is derived by taking into account the number and size of daughter nodes into which an attribute splits the dataset, disregarding any information about the class. In the situation shown in Figure 4.5, all counts have a value of 1, so the information value of the split is info([ , , , ]) log1 1 1 1 14 1 14 14… = − × × because the same fraction, 1/14, appears 14 times. This amounts to log 14, or 3.807 bits, which is a very high value. This is because the information value of a split is FIGURE 4.5 Tree stump for the ID code attribute. ID code no a no b yes c ... yes m no n 4.3 Divide-and-Conquer: Constructing Decision Trees 105 106 Table 4.6 Weather Data with Identification Codes ID code Outlook Temperature Humidity Windy Play a sunny hot high false no b sunny hot high true no c overcast hot high false yes d rainy mild high false yes e rainy cool normal false yes f rainy cool normal true no g overcast cool normal true yes h sunny mild high false no i sunny cool normal false yes j rainy mild normal false yes k sunny mild normal true yes l overcast mild high true yes m overcast hot normal false yes n rainy mild high true no Table 4.7 Gain Ratio Calculations for Figure 4.2 Tree Stumps Outlook Temperature Humidity Windy info: 0.693 info: 0.911 info: 0.788 info: 0.892 gain: 0.940–0.693 0.247 gain: 0.940–0.911 0.029 gain: 0.940–0.788 0.152 gain: 0.940–0.892 0.048 split info: info([5,4,5]) 1.577 split info: info([4,6,4]) 1.362 split info: info([7,7]) 1.000 split info: info([8,6]) 0.985 gain ratio: 0.247/1.577 0.156 gain ratio: 0.029/1.557 0.019 gain ratio: 0.152/1 0.152 gain ratio: 0.048/0.985 0.049 the number of bits needed to determine to which branch each instance is assigned, and the more branches there are, the greater this value. The gain ratio is calculated by dividing the original information gain, 0.940 in this case, by the information value of the attribute, 3.807—yielding a gain ratio value of 0.247 for the ID code attribute. Returning to the tree stumps for the weather data in Figure 4.2, outlook splits the dataset into three subsets of size 5, 4, and 5, and thus has an intrinsic information value of info([ , , ]) .54 5 1 577= without paying any attention to the classes involved in the subsets. As we have seen, this intrinsic information value is greater for a more highly branching attribute such as the hypothesized ID code. Again, we can correct the information gain by dividing by the intrinsic information value to get the gain ratio. The results of these calculations for the tree stumps of Figure 4.2 are summarized in Table 4.7. Outlook still comes out on top, but humidity is now a much closer contender because it splits the data into two subsets instead of three. In this particular example, the hypothetical ID code attribute, with a gain ratio of 0.247, would still be preferred to any of these four. However, its advantage is greatly reduced. In practical implementations, we can use an ad hoc test to guard against splitting on such a useless attribute. Unfortunately, in some situations the gain ratio modification overcompensates and can lead to preferring an attribute just because its intrinsic information is much lower than for the other attributes. A standard fix is to choose the attri- bute that maximizes the gain ratio, provided that the information gain for that attribute is at least as great as the average information gain for all the attributes examined. Discussion The divide-and-conquer approach to decision tree induction, sometimes called top- down induction of decision trees, was developed and refined over many years by J. Ross Quinlan at the University of Sydney in Australia. Although others have 4.3 Divide-and-Conquer: Constructing Decision Trees 107 108 CHAPTER 4 Algorithms: The Basic Methods worked on similar methods, Quinlan’s research has always been at the very forefront of decision tree induction. The scheme that has been described using the information gain criterion is essentially the same as one known as ID3. The use of the gain ratio was one of many improvements that were made to ID3 over several years; Quinlan described it as robust under a wide variety of circumstances. Although a practical solution, it sacrifices some of the elegance and clean theoretical motivation of the information gain criterion. A series of improvements to ID3 culminated in a practical and influential system for decision tree induction called C4.5. These improvements include methods for dealing with numeric attributes, missing values, noisy data, and generating rules from trees, and they are described in Section 6.1. 4.4 COVERING ALGORITHMS: CONSTRUCTING RULES As we have seen, decision tree algorithms are based on a divide-and-conquer approach to the classification problem. They work top-down, seeking at each stage an attribute to split on that best separates the classes, and then recursively processing the subproblems that result from the split. This strategy generates a decision tree, which can if necessary be converted into a set of classification rules—although if it is to produce effective rules, the conversion is not trivial. An alternative approach is to take each class in turn and seek a way of covering all instances in it, at the same time excluding instances not in the class. This is called a covering approach because at each stage you identify a rule that “covers” some of the instances. By its very nature, this covering approach leads to a set of rules rather than to a decision tree. The covering method can readily be visualized in a two-dimensional space of instances as shown in Figure 4.6(a). We first make a rule covering the a’s. For the first test in the rule, split the space vertically as shown in the center picture. This gives the beginnings of a rule: If x > 1.2 then class = a However, the rule covers many b’s as well as a’s, so a new test is added to it by further splitting the space horizontally as shown in the third diagram: If x > 1.2 and y > 2.6 then class = a This gives a rule covering all but one of the a’s. It’s probably appropriate to leave it at that, but if it were felt necessary to cover the final a, another rule would be needed, perhaps If x > 1.4 and y < 2.4 then class = a The same procedure leads to two rules covering the b’s: If x ≤ 1.2 then class = b If x > 1.2 and y ≤ 2.6 then class = b 4.4 Covering Algorithms: Constructing Rules 109 FIGURE 4.6 Covering algorithm: (a) covering the instances, and (b) decision tree for the same problem. y x x x a b b b b b bb b b b b b b b a a aa a y a b b b b b bb b b b b b b b a a aa a 1.2 y a b b b b b b b b b b bb b b a a aa a 1.2 2.6 (a) x > 1.2? b no y > 2.6? yes b no a yes (b) Again, one a is erroneously covered by these rules. If it were necessary to exclude it, more tests would have to be added to the second rule, and additional rules would be needed to cover the b’s that these new tests exclude. Rules versus Trees A top-down divide-and-conquer algorithm operates on the same data in a manner that is, at least superficially, quite similar to a covering algorithm. It might first split the dataset using the x attribute, and would probably end up splitting it at the same place, x = 1.2. However, whereas the covering algorithm is concerned only with covering a single class, the division would take both classes into account because divide-and-conquer algorithms create a single concept description that applies to all classes. The second split might also be at the same place, y = 2.6, leading to the decision tree in Figure 4.6(b). This tree corresponds exactly to the set of rules, and in this case there is no difference in effect between the covering and the divide-and-conquer algorithms. But in many situations there is a difference between rules and trees in terms of the perspicuity of the representation. For example, when we described the replicated subtree problem in Section 3.4, we noted that rules can be symmetric whereas trees must select one attribute to split on first, and this can lead to trees that are much 110 CHAPTER 4 Algorithms: The Basic Methods FIGURE 4.7 The instance space during operation of a covering algorithm. Space of examples Rule so far Rule after adding new term larger than an equivalent set of rules. Another difference is that, in the multiclass case, a decision tree split takes all classes into account in trying to maximize the purity of the split, whereas the rule-generating method concentrates on one class at a time, disregarding what happens to the other classes. A Simple Covering Algorithm Covering algorithms operate by adding tests to the rule that is under construction, always striving to create a rule with maximum accuracy. In contrast, divide-and-con- quer algorithms operate by adding tests to the tree that is under construction, always striving to maximize the separation between the classes. Each of these involves finding an attribute to split on. But the criterion for the best attribute is different in each case. Whereas divide-and-conquer algorithms such as ID3 choose an attribute to maximize the information gain, the covering algorithm we will describe chooses an attribute–value pair to maximize the probability of the desired classification. Figure 4.7 gives a picture of the situation, showing the space containing all the instances, a partially constructed rule, and the same rule after a new term has been added. The new term restricts the coverage of the rule: The idea is to include as many instances of the desired class as possible and exclude as many instances of other classes as possible. Suppose the new rule will cover a total of t instances, of which p are positive examples of the class and t – p are in other classes—that is, they are errors made by the rule. Then choose the new term to maximize the ratio p/t. An example will help. For a change, we use the contact lens problem of Table 1.1 (page 6). We will form rules that cover each of the three classes—hard, soft, and none—in turn. To begin, we seek a rule: If ? then recommendation = hard For the unknown term ?, we have nine choices: age = young 2/8 age = pre-presbyopic 1/8 age = presbyopic 1/8 spectacle prescription = myope 3/12 4.4 Covering Algorithms: Constructing Rules 111 spectacle prescription = hypermetrope 1/12 astigmatism = no 0/12 astigmatism = yes 4/12 tear production rate = reduced 0/12 tear production rate = normal 4/12 The numbers on the right show the fraction of “correct” instances in the set singled out by that choice. In this case, “correct” means that the recommendation is hard. For instance, age = young selects 8 instances, 2 of which recommend hard contact lenses, so the first fraction is 2/8. (To follow this, you will need to look back at the contact lens data in Table 1.1 (page 6) and count up the entries in the table.) We select the largest fraction, 4/12, arbitrarily choosing between the seventh and the last choice in the list, and create the rule: If astigmatism = yes then recommendation = hard This rule is quite inaccurate, getting only 4 instances correct out of the 12 that it covers, shown in Table 4.8. So we refine it further: If astigmatism = yes and ? then recommendation = hard Considering the possibilities for the unknown term, ? yields the following seven choices: age = young 2/4 age = pre-presbyopic 1/4 age = presbyopic 1/4 spectacle prescription = myope 3/6 spectacle prescription = hypermetrope 1/6 tear production rate = reduced 0/6 tear production rate = normal 4/6 (Again, count the entries in Table 4.8.) The last is a clear winner, getting 4 instances correct out of the 6 that it covers, and it corresponds to the rule If astigmatism = yes and tear production rate = normal then recommendation = hard Should we stop here? Perhaps. But let’s say we are going for exact rules, no matter how complex they become. Table 4.9 shows the cases that are covered by the rule so far. The possibilities for the next term are now age = young 2/2 age = pre-presbyopic 1/2 age = presbyopic 1/2 spectacle prescription = myope 3/3 spectacle prescription = hypermetrope 1/3 It is necessary for us to choose between the first and fourth. So far we have treated the fractions numerically, but although these two are equal (both evaluate to 1), they have different coverage: One selects just two correct instances and the other selects 112 Table 4.8 Part of Contact Lens Data for Which Astigmatism = yes Age Spectacle Prescription Astigmatism Tear Production Rate Recommended Lenses young myope yes reduced none young myope yes normal hard young hypermetrope yes reduced none young hypermetrope yes normal hard pre-presbyopic myope yes reduced none pre-presbyopic myope yes normal hard pre-presbyopic hypermetrope yes reduced none pre-presbyopic hypermetrope yes normal none presbyopic myope yes reduced none presbyopic myope yes normal hard presbyopic hypermetrope yes reduced none presbyopic hypermetrope yes normal none 113 Table 4.9 Part of Contact Lens Data for Which Astigmatism = yes and Tear Production rate = normal Age Spectacle Prescription Astigmatism Tear Production Rate Recommended Lenses young myope yes normal hard young hypermetrope yes normal hard pre-presbyopic myope yes normal hard pre-presbyopic hypermetrope yes normal none presbyopic myope yes normal hard presbyopic hypermetrope yes normal none 114 CHAPTER 4 Algorithms: The Basic Methods FIGURE 4.8 Pseudocode for a basic rule learner. For each class C Initialize E to the instance set While E contains instances in class C Create a rule R with an empty left-hand side that predicts class C Until R is perfect (or there are no more attributes to use) do For each attribute A not mentioned in R, and each value v, Consider adding the condition A = v to the LHS of R Select A and v to maximize the accuracy p/t (break ties by choosing the condition with the largest p) Add A = v to R Remove the instances covered by R from E three. In the event of a tie, we choose the rule with the greater coverage, giving the final rule: If astigmatism = yes and tear production rate = normal and spectacle prescription = myope then recommendation = hard This is indeed one of the rules given for the contact lens problem. But it only covers three out of the four hard recommendations. So we delete these three from the set of instances and start again, looking for another rule of the form: If ? then recommendation = hard Following the same process, we will eventually find that age = young is the best choice for the first term. Its coverage is one out of 7 the reason for the 7 is that 3 instances have been removed from the original set, leaving 21 instances altogether. The best choice for the second term is astigmatism = yes, selecting 1/3 (actually, this is a tie); tear production rate = normal is the best for the third, selecting 1/1. If age = young and astigmatism = yes and tear production rate = normal then recommendation = hard This rule actually covers two of the original set of instances, one of which is covered by the previous rule—but that’s all right because the recommendation is the same for each rule. Now that all the hard-lens cases are covered, the next step is to proceed with the soft-lens ones in just the same way. Finally, rules are generated for the none case— unless we are seeking a rule set with a default rule, in which case explicit rules for the final outcome are unnecessary. What we have just described is the PRISM method for constructing rules. It generates only correct or “perfect” rules. It measures the success of a rule by the accuracy formula p/t. Any rule with accuracy less than 100% is “incorrect” in that 4.4 Covering Algorithms: Constructing Rules 115 it assigns cases to the class in question that actually do not have that class. PRISM continues adding clauses to each rule until it is perfect—its accuracy is 100%. Figure 4.8 gives a summary of the algorithm. The outer loop iterates over the classes, gen- erating rules for each class in turn. Note that we reinitialize to the full set of examples each time around. Then we create rules for that class and remove the examples from the set until there are none of that class left. Whenever we create a rule, we start with an empty rule (which covers all the examples), and then restrict it by adding tests until it covers only examples of the desired class. At each stage we choose the most promising test—that is, the one that maximizes the accuracy of the rule. Finally, we break ties by selecting the test with greatest coverage. Rules versus Decision Lists Consider the rules produced for a particular class—that is, the algorithm in Figure 4.8 with the outer loop removed. It seems clear from the way that these rules are produced that they are intended to be interpreted in order—that is, as a decision list—testing the rules in turn until one applies and then using that. This is because the instances covered by a new rule are removed from the instance set as soon as the rule is com- pleted (in the last line of the code in Figure 4.8): Thus, subsequent rules are designed for instances that are not covered by the rule. However, although it appears that we are supposed to check the rules in turn, we do not have to do so. Consider that any subsequent rules generated for this class will have the same effect—they all predict the same class. This means that it does not matter what order they are executed in: Either a rule will be found that covers this instance, in which case the class in question is predicted, or no such rule is found, in which case the class is not predicted. Now return to the overall algorithm. Each class is considered in turn, and rules are generated that distinguish instances in that class from the others. No ordering is implied between the rules for one class and those for another. Consequently, the rules that are produced can be executed in any order. As described in Section 3.4, order-independent rules seem to provide more modularity by acting as independent nuggets of “knowledge,” but they suffer from the disadvantage that it is not clear what to do when conflicting rules apply. With rules generated in this way, a test example may receive multiple classifications—that is, it may satisfy rules that apply to different classes. Other test examples may receive no classification at all. A simple strategy to force a decision in ambiguous cases is to choose, from the classifications that are predicted, the one with the most training examples or, if no classification is predicted, to choose the category with the most training examples overall. These difficulties do not occur with decision lists because they are meant to be interpreted in order and execution stops as soon as one rule applies: The addition of a default rule at the end ensures that any test instance receives a classification. It is possible to generate good decision lists for the multi- class case using a slightly different method, as we will see in Section 6.2. Methods, such as PRISM, can be described as separate-and-conquer algorithms: You identify a rule that covers many instances in the class (and excludes ones not in the class), separate out the covered instances because they are already taken care of 116 CHAPTER 4 Algorithms: The Basic Methods by the rule, and continue with the process on those that remain. This contrasts with the divide-and-conquer approach of decision trees. The “separate” step results in an efficient method because the instance set continually shrinks as the operation proceeds. 4.5 MINING ASSOCIATION RULES Association rules are like classification rules. You could find them in the same way, by executing a divide-and-conquer rule-induction procedure for each possible expression that could occur on the right side of the rule. However, not only might any attribute occur on the right side with any possible value, but a single association rule often predicts the value of more than one attribute. To find such rules, you would have to execute the rule-induction procedure once for every possible combination of attributes, with every possible combination of values, on the right side. That would result in an enormous number of association rules, which would then have to be pruned down on the basis of their coverage (the number of instances that they predict correctly) and their accuracy (the same number expressed as a proportion of the number of instances to which the rule applies). This approach is quite infeasible. (Note that, as we mentioned in Section 3.4, what we are calling coverage is often called support and what we are calling accuracy is often called confidence.) Instead, we capitalize on the fact that we are only interested in association rules with high coverage. We ignore, for the moment, the distinction between the left and right sides of a rule and seek combinations of attribute–value pairs that have a prespecified minimum coverage. These are called item sets: An attribute–value pair is an item. The terminology derives from market basket analysis, in which the items are articles in your shopping cart and the supermarket manager is looking for associations among these purchases. Item Sets The first column of Table 4.10 shows the individual items for the weather data in Table 1.2 (page 10), with the number of times each item appears in the dataset given at the right. These are the one-item sets. The next step is to generate the two-item sets by making pairs of the one-item sets. Of course, there is no point in generating a set containing two different values of the same attribute (such as outlook = sunny and outlook = overcast) because that cannot occur in any actual instance. Assume that we seek association rules with minimum coverage 2; thus, we discard any item sets that cover fewer than two instances. This leaves 47 two-item sets, some of which are shown in the second column along with the number of times they appear. The next step is to generate the three-item sets, of which 39 have a coverage of 2 or greater. There are six four-item sets, and no five-item sets—for this data, a five-item set with coverage 2 or greater could only correspond to a repeated instance. The first rows of the table, for example, show that there are five days when outlook = sunny, two of which have temperature = hot, and, in fact, on both of those days humidity = high and play = no as well. 117 Table 4.10 Item Sets for Weather Data with Coverage 2 or Greater One-Item Sets Two-Item Sets Three-Item Sets Four-Item Sets 1 outlook = sunny 5 outlook = sunny temperature = mild 2 outlook = sunny temperature = hot humidity = high 2 outlook = sunny temperature = hot humidity = high play = no 2 2 outlook = overcast 4 outlook = sunny temperature = hot 2 outlook = sunny temperature = hot play = no 2 outlook = sunny humidity = high windy = false play = no 2 3 outlook = rainy 5 outlook = sunny humidity = normal 2 outlook = sunny humidity = normal play = yes 2 outlook = overcast temperature = hot windy = false play = yes 2 4 temperature = cool 4 outlook = sunny humidity = high 3 outlook = sunny humidity = high windy = false 2 outlook = rainy temperature = mild windy = false play = yes 2 5 temperature = mild 6 outlook = sunny windy = true 2 outlook = sunny humidity = high play = no 3 outlook = rainy humidity = normal windy = false play = yes 2 6 temperature = hot 4 outlook = sunny windy = false 3 outlook = sunny windy = false play = no 2 temperature = cool humidity = normal windy = false play = yes 2 7 humidity = normal 7 outlook = sunny play = yes 2 outlook = overcast temperature = hot windy = false 2 8 humidity = high 7 outlook = sunny play = no 3 outlook = overcast temperature = hot play = yes 2 ContinuedContinued 118 One-Item Sets Two-Item Sets Three-Item Sets Four-Item Sets 9 windy = true 6 outlook = overcast temperature = hot 2 outlook = overcast humidity = normal play = yes 2 10 windy = false 8 outlook = overcast humidity = normal 2 outlook = overcast humidity = high play = yes 2 11 play = yes 9 outlook = overcast humidity = high 2 outlook = overcast windy = true play = yes 2 12 play = no 5 outlook = overcast windy = true 2 outlook = overcast windy = false play = yes 2 13 outlook = overcast windy = false 2 outlook = rainy temperature = cool humidity = normal 2 … … … 38 humidity = normal windy = false 4 humidity = normal windy = false play = yes 4 39 humidity = normal play = yes 6 humidity = high windy = false play = no 2 40 humidity = high windy = true 3 … … 47 windy = false play = no 2 Table 4.10 Item Sets for Weather Data with Coverage or Greater (Continued) 4.5 Mining Association Rules 119 Association Rules Shortly we will explain how to generate these item sets efficiently. But first let us finish the story. Once all item sets with the required coverage have been generated, the next step is to turn each into a rule, or a set of rules, with at least the specified minimum accuracy. Some item sets will produce more than one rule; others will produce none. For example, there is one three-item set with a coverage of 4 (row 38 of Table 4.10): humidity = normal, windy = false, play = yes This set leads to seven potential rules: If humidity = normal and windy = false then play = yes 4/4 If humidity = normal and play = yes then windy = false 4/6 If windy = false and play = yes then humidity = normal 4/6 If humidity = normal then windy = false and play = yes 4/7 If windy = false then humidity = normal and play = yes 4/8 If play = yes then humidity = normal and windy = false 4/9 If – then humidity = normal and windy = false and play = yes 4/14 The figures at the right in this list show the number of instances for which all three conditions are true—that is, the coverage—divided by the number of instances for which the conditions in the antecedent are true. Interpreted as a fraction, they represent the proportion of instances on which the rule is correct—that is, its accuracy. Assuming that the minimum specified accuracy is 100%, only the first of these rules will make it into the final rule set. The denominators of the fractions are readily obtained by looking up the antecedent expression in Table 4.10 (although some are not shown in the table). The final rule above has no conditions in the antecedent, and its denominator is the total number of instances in the dataset. Table 4.11 shows the final rule set for the weather data, with minimum cover- age 2 and minimum accuracy 100%, sorted by coverage. There are 58 rules, 3 with coverage 4, 5 with coverage 3, and 50 with coverage 2. Only 7 have two conditions in the consequent, and none has more than two. The first rule comes from the item set described previously. Sometimes several rules arise from the same item set. For example, rules 9, 10, and 11 all arise from the four-item set in row 6 of Table 4.10: temperature = cool, humidity = normal, windy = false, play = yes which has coverage 2. Three subsets of this item set also have coverage 2: temperature = cool, windy = false temperature = cool, humidity = normal, windy = false temperature = cool, windy = false, play = yes and these lead to rules 9, 10, and 11, all of which are 100% accurate (on the training data). 120 CHAPTER 4 Algorithms: The Basic Methods Table 4.11 Association Rules for Weather Data Association Rule Coverage Accuracy 1 humidity = normal windy = false ⇒ play = yes 4 100% 2 temperature = cool ⇒ humidity = normal 4 100% 3 outlook = overcast ⇒ play = yes 4 100% 4 temperature = cool play = yes ⇒ humidity = normal 3 100% 5 outlook = rainy windy = false ⇒ play = yes 3 100% 6 outlook = rainy play = yes ⇒ windy = false 3 100% 7 outlook = sunny humidity = high ⇒ play = no 3 100% 8 outlook = sunny play = no ⇒ humidity = high 3 100% 9 temperature = cool windy = false ⇒ humidity = normal play = yes 2 100% 10 temperature = cool humidity = normal windy = false ⇒ play = yes 2 100% 11 temperature = cool windy = false play = yes ⇒ humidity = normal 2 100% 12 outlook = rainy humidity = normal windy = false ⇒ play = yes 2 100% 13 outlook = rainy humidity = normal play = yes ⇒ windy = false 2 100% 14 outlook = rainy temperature = mild windy = false ⇒ play = yes 2 100% 15 outlook = rainy temperature = mild play = yes ⇒ windy = false 2 100% 16 temperature = mild windy = false play = yes ⇒ outlook = rainy 2 100% 17 outlook = overcast temperature = hot ⇒ windy = false play = yes 2 100% 18 outlook = overcast windy = false ⇒ temperature = hot play = yes 2 100% 4.5 Mining Association Rules 121 Association Rule Coverage Accuracy 19 temperature = hot play = yes ⇒ outlook = overcast windy = false 2 100% 20 outlook = overcast temperature = hot windy = false ⇒ play = yes 2 100% 21 outlook = overcast temperature = hot play = yes ⇒ windy = false 2 100% 22 outlook = overcast windy = false play = yes ⇒ temperature = hot 2 100% 23 temperature = hot windy = false play = yes ⇒ outlook = overcast 2 100% 24 windy = false play = no ⇒ outlook = sunny humidity = high 2 100% 25 outlook = sunny humidity = high windy = false ⇒ play = no 2 100% 26 outlook = sunny windy = false play = no ⇒ humidity = high 2 100% 27 humidity = high windy = false play = no ⇒ outlook = sunny 2 100% 28 outlook = sunny temperature = hot ⇒ humidity = high play = no 2 100% 29 temperature = hot play = no ⇒ outlook = sunny humidity = high 2 100% 30 outlook = sunny temperature = hot humidity = high ⇒ play = no 2 100% 31 outlook = sunny temperature = hot play = no ⇒ humidity = high 2 100% … … … … 58 outlook = sunny temperature = hot ⇒ humidity = high 2 100% Table 4.11 Continued 122 CHAPTER 4 Algorithms: The Basic Methods Generating Rules Efficiently We now consider in more detail an algorithm for producing association rules with specified minimum coverage and accuracy. There are two stages: generating item sets with the specified minimum coverage, and from each item set determining the rules that have the specified minimum accuracy. The first stage proceeds by generating all one-item sets with the given minimum coverage (the first column of Table 4.10) and then using this to generate the two-item sets (second column), three-item sets (third column), and so on. Each operation involves a pass through the dataset to count the items in each set, and after the pass the surviving item sets are stored in a hash table—a standard data structure that allows elements stored in it to be found very quickly. From the one-item sets, can- didate two-item sets are generated, and then a pass is made through the dataset, counting the coverage of each two-item set; at the end the candidate sets with less than minimum coverage are removed from the table. The candidate two-item sets are simply all of the one-item sets taken in pairs, because a two-item set cannot have the minimum coverage unless both its constituent one-item sets have the minimum coverage, too. This applies in general: A three-item set can only have the minimum coverage if all three of its two-item subsets have minimum coverage as well, and similarly for four-item sets. An example will help to explain how candidate item sets are generated. Suppose there are five three-item sets—(A B C), (A B D), (A C D), (A C E), and (B C D)— where, for example, A is a feature such as outlook = sunny. The union of the first two, (A B C D), is a candidate four-item set because its other three-item subsets (A C D) and (B C D) have greater than minimum coverage. If the three-item sets are sorted into lexical order, as they are in this list, then we need only consider pairs with the same first two members. For example, we do not consider (A C D) and (B C D) because (A B C D) can also be generated from (A B C) and (A B D), and if these two are not candidate three-item sets, then (A B C D) cannot be a candidate four-item set. This leaves the pairs (A B C) and (A B D), which we have already explained, and (A C D) and (A C E). This second pair leads to the set (A C D E) whose three-item subsets do not all have the minimum coverage, so it is discarded. The hash table assists with this check: We simply remove each item from the set in turn and check that the remaining three-item set is indeed present in the hash table. Thus, in this example there is only one candidate four-item set, (A B C D). Whether or not it actually has minimum coverage can only be determined by checking the instances in the dataset. The second stage of the procedure takes each item set and generates rules from it, checking that they have the specified minimum accuracy. If only rules with a single test on the right side were sought, it would be simply a matter of considering each condition in turn as the consequent of the rule, deleting it from the item set, and dividing the coverage of the entire item set by the coverage of the resulting subset—obtained from the hash table—to yield the accuracy of the corresponding rule. Given that we are also interested in association rules with multiple tests in the 4.5 Mining Association Rules 123 consequent, it looks like we have to evaluate the effect of placing each subset of the item set on the right side, leaving the remainder of the set as the antecedent. This brute-force method will be excessively computation intensive unless item sets are small, because the number of possible subsets grows exponentially with the size of the item set. However, there is a better way. We observed when describing association rules in Section 3.4 that if the double-consequent rule If windy = false and play = no then outlook = sunny and humidity = high holds with a given minimum coverage and accuracy, then both single-consequent rules formed from the same item set must also hold: If humidity = high and windy = false and play = no then outlook = sunny If outlook = sunny and windy = false and play = no then humidity = high Conversely, if one or other of the single-consequent rules does not hold, there is no point in considering the double-consequent one. This gives a way of building up from single-consequent rules to candidate double-consequent ones, from double- consequent rules to candidate triple-consequent ones, and so on. Of course, each candidate rule must be checked against the hash table to see if it really does have more than the specified minimum accuracy. But this generally involves checking far fewer rules than the brute-force method. It is interesting that this way of building up candidate (n + 1)-consequent rules from actual n-consequent ones is really just the same as building up candidate (n + 1)-item sets from actual n-item sets, described earlier. Discussion Association rules are often sought for very large datasets, and efficient algorithms are highly valued. The method we have described makes one pass through the dataset for each different size of item set. Sometimes the dataset is too large to read in to main memory and must be kept on disk; then it may be worth reducing the number of passes by checking item sets of two consecutive sizes at the same time. For example, once sets with two items have been generated, all sets of three items could be generated from them before going through the instance set to count the actual number of items in the sets. More three-item sets than necessary would be considered, but the number of passes through the entire dataset would be reduced. In practice, the amount of computation needed to generate association rules depends critically on the minimum coverage specified. The accuracy has less influ- ence because it does not affect the number of passes that must be made through the dataset. In many situations we would like to obtain a certain number of rules—say 50—with the greatest possible coverage at a prespecified minimum accuracy level. One way to do this is to begin by specifying the coverage to be rather high and to 124 CHAPTER 4 Algorithms: The Basic Methods then successively reduce it, reexecuting the entire rule-finding algorithm for each of the coverage values and repeating until the desired number of rules has been generated. The tabular input format that we use throughout this book, and in particular the standard ARFF format based on it, is very inefficient for many association-rule problems. Association rules are often used in situations where attributes are binary— either present or absent—and most of the attribute values associated with a given instance are absent. This is a case for the sparse data representation described in Section 2.4; the same algorithm for finding association rules applies. 4.6 LINEAR MODELS The methods we have been looking at for decision trees and rules work most natu- rally with nominal attributes. They can be extended to numeric attributes either by incorporating numeric-value tests directly into the decision tree or rule-induction scheme, or by prediscretizing numeric attributes into nominal ones. We will see how in Chapters 6 and 7, respectively. However, there are methods that work most natu- rally with numeric attributes, namely the linear models introduced in Section 3.2; we examine them in more detail here. They can form components of more complex learning methods, which we will investigate later. Numeric Prediction: Linear Regression When the outcome, or class, is numeric, and all the attributes are numeric, linear regression is a natural technique to consider. This is a staple method in statistics. The idea is to express the class as a linear combination of the attributes, with pre- determined weights: x w w a w a w ak k= + + + +0 1 1 2 2 … where x is the class; a1, a2, …, ak are the attribute values; and w0, w1, …, wk are weights. The weights are calculated from the training data. Here the notation gets a little heavy, because we need a way of expressing the attribute values for each training instance. The first instance will have a class, say x(1), and attribute values a1 (1), a2 (1), … , ak (1), where the superscript denotes that it is the first example. Moreover, it is notationally convenient to assume an extra attribute a0, with a value that is always 1. The predicted value for the first instance’s class can be written as w a w a w a w a w ak k j j j k 0 0 1 1 1 1 2 2 1 1 1 0 ()()()()()+ + + + = = ∑… This is the predicted, not the actual, value for the class. Of interest is the difference between the predicted and actual values. The method of linear regression is to choose the 4.6 Linear Models 125 coefficients wj —there are k + 1 of them—to minimize the sum of the squares of these differences over all the training instances. Suppose there are n training instances; denote the ith one with a superscript (i). Then the sum of the squares of the differences is x w ai j j i j k i n ()()− == ∑∑ 01 2 where the expression inside the parentheses is the difference between the ith instance’s actual class and its predicted class. This sum of squares is what we have to minimize by choosing the coefficients appropriately. This is all starting to look rather formidable. However, the minimization technique is straightforward if you have the appropriate math background. Suffice it to say that given enough examples—roughly speaking, more examples than attributes—choosing weights to minimize the sum of the squared differences is really not difficult. It does involve a matrix inversion operation, but this is readily available as prepackaged software. Once the math has been accomplished, the result is a set of numeric weights, based on the training data, which can be used to predict the class of new instances. We saw an example of this when looking at the CPU performance data, and the actual numeric weights are given in Figure 3.4(a) (page 68). This formula can be used to predict the CPU performance of new test instances. Linear regression is an excellent, simple method for numeric prediction, and it has been widely used in statistical applications for decades. Of course, linear models suffer from the disadvantage of, well, linearity. If the data exhibits a nonlinear dependency, the best-fitting straight line will be found, where “best” is interpreted as the least mean-squared difference. This line may not fit very well. However, linear models serve well as building blocks for more complex learning methods. Linear Classification: Logistic Regression Linear regression can easily be used for classification in domains with numeric attributes. Indeed, we can use any regression technique, whether linear or nonlinear, for classification. The trick is to perform a regression for each class, setting the output equal to 1 for training instances that belong to the class and 0 for those that do not. The result is a linear expression for the class. Then, given a test example of unknown class, calculate the value of each linear expression and choose the one that is largest. This scheme is sometimes called multiresponse linear regression. One way of looking at multiresponse linear regression is to imagine that it approximates a numeric membership function for each class. The membership func- tion is 1 for instances that belong to that class and 0 for other instances. Given a new instance, we calculate its membership for each class and select the biggest. Multiresponse linear regression often yields good results in practice. However, it has two drawbacks. First, the membership values it produces are not proper prob- abilities because they can fall outside the range 0 to 1. Second, least-squares regres- sion assumes that the errors are not only statistically independent but are also 126 CHAPTER 4 Algorithms: The Basic Methods Suppose first that there are only two classes. Logistic regression replaces the original target variable Pr[ | , , , ]1 1 2a a ak… which cannot be approximated accurately using a linear function, by log[Pr[ | , , , ] ( Pr[ | , , , ])]1 1 11 2 1 2a a a a a ak k… …− The resulting values are no longer constrained to the interval from 0 to 1 but can lie anywhere between negative infinity and positive infinity. Figure 4.9(a) plots the transformation function, which is often called the logit transformation. The transformed variable is approximated using a linear function just like the ones generated by linear regression. The resulting model is Pr[ | , , , ] ( exp( ))1 1 11 2 0 1 1a a a w w a w ak… …= + − − − − k k with weights w. Figure 4.9(b) shows an example of this function in one dimension, with two weights w0 = –1.25 and w1 = 0.5. Just as in linear regression, weights must be found that fit the training data well. Linear regression measures goodness of fit using the squared error. In logistic regression the log-likelihood of the model is used instead. This is given by ( )log( Pr[ | , , , ]) log(Pr[ |()()()()()(1 1 1 11 1 2 2 1 1− − +x a a a x ai k k i… )) ( ) ( ), , , ])a ak k i n 2 2 1 … = ∑ where the x(i) are either 0 or 1. The weights wi need to be chosen to maximize the log-likelihood. There are several methods for solving this maximization problem. A simple one is to iteratively solve a sequence of weighted least-squares regression problems until the log-likelihood converges to a maximum, which usually happens in a few iterations. To generalize logistic regression to several classes, one possibility is to proceed in the way described above for multiresponse linear regression by performing logistic regression independently for each class. Unfortunately, the resulting probability estimates will not sum to 1. To obtain proper probabilities it is necessary to couple the individual models for each class. This yields a joint optimization problem, and there are efficient solution methods for this. normally distributed with the same standard deviation, an assumption that is bla- tently violated when the method is applied to classification problems because the observations only ever take on the values 0 and 1. A related statistical technique called logistic regression does not suffer from these problems. Instead of approximating the 0 and 1 values directly, thereby risking illegitimate probability values when the target is overshot, logistic regression builds a linear model based on a transformed target variable. The use of linear functions for classification can easily be visualized in instance space. The decision boundary for two-class logistic regression lies where the predic- tion probability is 0.5—that is: Pr[ | , , , ] ( exp( )) .1 1 1 0 51 2 0 1 1a a a w w a w ak… …= + − − − − =k k 4.6 Linear Models 127 FIGURE 4.9 Logistic regression: (a) the logit transformation and (b) example logistic regression function. –5 –4 –3 –2 –1 0 1 2 3 4 5 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 –10 –5 0 5 10 (a) (b) This occurs when − − − − =w w a w a0 1 1 0… k k Because this is a linear equality in the attribute values, the boundary is a plane, or hyperplane, in instance space. It is easy to visualize sets of points that cannot be separated by a single hyperplane, and these cannot be discriminated correctly by logistic regression. Multiresponse linear regression suffers from the same problem. Each class receives a weight vector calculated from the training data. Focus for the moment on a particular pair of classes. Suppose the weight vector for class 1 is w w a w a w ak k0 1 1 1 1 2 1 2 1()()()()+ + +…+ and the same for class 2 with appropriate superscripts. Then an instance will be assigned to class 1 rather than class 2 if w w a w a w w a w ak k k k0 1 1 1 1 1 0 2 1 2 1 2()()()()()()+ + + > + + +… … In other words, it will be assigned to class 1 if ()()()()()()()()()w w w w a w w ak k k0 1 0 2 1 1 1 2 1 1 2 0− + − +…+ − > This is a linear inequality in the attribute values, so the boundary between each pair of classes is a hyperplane. Linear Classification Using the Perceptron Logistic regression attempts to produce accurate probability estimates by maximiz- ing the probability of the training data. Of course, accurate probability estimates 128 CHAPTER 4 Algorithms: The Basic Methods lead to accurate classifications. However, it is not necessary to perform probability estimation if the sole purpose of the model is to predict class labels. A different approach is to learn a hyperplane that separates the instances pertaining to the dif- ferent classes—let’s assume that there are only two of them. If the data can be sepa- rated perfectly into two groups using a hyperplane, it is said to be linearly separable. It turns out that if the data is linearly separable, there is a very simple algorithm for finding a separating hyperplane. The algorithm is called the perceptron learning rule. Before looking at it in detail, let’s examine the equation for a hyperplane again: w a w a w a w ak k0 0 1 1 2 2 0+ + +…+ = Here, a1, a2, … , ak are the attribute values, and w0, w1, … , wk are the weights that define the hyperplane. We will assume that each training instance a1, a2, … is extended by an additional attribute a0 that always has the value 1 (as we did in the case of linear regression). This extension, which is called the bias, just means that we don’t have to include an additional constant element in the sum. If the sum is greater than 0, we will predict the first class; otherwise, we will predict the second class. We want to find values for the weights so that the training data is correctly classified by the hyperplane. Figure 4.10(a) gives the perceptron learning rule for finding a separating hyper- plane. The algorithm iterates until a perfect solution has been found, but it will only work properly if a separating hyperplane exists—that is, if the data is linearly separable. Each iteration goes through all the training instances. If a misclassified instance is encountered, the parameters of the hyperplane are changed so that the misclassified instance moves closer to the hyperplane or maybe even across the hyperplane onto the correct side. If the instance belongs to the first class, this is done by adding its attribute values to the weight vector; otherwise, they are subtracted from it. To see why this works, consider the situation after an instance a pertaining to the first class has been added: ()()()()waawaawaa waak k k0 0 0 1 1 1 2 2 2+ + + + + + + +… This means that the output for a has increased by a a a a a a a ak k0 0 1 1 2 2× + × + × +…+ × This number is always positive. Thus, the hyperplane has moved in the correct direction for classifying instance a as positive. Conversely, if an instance belonging to the second class is misclassified, the output for that instance decreases after the modification, again moving the hyperplane in the correct direction. These corrections are incremental, and can interfere with earlier updates. However, it can be shown that the algorithm converges in a finite number of itera- tions if the data is linearly separable. Of course, if the data is not linearly separable, the algorithm will not terminate, so an upper bound needs to be imposed on the number of iterations when this method is applied in practice. 4.6 Linear Models 129 FIGURE 4.10 The perceptron: (a) learning rule, and (b) representation as a neural network. attribute a1 attribute a2 attribute ak 1 (“bias”) Set all weights to zero Until all instances in the training data are classified correctly For each instance I in the training data If I is classified incorrectly by the perceptron If I belongs to the first class add it to the weight vector else subtract it from the weight vector (a) (b) w0 w1 w2 wk The resulting hyperplane is called a perceptron, and it’s the grandfather of neural networks (we return to neural networks in Section 6.4). Figure 4.10(b) represents the perceptron as a graph with nodes and weighted edges, imaginatively termed a “network” of “neurons.” There are two layers of nodes: input and output. The input layer has one node for every attribute, plus an extra node that is always set to 1. The output layer consists of just one node. Every node in the input layer is connected to the output layer. The connections are weighted, and the weights are those numbers found by the perceptron learning rule. When an instance is presented to the perceptron, its attribute values serve to “activate” the input layer. They are multiplied by the weights and summed up at the output node. If the weighted sum is greater than 0 the output signal is 1, representing the first class; otherwise, it is –1, representing the second. Linear Classification Using Winnow The perceptron algorithm is not the only method that is guaranteed to find a sepa- rating hyperplane for a linearly separable problem. For datasets with binary attri- butes there is an alternative known as Winnow, which is illustrated in Figure 4.11(a). 130 CHAPTER 4 Algorithms: The Basic Methods a a using the current weights a belongs to the first class ai that is 1, multiply wi by α ai is 0, leave wi unchanged) ai that is 1, divide wi by α ai is 0, leave wi unchanged) a a using the current weights a belongs to the first class ai that is 1, wi + by α wi – by α ai is 0, leave wi + and wi - unchanged) wi – by α wi + by α While some instances are misclassified for every instance classify if the predicted class is incorrect if for each (if otherwise for each (if While some instances are misclassified for every instance classify if the predicted class is incorrect if for each multiply divide (if otherwise multiply divide (if ai is 0, leave wi + and wi - unchanged) (a) (b) FIGURE 4.11 The Winnow algorithm: (a) unbalanced version and (b) balanced version. The structure of the two algorithms is very similar. Like the perceptron, Winnow only updates the weight vector when a misclassified instance is encountered—it is mistake driven. The two methods differ in how the weights are updated. The perceptron rule employs an additive mechanism that alters the weight vector by adding (or subtract- ing) the instance’s attribute vector. Winnow employs multiplicative updates and alters weights individually by multiplying them by a user-specified parameter α (or its inverse). The attribute values ai are either 0 or 1 because we are working with binary data. Weights are unchanged if the attribute value is 0, because then they do not participate in the decision. Otherwise, the multiplier is α if that attribute helps to make a correct decision and 1/α if it does not. Another difference is that the threshold in the linear function is also a user- specified parameter. We call this threshold θ and classify an instance as belonging to class 1 if and only if w a w a w a w ak k0 0 1 1 2 2+ + + + >… θ 4.7 Instance-Based Learning 131 The multiplier α needs to be greater than 1, and the wi are set to a constant at the start. The algorithm we have described doesn’t allow for negative weights, which— depending on the domain—can be a drawback. However, there is a version, called Balanced Winnow, which does allow them. This version maintains two weight vectors, one for each class. An instance is classified as belonging to class 1 if ()()()w w a w w a w w ak k k0 0 0 1 1 1 + − + − + −− + − +…+ − > θ Figure 4.11(b) shows the balanced algorithm. Winnow is very effective in homing in on the relevant features in a dataset; therefore, it is called an attribute-efficient learner. That means that it may be a good candidate algorithm if a dataset has many (binary) features and most of them are irrelevant. Both Winnow and the perceptron algorithm can be used in an online setting in which new instances arrive continuously, because they can incrementally update their hypotheses as new instances arrive. 4.7 INSTANCE-BASED LEARNING In instance-based learning the training examples are stored verbatim, and a distance function is used to determine which member of the training set is closest to an unknown test instance. Once the nearest training instance has been located, its class is predicted for the test instance. The only remaining problem is defining the distance function, and that is not very difficult to do, particularly if the attributes are numeric. Distance Function Although there are other possible choices, most instance-based learners use Euclid- ean distance. The distance between an instance with attribute values a1 (1), a2 (1), … , ak (1) (where k is the number of attributes) and one with values a1 (2), a2 (2), … , ak (2) is defined as ()()()()()()()()()a a a a a ak k1 1 1 2 2 2 1 2 2 2 1 2 2− + − + + −… When comparing distances it is not necessary to perform the square root operation—the sums of squares can be compared directly. One alternative to the Euclidean distance is the Manhattan, or city-block, metric, where the difference between attribute values is not squared but just added up (after taking the absolute value). Others are obtained by taking powers higher than the square. Higher powers increase the influence of large differences at the expense of small differences. Gener- ally, the Euclidean distance represents a good compromise. Other distance metrics may be more appropriate in special circumstances. The key is to think of actual instances and what it means for them to be separated by a certain distance—what would twice that distance mean, for example? 132 CHAPTER 4 Algorithms: The Basic Methods Different attributes are often measured on different scales, so if the Euclidean distance formula were used directly, the effect of some attributes might be com- pletely dwarfed by others that had larger scales of measurement. Consequently, it is usual to normalize all attribute values to lie between 0 and 1 by calculating a v v v vi i i i i = − − min max min where vi is the actual value of attribute i, and the maximum and minimum are taken over all instances in the training set. These formulae implicitly assume numeric attributes. Here the difference between two values is just the numerical difference between them, and it is this difference that is squared and added to yield the distance function. For nominal attributes that take on values that are symbolic rather than numeric, the difference between two values that are not the same is often taken to be 1, whereas if the values are the same the difference is 0. No scaling is required in this case because only the values 0 and 1 are used. A common policy for handling missing values is as follows. For nominal attri- butes, assume that a missing feature is maximally different from any other feature value. Thus, if either or both values are missing, or if the values are different, the difference between them is taken as 1; the difference is 0 only if they are not missing and both are the same. For numeric attributes, the difference between two missing values is also taken as 1. However, if just one value is missing, the difference is often taken as either the (normalized) size of the other value or 1 minus that size, whichever is larger. This means that if values are missing, the difference is as large as it can possibly be. Finding Nearest Neighbors Efficiently Although instance-based learning is simple and effective, it is often slow. The obvious way to find which member of the training set is closest to an unknown test instance is to calculate the distance from every member of the training set and select the smallest. This procedure is linear in the number of training instances. In other words, the time it takes to make a single prediction is pro- portional to the number of training instances. Processing an entire test set takes time proportional to the product of the number of instances in the training and test sets. Nearest neighbors can be found more efficiently by representing the training set as a tree, although it is not quite obvious how. One suitable structure is a kD-tree. This is a binary tree that divides the input space with a hyperplane and then splits each partition again, recursively. All splits are made parallel to one of the axes, either vertically or horizontally, in the two-dimensional case. The data structure is called a kD-tree because it stores a set of points in k-dimensional space, with k being the number of attributes. FIGURE 4.12 A kD-tree for four training instances: (a) the tree and (b) instances and splits. (a) (3,8) (2,2) (6,7) a2 a1 (7,4) (7,4); h (2,2) (3,8) (6,7); v (b) Figure 4.12(a) gives a small example with k = 2, and Figure 4.12(b) shows the four training instances it represents, along with the hyperplanes that constitute the tree. Note that these hyperplanes are not decision boundaries: Decisions are made on a nearest-neighbor basis as explained later. The first split is horizontal (h), through the point (7,4)—this is the tree’s root. The left branch is not split further: It contains the single point (2,2), which is a leaf of the tree. The right branch is split vertically (v) at the point (6,7). Its right child is empty, and its left child contains the point (3,8). As this example illustrates, each region contains just one point—or, perhaps, no points. Sibling branches of the tree—for example, the two daughters of the root in Figure 4.12(a)—are not necessarily developed to the same depth. Every point in the training set corresponds to a single node, and up to half are leaf nodes. How do you build a kD-tree from a dataset? Can it be updated efficiently as new training examples are added? And how does it speed up nearest-neighbor calcula- tions? We tackle the last question first. To locate the nearest neighbor of a given target point, follow the tree down from its root to locate the region containing the target. Figure 4.13 shows a space like that of Figure 4.12(b) but with a few more instances and an extra boundary. The target, which is not one of the instances in the tree, is marked by a star. The leaf node of the region containing the target is colored black. This is not necessarily the target’s closest neighbor, as this example illustrates, but it is a good first approximation. In particular, any nearer neighbor must lie closer—within the dashed circle in Figure 4.13. To determine whether one exists, first check whether it is possible for a closer neighbor to lie within the node’s sibling. The black node’s sibling is shaded in Figure 4.13, and the circle does not intersect it, so the sibling cannot contain a closer 4.7 Instance-Based Learning 133 134 CHAPTER 4 Algorithms: The Basic Methods FIGURE 4.13 Using a kD-tree to find the nearest neighbor of the star. neighbor. Then back up to the parent node and check its sibling, which here covers everything above the horizontal line. In this case it must be explored because the area it covers intersects with the best circle so far. To explore it, find its daughters (the original point’s two aunts); check whether they intersect the circle (the left one does not, but the right one does); and descend to see if it contains a closer point (it does). In a typical case, this algorithm is far faster than examining all points to find the nearest neighbor. The work involved in finding the initial approximate nearest neighbor—the black point in Figure 4.13—depends on the depth of the tree, given by the logarithm of the number of nodes, log2n if the tree is well balanced. The amount of work involved in backtracking to check whether this really is the nearest neighbor depends a bit on the tree, and on how good the initial approximation is. But for a well-constructed tree with nodes that are approximately square rather than long skinny rectangles, it can also be shown to be logarithmic in the number of nodes (if the number of attributes in the dataset is not too large). How do you build a good tree for a set of training examples? The problem boils down to selecting the first training instance to split at and the direction of the split. Once you can do that, apply the same method recursively to each child of the initial split to construct the entire tree. To find a good direction for the split, calculate the variance of the data points along each axis individually, select the axis with the greatest variance, and create a splitting hyperplane perpendicular to it. To find a good place for the hyperplane, locate the median value along that axis and select the corresponding point. This makes the split perpendicular to the direction of greatest spread, with half the points lying on either side. This produces a well-balanced tree. To avoid long skinny regions it is best for successive splits to be along different axes, which is likely because the dimension of greatest variance is chosen at each stage. However, if the distribution of points is badly skewed, choosing the median value may generate several successive splits in the same direction, yielding long, skinny hyperrectangles. A better strategy is to calculate the mean rather than the median and use the point closest to that. The tree will not be perfectly balanced, but its regions will tend to be squarish because there is a greater chance that different directions will be chosen for successive splits. An advantage of instance-based learning over most other machine learning methods is that new examples can be added to the training set at any time. To retain this advantage when using a kD-tree, we need to be able to update it incrementally with new data points. To do this, determine which leaf node contains the new point and find its hyperrectangle. If it is empty, simply place the new point there. Other- wise, split the hyperrectangle along its longest dimension to preserve squareness. This simple heuristic does not guarantee that adding a series of points will preserve the tree’s balance, nor that the hyperrectangles will be well shaped for a nearest- neighbor search. It is a good idea to rebuild the tree from scratch occasionally—for example, when its depth grows to twice the best possible depth. As we have seen, kD-trees are good data structures for finding nearest neighbors efficiently. However, they are not perfect. Skewed datasets present a basic conflict between the desire for the tree to be perfectly balanced and the desire for regions to be squarish. More important, rectangles—even squares—are not the best shape to use anyway, because of their corners. If the dashed circle in Figure 4.13 were any bigger, which it would be if the black instance were a little further from the target, it would intersect the lower right corner of the rectangle at the top left and then that rectangle would have to be investigated, too—despite the fact that the training instances that define it are a long way from the corner in question. The corners of rectangular regions are awkward. The solution? Use hyperspheres, not hyperrectangles. Neighboring spheres may overlap, whereas rectangles can abut, but this is not a problem because the nearest- neighbor algorithm for kD-trees does not depend on the regions being disjoint. A data structure called a ball tree defines k-dimensional hyperspheres (“balls”) that cover the data points, and arranges them into a tree. Figure 4.14(a) shows 16 training instances in two-dimensional space, overlaid by a pattern of overlapping circles, and Figure 4.14(b) shows a tree formed from these circles. Circles at different levels of the tree are indicated by different styles of dash, and the smaller circles are drawn in shades of gray. Each node of the tree represents a ball, and the node is dashed or shaded according to the same convention 4.7 Instance-Based Learning 135 136 CHAPTER 4 Algorithms: The Basic Methods FIGURE 4.14 Ball tree for 16 training instances: (a) instances and balls and (b) the tree. (a) (b) 16 6 4 4 42 2 2 2 2 2 2 2 6 10 so that you can identify which level the balls are at. To help you understand the tree, numbers are placed on the nodes to show how many data points are deemed to be inside that ball. But be careful: This is not necessarily the same as the number of points falling within the spatial region that the ball represents. The regions at each level sometimes overlap, but points that fall into the overlap area are assigned to only one of the overlapping balls (the diagram does not show which one). Instead of the occupancy counts in Figure 4.14(b), the nodes of actual ball trees store the center and radius of their ball; leaf nodes record the points they contain as well. To use a ball tree to find the nearest neighbor to a given target, start by traversing the tree from the top down to locate the leaf that contains the target and find the closest point to the target in that ball. This gives an upper bound for the target’s distance from its nearest neighbor. Then, just as for the kD-tree, examine the sibling node. If the distance from the target to the sibling’s center exceeds its radius plus the current upper bound, it cannot possibly contain a closer point; otherwise, the sibling must be examined by descending the tree further. In Figure 4.15 the target is marked with a star and the black dot is its closest cur- rently known neighbor. The entire contents of the gray ball can be ruled out: It cannot contain a closer point because its center is too far away. Proceed recursively back up the tree to its root, examining any ball that may possibly contain a point nearer than the current upper bound. Ball trees are built from the top down, and as with kD-trees the basic problem is to find a good way of splitting a ball containing a set of data points into two. In prac- tice, you do not have to continue until the leaf balls contain just two points: You can stop earlier, once a predetermined minimum number is reached—and the same goes for kD-trees. Here is one possible splitting method. Choose the point in the ball that FIGURE 4.15 Ruling out an entire ball (the gray one) based on a target point (star) and its current nearest neighbor. is farthest from its center, and then a second point that is farthest from the first one. Assign all data points in the ball to the closest one of these two cluster centers; then compute the centroid of each cluster and the minimum radius required for it to enclose all the data points it repre- sents. This method has the merit that the cost of splitting a ball con- taining n points is only linear in n. There are more elaborate algo- rithms that produce tighter balls, but they require more computation. We will not describe sophisticated algorithms for constructing ball trees or updating them incremen- tally as new training instances are encountered. Discussion Nearest-neighbor instance-based learning is simple and often works very well. In the scheme we have described, each attribute has exactly the same influence on the decision, just as it does in the Naïve Bayes method. Another problem is that the database can easily become corrupted by noisy exemplars. One solution is to adopt the k-nearest-neighbor strategy, where some fixed, small number k of nearest neighbors—say five—are located and used together to determine the class of the test instance through a simple majority vote. (Note that earlier we used k to denote the number of attributes; this is a different, independent usage.) Another way of proofing the database against noise is to choose the exemplars that are added to it selectively and judiciously. Improved procedures, which are described in Chapter 6, address these shortcomings. The nearest-neighbor method originated many decades ago, and statisticians analyzed k-nearest-neighbor schemes in the early 1950s. If the number of training instances is large, it makes intuitive sense to use more than one nearest neighbor, but clearly this is dangerous if there are few instances. It can be shown that when k and the number n of instances both become infinite in such a way that k/n → 0, the probability of error approaches the theoretical minimum for the dataset. The nearest- neighbor method was adopted as a classification scheme in the early 1960s and has been widely used in the field of pattern recognition for almost half a century. Nearest-neighbor classification was notoriously slow until kD-trees began to be applied in the early 1990s, although the data structure itself was developed much earlier. In practice, these trees become inefficient when the dimension of the space 4.7 Instance-Based Learning 137 138 CHAPTER 4 Algorithms: The Basic Methods increases and they are only worthwhile when the number of attributes is small—up to 10. Ball trees were developed much more recently and are an instance of a more general structure called a metric tree. Sophisticated algorithms can create metric trees that deal successfully with thousands of dimensions. Instead of storing all training instances, you can compress them into regions. A very simple technique, mentioned at the end of Section 4.1, is to just record the range of values observed in the training data for each attribute and category. Given a test instance, you work out which ranges the attribute values fall into and choose the category with the greatest number of correct ranges for that instance. A slightly more elaborate technique is to construct intervals for each attribute and use the training set to count the number of times each class occurs for each interval on each attribute. Numeric attributes can be discretized into intervals, and “intervals” con- sisting of a single point can be used for nominal ones. Then, given a test instance, you can determine which intervals the instance resides in and classify it by voting, a method called voting feature intervals. These methods are very approximate, but very fast, and can be useful for initial analysis of large datasets. 4.8 CLUSTERING Clustering techniques apply when there is no class to be predicted but the instances are to be divided into natural groups. These clusters presumably reflect some mechanism that is at work in the domain from which instances are drawn, a mechanism that causes some instances to bear a stronger resemblance to each other than they do to the remaining instances. Clustering naturally requires dif- ferent techniques to the classification and association learning methods that we have considered so far. As we saw in Section 3.6, there are different ways in which the result of cluster- ing can be expressed. The groups that are identified may be exclusive: Any instance belongs in only one group. Or they may be overlapping: An instance may fall into several groups. Or they may be probabilistic: An instance belongs to each group with a certain probability. Or they may be hierarchical: A rough division of instances into groups at the top level and each group refined further—perhaps all the way down to individual instances. Really, the choice among these possibilities should be dictated by the nature of the mechanisms that are thought to underlie the particular clustering phenomenon. However, because these mechanisms are rarely known—the very existence of clusters is, after all, something that we’re trying to discover—and for pragmatic reasons too, the choice is usually dictated by the clustering tools that are available. We will examine an algorithm that works in numeric domains, partitioning instances into disjoint clusters. Like the basic nearest-neighbor method of instance- based learning, it is a simple and straightforward technique that has been used for several decades. In Chapter 6 we examine newer clustering methods that perform incremental and probabilistic clustering. 4.8 Clustering 139 Iterative Distance-Based Clustering The classic clustering technique is called k-means. First, you specify in advance how many clusters are being sought: This is the parameter k. Then k points are chosen at random as cluster centers. All instances are assigned to their closest cluster center according to the ordinary Euclidean distance metric. Next the centroid, or mean, of the instances in each cluster is calculated—this is the “means” part. These centroids are taken to be new center values for their respective clusters. Finally, the whole process is repeated with the new cluster centers. Iteration continues until the same points are assigned to each cluster in consecutive rounds, at which stage the cluster centers have stabilized and will remain the same forever. This clustering method is simple and effective. It is easy to prove that choosing the cluster center to be the centroid minimizes the total squared distance from each of the cluster’s points to its center. Once the iteration has stabilized, each point is assigned to its nearest cluster center, so the overall effect is to minimize the total squared distance from all points to their cluster centers. But the minimum is a local one; there is no guarantee that it is the global minimum. The final clusters are quite sensitive to the initial cluster centers. Completely different arrangements can arise from small changes in the initial random choice. In fact, this is true of all practical clustering techniques: It is almost always infeasible to find globally optimal clusters. To increase the chance of finding a global minimum people often run the algorithm several times with different initial choices and choose the best final result—the one with the smallest total squared distance. It is easy to imagine situations in which k-means fails to find a good clustering. Consider four instances arranged at the vertices of a rectangle in two-dimensional space. There are two natural clusters, formed by grouping together the two vertices at either end of a short side. But suppose the two initial cluster centers happen to fall at the midpoints of the long sides. This forms a stable configuration. The two clusters each contain the two instances at either end of a long side—no matter how great the difference between the long and the short sides. k-means clustering can be dramatically improved by careful choice of the initial cluster centers, often called seeds. Instead of beginning with an arbitrary set of seeds, here is a better procedure. Choose the initial seed at random from the entire space, with a uniform probability distribution. Then choose the second seed with a probability that is proportional to the square of the distance from the first. Proceed, at each stage choosing the next seed with a probability proportional to the square of the distance from the closest seed that has already been chosen. This procedure, called k-means++, improves both speed and accuracy over the original algorithm with random seeds. Faster Distance Calculations The k-means clustering algorithm usually requires several iterations, each involv- ing finding the distance of the k cluster centers from every instance to determine 140 CHAPTER 4 Algorithms: The Basic Methods its cluster. There are simple approximations that speed this up considerably. For example, you can project the dataset and make cuts along selected axes, instead of using the arbitrary hyperplane divisions that are implied by choosing the nearest cluster center. But this inevitably compromises the quality of the resulting clusters. Here’s a better way of speeding things up. Finding the closest cluster center is not so different from finding nearest neighbors in instance-based learning. Can the same efficient solutions—kD-trees and ball trees—be used? Yes! Indeed, they can be applied in an even more efficient way, because in each iteration of k-means all the data points are processed together whereas, in instance-based learning, test instances are processed individually. First, construct a kD-tree or ball tree for all the data points, which will remain static throughout the clustering procedure. Each iteration of k-means produces a set of cluster centers, and all data points must be examined and assigned to the nearest center. One way of processing the points is to descend the tree from the root until reaching a leaf and check each individual point in the leaf to find its closest cluster center. But it may be that the region represented by a higher interior node falls entirely within the domain of a single cluster center. In that case, all the data points under that node can be processed in one blow! The aim of the exercise, after all, is to find new positions for the cluster centers by calculating the centroid of the points they contain. The centroid can be calculated by keeping a running vector sum of the points in the cluster, and a count of how many there are so far. At the end, just divide one by the other to find the centroid. Suppose that with each node of the tree we store the vector sum of the points within that node and a count of the number of points. If the whole node falls within the ambit of a single cluster, the running totals for that cluster can be updated immediately. If not, look inside the node by proceeding recursively down the tree. Figure 4.16 shows the same instances and ball tree as in Figure 4.14, but with two cluster centers marked as black stars. Because all instances are assigned to the closest center, the space is divided in two by the thick line shown in Figure 4.16(a). Begin at the root of the tree in Figure 4.16(b), with initial values for the vector sum and counts for each cluster; all initial values are 0. Proceed recursively down the tree. When node A is reached, all points within it lie in cluster 1, so cluster 1’s sum and count can be updated with the sum and count for node A, and we need not descend any further. Recursing back to node B, its ball straddles the boundary between the clusters, so its points must be examined individually. When node C is reached, it falls entirely within cluster 2; again, we can update cluster 2 immediately and we need not descend any further. The tree is only examined down to the frontier marked by the dashed line in Figure 4.16(b), and the advantage is that the nodes below need not be opened—at least not on this particular iteration of k-means. Next time, the cluster centers will have changed and things may be different. 4.9 Multi-Instance learning 141 FIGURE 4.16 A ball tree: (a) two cluster centers and their dividing line and (b) the corresponding tree. (a) (b) 16 6 4 2 6 4 10 2 2 24 2 2 22 Discussion Many variants of the basic k-means procedure have been developed. Some produce a hierarchical clustering by applying the algorithm with k = 2 to the overall dataset and then repeating, recursively, within each cluster. How do you choose k? Often nothing is known about the likely number of clus- ters, and the whole point of clustering is to find out. One way is to try different values and choose the best. To do this you need to learn how to evaluate the success of machine learning, which is what Chapter 5 is about. We return to clustering in Section 6.8. 4.9 MULTI-INSTANCE LEARNING In Chapter 2 we introduced multi-instance learning, where each example in the data comprises several different instances. We call these examples bags (we noted the difference between bags and sets in Section 4.2). In supervised multi-instance learning, a class label is associated with each bag, and the goal of learning is to determine how the class can be inferred from the instances that make up the bag. While advanced algorithms have been devised to tackle such problems, it turns out that the simplicity-first methodology can be applied here with surprisingly good results. A simple but effective approach is to manipulate the input data to transform it into a single-instance learning problem and then apply standard learning methods, 142 CHAPTER 4 Algorithms: The Basic Methods such as the ones described in this chapter. Two such approaches are described in the following sections. Aggregating the Input You can convert a multiple-instance problem to a single-instance one by calculating values such as mean, mode, minimum, and maximum that summarize the instances in the bag and adding these as new attributes. Each “summary” instance retains the class label of the bag it was derived from. To classify a new bag the same process is used: A single aggregated instance is created with attributes that summarize the instances in the bag. Surprisingly, for the original drug activity dataset that spurred the development of multi-instance learning, results comparable with special-purpose multi-instance learners can be obtained using just the minimum and maximum values of each attribute for each bag, combined with a support vector machine clas- sifier (see Chapter 6). One potential drawback of this approach is that the best summary statistics to compute depend on the problem at hand. However, the addi- tional computational cost associated with exploring combinations of different summary statistics is offset by the fact that the summarizing process means that fewer instances are processed by the learning algorithm. Aggregating the Output Instead of aggregating the instances in each bag, another approach is to learn a clas- sifier directly from the original instances that comprise the bag. To achieve this, the instances in a given bag are all assigned the bag’s class label. At classification time, a prediction is produced for each instance in the bag to be predicted, and the predic- tions are aggregated in some fashion to form a prediction for the bag as a whole. One approach is to treat the predictions as votes for the various class labels. If the classifier is capable of assigning probabilities to the class labels, these could be averaged to yield an overall probability distribution for the bag’s class label. This method treats the instances independently and gives them equal influence on the predicted class label. One problem is that the bags in the training data can contain different numbers of instances. Ideally, each bag should have the same influence on the final model that is learned. If the learning algorithm can accept instance-level weights, this can be achieved by assigning each instance in a given bag a weight inversely propor- tional to the bag’s size. If a bag contains n instances, giving each one a weight of 1/n ensures that the instances contribute equally to the bag’s class label and each bag receives a total weight of 1. Discussion Both methods described previously for tackling multi-instance problems disregard the original multi-instance assumption that a bag is positive if and only if at least one 4.10 Further Reading 143 of its instances is positive. Instead, making each instance in a bag contribute equally to its label is the key element that allows standard learning algorithms to be applied. Otherwise, it is necessary to try to identify the “special” instances that are the key to determining the bag’s label. 4.10 FURTHER READING The 1R scheme was proposed and thoroughly investigated by Holte (1993). It was never really intended as a machine learning “method.” The point was more to demonstrate that very simple structures underlie most of the practical datasets being used to evaluate machine learning schemes at the time and that putting high-powered inductive inference schemes to work on simple datasets was like using a sledgehammer to crack a nut. Why grapple with a complex decision tree when a simple rule will do? The scheme that generates one simple rule per class is due to Lucio de Souza Coelho of Brazil and Len Trigg of New Zealand, and it has been dubbed hyperpipes. A very simple algorithm, it has the advantage of being extremely fast and is quite feasible even with an enormous number of attributes. Bayes was an eighteenth-century English philosopher who set out his theory of probability in an “Essay towards solving a problem in the doctrine of chances,” published in the Philosophical Transactions of the Royal Society of London (Bayes, 1763). The rule that bears his name has been a cornerstone of probability theory ever since. The difficulty with the application of Bayes’ rule in practice is the assignment of prior probabilities. Some statisticians, dubbed Bayesians, take the rule as gospel and insist that people make serious attempts to estimate prior probabilities accurately—although such estimates are often subjective. Others, non-Bayesians, prefer the kind of prior- free analysis that typically generates statistical confidence intervals, which we will see in Chapter 5. With a particular dataset, prior probabilities for Naïve Bayes are usually reasonably easy to estimate, which encourages a Bayesian approach to learn- ing. The independence assumption made by the Naïve Bayes method is a great stumbling block, however, and efforts are being made to apply Bayesian analysis without assuming independence. The resulting models are called Bayesian networks (Heckerman et al., 1995), and we describe them in Section 6.7. Bayesian techniques had been used in the field of pattern recognition (Duda and Hart, 1973) for 20 years before they were adopted by machine learning researchers (e.g., Langley et al., 1992) and made to work on datasets with redun- dant attributes (Langley and Sage 1994) and numeric attributes (John and Langley, 1995). The label Naïve Bayes is unfortunate because it is hard to use this method without feeling simpleminded. However, there is nothing naïve about its use in appropriate circumstances. The multinomial Naïve Bayes model, which is particu- larly useful for text classification, was investigated by McCallum and Nigam (1998). 144 CHAPTER 4 Algorithms: The Basic Methods The classic paper on decision tree induction is Quinlan (1986), who describes the basic ID3 procedure developed in this chapter. A comprehensive description of the method, including the improvements that are embodied in C4.5, appears in a classic book by Quinlan (1993), which gives a listing of the complete C4.5 system, written in the C programming language. PRISM was developed by Cendrowska (1987), who also introduced the contact lens dataset. Association rules are introduced and described in the database literature rather than in the machine learning literature. Here the emphasis is very much on dealing with huge amounts of data rather than on sensitive ways of testing and evaluating algorithms on limited datasets. The algorithm introduced in this chapter is the Apriori method developed by Agrawal and his associates (Agrawal et al., 1993a, 1993b; Agrawal and Srikant, 1994). A survey of association-rule mining appears in an article by Chen et al. (1996). Linear regression is described in most standard statistical texts, and a particularly comprehensive treatment can be found in Lawson and Hanson (1995). The use of linear models for classification enjoyed a great deal of popularity in the 1960s; Nilsson (1965) is an excellent reference. He defines a linear threshold unit as a binary test of whether a linear function is greater or less than zero and a linear machine as a set of linear functions, one for each class, whose value for an unknown example is compared and the largest chosen as its predicted class. In the distant past, perceptrons fell out of favor on publication of an influential book that showed that they had fundamental limitations (Minsky and Papert, 1969); however, more complex systems of linear functions have enjoyed a resurgence in recent years in the form of neural networks, described in Section 6.4. The Winnow algorithms were introduced by Nick Littlestone in his Ph.D. thesis (Littlestone, 1988, 1989). Mul- tiresponse linear classifiers have found application in an operation called stacking that combines the output of other learning algorithms, described in Chapter 8 (see Wolpert, 1992). Fix and Hodges (1951) performed the first analysis of the nearest-neighbor method, and Johns (1961) pioneered its use in classification problems. Cover and Hart (1967) obtained the classic theoretical result that, for large enough datasets, its probability of error never exceeds twice the theoretical minimum. Devroye et al. (1996) showed that k-nearest neighbor is asymptotically optimal for large k and n with k/n → 0. Nearest-neighbor methods gained popularity in machine learning through the work of Aha (1992), who showed that instance-based learning can be combined with noisy exemplar pruning and attribute weighting and that the resulting methods perform well in comparison with other learning methods. We take this up again in Chapter 6. The kD-tree data structure was developed by Friedman et al. (1977). Our descrip- tion closely follows an explanation given by Andrew Moore in his Ph.D. thesis (Moore, 1991). Moore, who, along with Omohundro (1987), pioneered its use in machine learning. Moore (2000) describes sophisticated ways of constructing ball trees that perform well even with thousands of attributes. We took our ball tree example from lecture notes by Alexander Gray of Carnegie-Mellon University. The 4.11 Weka Implementations 145 voting feature interval method mentioned in the Discussion section at the end of Section 4.7 is described by Demiroz and Guvenir (1997). The k-means algorithm is a classic technique, and many descriptions and varia- tions are available (e.g., Hartigan, 1975). The k-means++ variant, which yields a significant improvement by choosing the initial seeds more carefully, was introduced as recently as 2007 by Arthur and Vassilvitskii (2007). The clever use of kD-trees to speed up k-means clustering, which we have chosen to illustrate using ball trees instead, was pioneered by Moore and Pelleg (2000) in their X-means clustering algorithm. That algorithm contains some other innovations, described in Section 6.8. The method of dealing with multi-instance learning problems by applying stan- dard single-instance learners to summarized bag-level data was applied in conjunc- tion with support vector machines by Gärtner et al. (2002). The alternative approach of aggregating the output is explained by Frank and Xu (2003). 4.11 WEKA IMPLEMENTATIONS For classifiers, see Section 11.4 and Table 11.5. • Inferring rudimentary rules: OneR, HyperPipes (learns one rule per class) • Statistical modeling: • NaïveBayes and many variants, including NaiveBayesMultinomial • Decision trees: Id3 • Decision rules: Prism • Association rules (see Section 11.7 and Table 11.8): a priori • Linear models: • SimpleLinearRegression, LinearRegression, Logistic (regression) • VotedPerceptron, Winnow • Instance-based learning: IB1, VFI (voting feature intervals) • Clustering (see Section 11.6 and Table 11.7): SimpleKMeans • Multi-instance learning: SimpleMI, MIWrapper This page intentionally left blank 147Data Mining: Practical Machine Learning Tools and Techniques Copyright © 2011 Elsevier Inc. All rights of reproduction in any form reserved. CHAPTER 5 Credibility: Evaluating What’s Been Learned Evaluation is the key to making real progress in data mining. There are lots of ways of inferring structure from data: We have encountered many already and will see further refinements, and new methods, in Chapter 6. However, in order to determine which ones to use on a particular problem we need systematic ways to evaluate how different methods work and to compare one with another. But evaluation is not as simple as it might appear at first sight. What’s the problem? We have the training set; surely we can just look at how well different methods do on that. Well, no: As we will see very shortly, performance on the training set is definitely not a good indicator of performance on an indepen- dent test set. We need ways of predicting performance bounds in practice, based on experiments with whatever data can be obtained. When a vast supply of data is available, this is no problem: Just make a model based on a large training set, and try it out on another large test set. But although data mining sometimes involves “big data”—particularly in marketing, sales, and customer support applications—it is often the case that data, quality data, is scarce. The oil slicks mentioned in Chapter 1 (page 23) had to be detected and marked manually—a skilled and labor-intensive process—before being used as training data. Even in the personal loan application data (page 22), there turned out to be only 1000 training examples of the appropriate type. The electricity supply data (page 24) went back 15 years, 5000 days—but only 15 Christmas days and Thanks- givings, and just four February 29s and presidential elections. The electromechanical diagnosis application (page 25) was able to capitalize on 20 years of recorded experience, but this yielded only 300 usable examples of faults. The marketing and sales applications (page 26) certainly involve big data, but many others do not: Training data frequently relies on specialist human expertise—and that is always in short supply. The question of predicting performance based on limited data is an interesting, and still controversial, one. We will encounter many different techniques, of which one—repeated cross-validation—is probably the method of choice in most practical limited-data situations. Comparing the performance of different machine learning schemes on a given problem is another matter that is not as easy as it sounds: To be sure that apparent differences are not caused by chance effects, statistical tests are needed. 148 CHAPTER 5 Credibility: Evaluating What’s Been Learned So far we have tacitly assumed that what is being predicted is the ability to clas- sify test instances accurately; however, some situations involve predicting class probabilities rather than the classes themselves, and others involve predicting numeric rather than nominal values. Different methods are needed in each case. Then we look at the question of cost. In most practical data mining situations, the cost of a misclassification error depends on the type of error it is—whether, for example, a positive example was erroneously classified as negative or vice versa. When doing data mining, and evaluating its performance, it is often essential to take these costs into account. Fortunately, there are simple techniques to make most learning schemes cost sensitive without grappling with the algorithm’s internals. Finally, the whole notion of evaluation has fascinating philosophical connections. For 2000 years, philosophers have debated the question of how to evaluate scientific theories, and the issues are brought into sharp focus by data mining because what is extracted is essentially a “theory” of the data. 5.1 TRAINING AND TESTING For classification problems, it is natural to measure a classifier’s performance in terms of the error rate. The classifier predicts the class of each instance: If it is correct, that is counted as a success; if not, it is an error. The error rate is just the proportion of errors made over a whole set of instances, and it measures the overall performance of the classifier. Of course, what we are interested in is the likely future performance on new data, not the past performance on old data. We already know the classifications of each instance in the training set, which after all is why we can use it for training. We are not generally interested in learning about those classifications—although we might be if our purpose is data cleansing rather than prediction. So the question is, is the error rate on old data likely to be a good indicator of the error rate on new data? The answer is a resounding no—not if the old data was used during the learning process to train the classifier. This is a surprising fact, and a very important one. The error rate on the training set is not likely to be a good indicator of future performance. Why? Because the classifier has been learned from the very same training data, any estimate of perfor- mance based on that data will be optimistic, even hopelessly optimistic. We have already seen an example of this in the labor relations dataset. Figure 1.3(b) (page 18) was generated directly from the training data, and Figure 1.3(a) was obtained from it by a process of pruning. The former is potentially more accurate on the data that was used to train the classifier, but may perform less well on inde- pendent test data because it is overfitted to the training data. The first tree will look good according to the error rate on the training data, better than the second tree. But this does not necessarily reflect how they will perform on independent test data. The error rate on the training data is called the resubstitution error because it is calculated by resubstituting the training instances into a classifier that was 5.1 Training and Testing 149 constructed from them. Although it is not a reliable predictor of the true error rate on new data, it is nevertheless often useful to know. To predict the performance of a classifier on new data, we need to assess its error rate on a dataset that played no part in the formation of the classifier. This indepen- dent dataset is called the test set. We assume that both the training data and the test data are representative samples of the underlying problem. In some cases the test data might be distinct in nature from the training data. Consider, for example, the credit risk problem from Section 1.3 (page 22). Suppose the bank had training data from branches in New York and Florida and wanted to know how well a classifier trained on one of these datasets would perform in a new branch in Nebraska. It should probably use the Florida data as test data for evaluating the New York–trained classifier and the New York data to evaluate the Florida-trained classifier. If the datasets were amalgamated before training, perfor- mance on the test data would probably not be a good indicator of performance on future data in a completely different state. It is important that the test data is not used in any way to create the classifier. For example, some learning schemes involve two stages, one to come up with a basic structure and the second to optimize parameters involved in that structure, and separate sets of data may be needed in the two stages. Or you might try out several learning schemes on the training data and then evaluate them—on a fresh dataset, of course—to see which one works best. But none of this data may be used to determine an estimate of the future error rate. In such situations people often talk about three datasets: the training data, the validation data, and the test data. The training data is used by one or more learning schemes to come up with classifiers. The validation data is used to optimize param- eters of those classifier, or to select a particular one. Then the test data is used to calculate the error rate of the final, optimized, method. Each of the three sets must be chosen independently: The validation set must be different from the training set to obtain good performance in the optimization or selection stage, and the test set must be different from both to obtain a reliable estimate of the true error rate. It may be that once the error rate has been determined, the test data is bundled back into the training data to produce a new classifier for actual use. There is nothing wrong with this: It is just a way of maximizing the amount of data used to generate the classifier that will actually be employed in practice. With well-behaved learning schemes, this should not decrease predictive performance. Also, once the validation data has been used—maybe to determine the best type of learning scheme to use— then it can be bundled back into the training data to retrain that learning scheme, maximizing the use of data. If lots of data is available, there is no problem: We take a large sample and use it for training; then another, independent large sample of different data and use it for testing. Provided both samples are representative, the error rate on the test set will give a good indication of future performance. Generally, the larger the training sample, the better the classifier, although the returns begin to diminish once a certain volume of training data is exceeded. And the larger the test sample, the more accurate 150 CHAPTER 5 Credibility: Evaluating What’s Been Learned the error estimate. The accuracy of the error estimate can be quantified statistically, as we will see in Section 5.2. The real problem occurs when there is not a vast supply of data available. In many situations the training data must be classified manually—and so must the test data, of course, to obtain error estimates. This limits the amount of data that can be used for training, validation, and testing, and the problem becomes how to make the most of a limited dataset. From this dataset, a certain amount is held over for testing—this is called the holdout procedure—and the remainder used for training (and, if necessary, part of that is set aside for validation). There’s a dilemma here: To find a good classifier, we want to use as much of the data as possible for training; to obtain a good error estimate, we want to use as much of it as possible for testing. Sections 5.3 and 5.4 review widely used methods for dealing with this dilemma. 5.2 PREDICTING PERFORMANCE Suppose we measure the error of a classifier on a test set and obtain a certain numeri- cal error rate—say 25%. Actually, in this section we talk about success rate rather than error rate, so this corresponds to a success rate of 75%. Now, this is only an estimate. What can you say about the true success rate on the target population? Sure, it’s expected to be close to 75%. But how close—within 5 or 10%? It must depend on the size of the test set. Naturally, we would be more confident of the 75% figure if it were based on a test set of 10,000 instances rather than a test set of 100 instances. But how much more confident would we be? To answer these questions, we need some statistical reasoning. In statistics, a succession of independent events that either succeed or fail is called a Bernoulli process. The classic example is coin tossing. Each toss is an independent event. Let’s say we always predict heads; but rather than “heads” or “tails,” each toss is consid- ered a “success” or a “failure.” Let’s say the coin is biased, but we don’t know what the probability of heads is. Then, if we actually toss the coin 100 times and 75 of the tosses are heads, we have a situation very like the one just described for a clas- sifier with an observed 75% success rate on a test set. What can we say about the true success probability? In other words, imagine that there is a Bernoulli process—a biased coin—with a true (but unknown) success rate of p. Suppose that out of N trials, S are successes; thus, the observed success rate is f = S/N. The question is, what does this tell you about the true success rate p? The answer to this question is usually expressed as a confidence interval—that is, p lies within a certain specified interval with a certain specified confidence. For example, if S = 750 successes are observed out of N = 1000 trials, this indicates that the true success rate must be around 75%. But how close to 75%? It turns out that with 80% confidence, the true success rate p lies between 73.2% and 76.7%. If S = 75 successes are observed out of N = 100 trials, this also indicates that the true success rate must be around 75%. But the experiment is smaller, and so the 80% confidence interval for p is wider, stretching from 69.1 to 80.1%. 5.2 Predicting Performance 151 These figures are easy to relate to qualitatively, but how are they derived quantitatively? We reason as follows: The mean and variance of a single Bernoulli trial with success rate p are p and p(1 − p), respectively. If N trials are taken from a Bernoulli process, the expected success rate f = S/N is a random variable with the same mean p; the variance is reduced by a factor of N to p(1 − p)/N. For large N, the distribution of this random variable approaches the normal distribution. These are all facts of statistics—we will not go into how they are derived. The probability that a random variable X, with zero mean, lies within a certain confidence range of width 2z is Pr − ≤ ≤[] =z X z c For a normal distribution, values of c and corresponding values of z are given in tables printed at the back of most statistical texts. However, the tabulations conventionally take a slightly different form: They give the confidence that X will lie outside the range, and they give it for the upper part of the range only: Pr X z≥[] This is called a one-tailed probability because it refers only to the upper “tail” of the distribution. Normal distributions are symmetric, so the probabilities for the lower tail Pr X z≤ −[] are just the same. Table 5.1 gives an example. Like other tables for the normal distribution, this assumes that the random variable X has a mean of 0 and a variance of 1. Alternatively, you might say that the z figures are measured in standard deviations from the mean. Thus, the figure for Pr[X ≥ z] = 5% implies that there is a 5% chance that X lies more than 1.65 standard deviations above the mean. Because the distribution is symmetric, the chance that X lies more than 1.65 standard deviations from the mean (above or below) is 10%, or Pr . . %− ≤ ≤[ ] =1 65 1 65 90X Now all we need to do is reduce the random variable f to have zero mean and unit variance. We do this by subtracting the mean p and dividing by the standard deviation p p N( )1− . This leads to Pr ()− < − − < =z f p p p N z c 1 Here is the procedure for finding confidence limits. Given a particular confidence figure c, consult Table 5.1 for the corresponding z value. To use the table you will first have to subtract c from 1 and then halve the result, so that for c = 90% you use the table entry for 5%. Linear interpolation can be used for intermediate confidence levels. Then write the inequality in the preceding expression as an equality and invert it to find an expression for p. The final step involves solving a quadratic equation. Although this is not hard to do, it leads to an unpleasantly formidable expression for the confidence limits: p f z N z f N f N z N z N= + ± − + + 2 2 2 2 2 2 4 1 The ± in this expression gives two values for p that represent the upper and lower confidence boundaries. Although the formula looks complicated, it is not hard to work out in particular cases. 152 CHAPTER 5 Credibility: Evaluating What’s Been Learned This result can be used to obtain the values in the numeric example given earlier. Setting f = 75%, N = 1000, and c = 80% (so that z = 1.28) leads to the interval [0.732, 0.767] for p, and N = 100 leads to [0.691, 0.801] for the same level of confidence. Note that the normal distribution assumption is only valid for large N (say, N > 100). Thus, f = 75% and N = 10 leads to confidence limits [0.549, 0.881], but these should be taken with a grain of salt. Table 5.1 Confidence Limits for the Normal Distribution Pr[X ≥ z] z 0.1% 3.09 0.5% 2.58 1% 2.33 5% 1.65 10% 1.28 20% 0.84 40% 0.25 5.3 CROSS-VALIDATION Now consider what to do when the amount of data for training and testing is limited. The holdout method reserves a certain amount for testing and uses the remainder for training (and sets part of that aside for validation, if required). In practical terms, it is common to hold out one-third of the data for testing and use the remaining two-thirds for training. Of course, you may be unlucky: The sample used for training (or testing) might not be representative. In general, you cannot tell whether a sample is representative or not. But there is one simple check that might be worthwhile: Each class in the full dataset should be represented in about the right proportion in the training and testing sets. If, by bad luck, all examples with a certain class were omitted from the training set, you could hardly expect a classifier learned from that data to perform well on examples of that class—and the situation would be exacerbated by the fact that the class would necessarily be overrepresented in the test set because none of its instances made it into the training set! Instead, you should ensure that the random sampling is done in a way that guarantees that each class is properly represented in both training and test sets. This procedure is called stratification, and we might speak of stratified holdout. While it is generally well worth doing, stratification provides only a primitive safeguard against uneven representation in training and test sets. A more general way to mitigate any bias caused by the particular sample chosen for holdout is to repeat the whole process, training and testing, several times with different random samples. In each iteration a certain proportion, say two-thirds, of 5.3 Cross-Validation 153 the data is randomly selected for training, possibly with stratification, and the remainder is used for testing. The error rates on the different iterations are averaged to yield an overall error rate. This is the repeated holdout method of error rate estimation. In a single holdout procedure, you might consider swapping the roles of the testing and training data—that is, train the system on the test data and test it on the training data—and average the two results, thus reducing the effect of uneven representation in training and test sets. Unfortunately, this is only really plausible with a 50:50 split between training and test data, which is generally not ideal—it is better to use more than half the data for training even at the expense of test data. However, a simple variant forms the basis of an important statistical technique called cross-validation. In cross-validation, you decide on a fixed number of folds, or partitions, of the data. Suppose we use three. Then the data is split into three approximately equal partitions; each in turn is used for testing and the remainder is used for training. That is, use two-thirds of the data for training and one-third for testing, and repeat the procedure three times so that in the end, every instance has been used exactly once for testing. This is called threefold cross-validation, and if stratification is adopted as well—which it often is—it is stratified threefold cross-validation. The standard way of predicting the error rate of a learning technique given a single, fixed sample of data is to use stratified tenfold cross-validation. The data is divided randomly into 10 parts in which the class is represented in approximately the same proportions as in the full dataset. Each part is held out in turn and the learning scheme trained on the remaining nine-tenths; then its error rate is calculated on the holdout set. Thus, the learning procedure is executed a total of 10 times on different training sets (each set has a lot in common with the others). Finally, the 10 error estimates are averaged to yield an overall error estimate. Why 10? Extensive tests on numerous different datasets, with different learning techniques, have shown that 10 is about the right number of folds to get the best estimate of error, and there is also some theoretical evidence that backs this up. Although these arguments are by no means conclusive, and debate continues to rage in machine learning and data mining circles about what is the best scheme for evaluation, tenfold cross-validation has become the standard method in practi- cal terms. Tests have also shown that the use of stratification improves results slightly. Thus, the standard evaluation technique in situations where only limited data is available is stratified tenfold cross-validation. Note that neither the strati- fication nor the division into 10 folds has to be exact: It is enough to divide the data into 10 approximately equal sets in which the various class values are rep- resented in approximately the right proportion. Moreover, there is nothing magic about the exact number 10: 5-fold or 20-fold cross-validation is likely to be almost as good. A single tenfold cross-validation might not be enough to get a reliable error estimate. Different tenfold cross-validation experiments with the same learning scheme and dataset often produce different results because of the effect of random 154 CHAPTER 5 Credibility: Evaluating What’s Been Learned variation in choosing the folds themselves. Stratification reduces the variation, but it certainly does not eliminate it entirely. When seeking an accurate error estimate, it is standard procedure to repeat the cross-validation process 10 times—that is, 10 times tenfold cross-validation—and average the results. This involves invoking the learning algorithm 100 times on datasets that are all nine-tenths the size of the original. Getting a good measure of performance is a computation-intensive undertaking. 5.4 OTHER ESTIMATES Tenfold cross-validation is the standard way of measuring the error rate of a learning scheme on a particular dataset; for reliable results, 10 times tenfold cross-validation. But many other methods are used instead. Two that are particularly prevalent are leave-one-out cross-validation and the bootstrap. Leave-One-Out Cross-Validation Leave-one-out cross-validation is simply n-fold cross-validation, where n is the number of instances in the dataset. Each instance in turn is left out, and the learning scheme is trained on all the remaining instances. It is judged by its correctness on the remaining instance—one or zero for success or failure, respectively. The results of all n judgments, one for each member of the dataset, are averaged, and that average represents the final error estimate. This procedure is an attractive one for two reasons. First, the greatest possible amount of data is used for training in each case, which presumably increases the chance that the classifier is an accurate one. Second, the procedure is deterministic: No random sampling is involved. There is no point in repeating it 10 times, or repeating it at all: The same result will be obtained each time. Set against this is the high computational cost, because the entire learning procedure must be executed n times and this is usually infeasible for large datasets. Nevertheless, leave-one-out seems to offer a chance of squeezing the maximum out of a small dataset and getting as accurate an estimate as possible. But there is a disadvantage to leave-one-out cross-validation, apart from the computational expense. By its very nature, it cannot be stratified—worse than that, it guarantees a nonstratified sample. Stratification involves getting the correct pro- portion of examples in each class into the test set, and this is impossible when the test set contains only a single example. A dramatic, although highly artificial, illus- tration of the problems this might cause is to imagine a completely random dataset that contains exactly the same number of instances of each of two classes. The best that an inducer can do with random data is to predict the majority class, giving a true error rate of 50%. But in each fold of leave-one-out, the opposite class to the test instance is in the majority—and therefore the predictions will always be incor- rect, leading to an estimated error rate of 100%! 5.4 Other Estimates 155 The Bootstrap The second estimation method we describe, the bootstrap, is based on the statistical procedure of sampling with replacement. Previously, whenever a sample was taken from the dataset to form a training or test set, it was drawn without replacement. That is, the same instance, once selected, could not be selected again. It is like picking teams for football: You cannot choose the same person twice. But dataset instances are not like people. Most learning schemes can use the same instance twice, and it makes a difference in the result of learning if it is present in the training set twice. (Mathematical sticklers will notice that we should not really be talking about “sets” at all if the same object can appear more than once.) The idea of the bootstrap is to sample the dataset with replacement to form a training set. We will describe a particular variant, mysteriously (but for a reason that will soon become apparent) called the 0.632 bootstrap. For this, a dataset of n instances is sampled n times, with replacement, to give another dataset of n instances. Because some elements in this second dataset will (almost certainly) be repeated, there must be some instances in the original dataset that have not been picked—we will use these as test instances. What is the chance that a particular instance will not be picked for the training set? It has a 1/n probability of being picked each time and so a 1 – 1/n probability of not being picked. Multiply these probabilities together for a sufficient number of picking opportunities, n, and the result is a figure of 1 1 0 3681− ≈ =− n e n . where e is the base of natural logarithms, 2.7183 (not the error rate!) This gives the chance of a particular instance not being picked at all. Thus, for a reasonably large dataset, the test set will contain about 36.8% of the instances and the training set will contain about 63.2% of them (now you can see why it’s called the 0.632 bootstrap). Some instances will be repeated in the training set, bringing it up to a total size of n, the same as in the original dataset. The figure obtained by training a learning system on the training set and cal- culating its error over the test set will be a pessimistic estimate of the true error rate because the training set, although its size is n, nevertheless contains only 63% of the instances, which is not a great deal compared, for example, with the 90% used in tenfold cross-validation. To compensate for this, we combine the test-set error rate with the resubstitution error on the instances in the training set. The resubstitution figure, as we warned earlier, gives a very optimistic estimate of the true error and should certainly not be used as an error figure on its own. But the bootstrap procedure combines it with the test error rate to give a final estimate e as follows: 156 CHAPTER 5 Credibility: Evaluating What’s Been Learned e e e= × + ×0 632 0 368. .test instances training instances Then, the whole bootstrap procedure is repeated several times, with different replacement samples for the training set, and the results are averaged. The bootstrap procedure may be the best way of estimating the error rate for very small datasets. However, like leave-one-out cross-validation, it has disadvan- tages that can be illustrated by considering a special, artificial situation. In fact, the very dataset we considered above will do: a completely random dataset with two classes of equal size. The true error rate is 50% for any prediction rule. But a scheme that memorized the training set would give a perfect resubstitution score of 100%, so that etraining instances = 0, and the 0.632 bootstrap will mix this in with a weight of 0.368 to give an overall error rate of only 31.6% (0.632 × 50% + 0.368 × 0%), which is misleadingly optimistic. 5.5 COMPARING DATA MINING SCHEMES We often need to compare two different learning schemes on the same problem to see which is the better one to use. It seems simple: Estimate the error using cross- validation (or any other suitable estimation procedure), perhaps repeated several times, and choose the scheme with the smaller estimate. This is quite sufficient in many practical applications: If one scheme has a lower estimated error than another on a particular dataset, the best we can do is to use the former scheme’s model. However, it may be that the difference is simply due to estimation error, and in some circumstances it is important to determine whether one scheme is really better than another on a particular problem. This is a standard challenge for machine learning researchers. If a new learning algorithm is proposed, its proponents must show that it improves on the state of the art for the problem at hand and demonstrate that the observed improvement is not just a chance effect in the estimation process. This is a job for a statistical test based on confidence bounds, the kind we met previously when trying to predict true performance from a given test-set error rate. If there were unlimited data, we could use a large amount for training and evaluate performance on a large independent test set, obtaining confidence bounds just as before. However, if the difference turns out to be significant we must ensure that this is not just because of the particular dataset we happened to base the experiment on. What we want to determine is whether one scheme is better or worse than another on average, across all possible training and test datasets that can be drawn from the domain. Because the amount of training data naturally affects performance, all datasets should be the same size. Indeed, the experiment might be repeated with different sizes to obtain a learning curve. For the moment, assume that the supply of data is unlimited. For definiteness, suppose that cross-validation is being used to obtain the error estimates (other esti- mators, such as repeated cross-validation, are equally viable). For each learning scheme we can draw several datasets of the same size, obtain an accuracy estimate 5.5 Comparing Data Mining Schemes 157 for each dataset using cross-validation, and compute the mean of the estimates. Each cross-validation experiment yields a different, independent error estimate. What we are interested in is the mean accuracy across all possible datasets of the same size, and whether this mean is greater for one scheme or the other. From this point of view, we are trying to determine whether the mean of a set of samples—cross-validation estimates for the various datasets that we sampled from the domain—is significantly greater than, or significantly less than, the mean of another. This is a job for a statistical device known as the t-test, or Student’s t-test. Because the same cross-validation experiment can be used for both learning schemes to obtain a matched pair of results for each dataset, a more sensitive version of the t-test known as a paired t-test can be used. We need some notation. There is a set of samples x1, x2, …, xk obtained by successive tenfold cross-validations using one learning scheme, and a second set of samples y1, y2, …, yk obtained by successive tenfold cross-validations using the other. Each cross- validation estimate is generated using a different dataset, but all datasets are of the same size and from the same domain. We will get best results if exactly the same cross- validation partitions are used for both schemes, so that x1 and y1 are obtained using the same cross-validation split, as are x2 and y2, and so on. Denote the mean of the first set of samples by x and the mean of the second set by y . We are trying to determine whether x is significantly different from y . If there are enough samples, the mean (x ) of a set of independent samples (x1, x2, …, xk) has a normal (i.e., Gaussian) distribution, regardless of the distribution underlying the samples themselves. Call the true value of the mean µ. If we knew the variance of that normal distribution, so that it could be reduced to have zero mean and unit variance, we could obtain confidence limits on µ given the mean of the samples (x ). However, the variance is unknown, and the only way we can obtain it is to estimate it from the set of samples. That is not hard to do. The variance of x can be estimated by dividing the variance calculated from the samples x1, x2, …, xk—call it σx 2—by k. We can reduce the distribution of x to have zero mean and unit variance by using x kx − µ σ 2 The fact that we have to estimate the variance changes things somewhat. Because the variance is only an estimate, this does not have a normal distribution (although it does become normal for large values of k). Instead, it has what is called a Student’s distribution with k – 1 degrees of freedom. What this means in practice is that we have to use a table of confidence intervals for the Student’s distribution rather than the confidence table for the normal distribution given earlier. For 9 degrees of freedom (which is the correct number if we are using the average of 10 cross-validations) the appropriate confidence limits are shown in Table 5.2. If you compare them with Table 5.1 you will see that the Student’s figures are slightly more conservative—for a given degree of confidence, the interval is slightly wider—and this reflects the additional uncertainty caused by having to estimate the variance. Different tables are needed for different numbers of degrees of freedom, and if there are more than 100 degrees of freedom the confidence limits are very close to those for the normal distribution. Like Table 5.1, the figures in Table 5.2 are for a “one-sided” confidence interval. 158 CHAPTER 5 Credibility: Evaluating What’s Been Learned To decide whether the means x and y , each an average of the same number k of samples, are the same or not, we consider the differences di between corresponding observations, di = xi − yi. This is legitimate because the observations are paired. The mean of this difference is just the difference between the two means, d x y= − , and, like the means themselves, it has a Student’s distribution with k – 1 degrees of freedom. If the means are the same, the difference is zero (this is called the null hypothesis); if they’re significantly different, the difference will be significantly different from zero. So for a given confidence level, we will check whether the actual difference exceeds the confidence limit. First, reduce the difference to a zero-mean, unit-variance variable called the t-statistic, t d kd = σ 2 where σd 2 is the variance of the difference samples. Then, decide on a confidence level—generally, 5% or 1% is used in practice. From this, the confidence limit z is determined using Table 5.2 if k is 10; if it is not, a confidence table of the Student distribution for the k value in question is used. A two-tailed test is appropriate because we do not know in advance whether the mean of the x’s is likely to be greater than that of the y’s or vice versa; thus, for a 1% test we use the value corresponding to 0.5% in Table 5.2. If the value of t according to the last formula is greater than z, or less than –z, we reject the null hypothesis that the means are the same and conclude that there really is a significant difference between the two learning methods on that domain for that dataset size. Two observations are worth making on this procedure. The first is technical: What if the observations were not paired? That is, what if we were unable, for some reason, to assess the error of each learning scheme on the same datasets? What if the number of datasets for each scheme was not even the same? These conditions could arise if someone else had evaluated one of the schemes and published several different estimates for a particular domain and dataset size—or perhaps just their mean and variance—and we wished to compare this with a different learning scheme. Then it is necessary to use a regular, nonpaired t-test. Instead of taking the mean of the difference, d , we use the difference of the means, x y− . Of course, that’s the same thing: The mean of the difference is the difference of the means. But the variance of the difference d is not the same. If the variance of the samples x1, x2, …, xk is σx 2 and the variance of the samples y1, y2, …, y is σy 2, σ σx y k 2 2 + is a good estimate of the variance of the difference of the means. It is this variance (or rather its square root) that should be used as the denominator of the t-statistic given previously. The degrees of freedom, necessary for consulting Student’s confidence tables, should be taken conservatively to be the minimum of the degrees of freedom of the two samples. Essentially, knowing that the observations are paired allows the use of a better estimate for the variance, which will produce tighter confidence bounds. The second observation concerns the assumption that there is essentially unlimited data, so that several independent datasets of the right size can be used. In practice, there is usually only a single dataset of limited size. What can be done? We could split the data into subsets (perhaps 10) and perform a cross-validation on each one. However, the overall result will only tell us whether a learning scheme is preferable for that particular size—one-tenth of the original dataset. Alternatively, the original dataset could be reused—for example, with different randomizations of the dataset for each cross- validation. However, the resulting cross-validation estimates will not be independent 5.6 Predicting Probabilities 159 because they are not based on independent datasets. In practice, this means that a difference may be judged to be significant when in fact it is not. Indeed, just increasing the number of samples k—that is, the number of cross-validation runs—will eventually yield an apparently significant difference because the value of the t-statistic increases without bound. Various modifications of the standard t-test have been proposed to circumvent this problem, all of them heuristic and somewhat lacking in theoretical justification. One that appears to work well in practice is the corrected resampled t-test. Assume for the moment that the repeated holdout method is used instead of cross-validation, repeated k times on different random splits of the same dataset to obtain accuracy estimates for two learning schemes. Each time, n1 instances are used for training and n2 for testing, and differences di are computed from performance on the test data. The corrected resampled t-test uses the modified statistic t d k n n d = + 1 2 1 2σ in exactly the same way as the standard t-statistic. A closer look at the formula shows that its value cannot be increased simply by increasing k. The same modified statistic can be used with repeated cross-validation, which is just a special case of repeated holdout in which the individual test sets for one cross-validation do not overlap. For tenfold cross- validation repeated 10 times, k =100, n2/n1 = 0.1/0.9, and σd 2 is based on 100 differences. Table 5.2 Confidence Limits for Student’s Distribution with 9 Degrees of Freedom Pr[X ≥ z] z 0.1% 4.30 0.5% 3.25 1% 2.82 5% 1.83 10% 1.38 20% 0.88 5.6 PREDICTING PROBABILITIES Throughout this chapter we have tacitly assumed that the goal is to maximize the success rate of the predictions. The outcome for each test instance is either correct, if the prediction agrees with the actual value for that instance, or incorrect, if it does not. There are no grays: Everything is black or white, correct or incorrect. In many situations, this is the most appropriate perspective. If the learning scheme, when it is actually applied, results in either a correct or an incorrect prediction, success is 160 CHAPTER 5 Credibility: Evaluating What’s Been Learned the right measure to use. This is sometimes called a 0 – 1 loss function: The “loss” is either 0 if the prediction is correct or 1 if it is not. The use of loss is conventional, although a more optimistic terminology might couch the outcome in terms of profit instead. Other situations are softer-edged. Most learning schemes can associate a prob- ability with each prediction (as the Naïve Bayes scheme does). It might be more natural to take this probability into account when judging correctness. For example, a correct outcome predicted with a probability of 99% should perhaps weigh more heavily than one predicted with a probability of 51%, and, in a two-class situation, perhaps the latter is not all that much better than an incorrect outcome predicted with probability 51%. Whether it is appropriate to take prediction probabilities into account depends on the application. If the ultimate application really is just a predic- tion of the outcome, and no prizes are awarded for a realistic assessment of the likelihood of the prediction, it does not seem appropriate to use probabilities. If the prediction is subject to further processing, however—perhaps involving assessment by a person, or a cost analysis, or maybe even serving as input to a second-level learning process—then it may well be appropriate to take prediction probabilities into account. Quadratic Loss Function Suppose for a single instance there are k possible outcomes, or classes, and for a given instance the learning scheme comes up with a probability vector p1, p2, …, pk for the classes (where these probabilities sum to 1). The actual outcome for that instance will be one of the possible classes. However, it is convenient to express it as a vector a1, a2, …, ak whose ith component, where i is the actual class, is 1 and all other components are 0. We can express the penalty associated with this situation as a loss function that depends on both the p vector and the a vector. One criterion that is frequently used to evaluate probabilistic prediction is the quadratic loss function: ()p aj jj −∑ 2 Note that this is for a single instance: The summation is over possible outputs, not over different instances. Just one of the a’s will be 1 and the rest 0, so the sum contains contributions of pj 2 for the incorrect predictions and (1– pi)2 for the correct one. Consequently, it can be written as 1 2 2− + ∑p pi jj where i is the correct class. When the test set contains several instances, the loss function is summed over them all. 5.6 Predicting Probabilities 161 It is an interesting theoretical fact that if you seek to minimize the value of the quadratic loss function in a situation where the actual class is generated probabilistically, the best strategy is to choose for the p vector the actual probabilities of the different outcomes— that is, pi = Pr[class = i ]. If the true probabilities are known, they will be the best values for p. If they are not, a system that strives to minimize the quadratic loss function will be encouraged to use its best estimate of Pr[class = i ] as the value for pi. This is quite easy to see. Denote the true probabilities by p1*, p2*, …, pk* so that pi* = Pr[class = i ]. The expected value of the quadratic loss function over test instances can be rewritten as E p a E p E p a E a p p p p j jj j j j jj j j j j ( ) ( [ ] [ ] [ ]) ( − = − + = − + ∑ ∑2 2 2 2 2 2 * *)) (( ) ( )) j j j j jj p p p p ∑ ∑= − + −***2 1 The first stage involves bringing the expectation inside the sum and expanding the square. For the second, pj is just a constant and the expected value of aj is simply pj*; moreover, because aj is either 0 or 1, aj 2 = aj and its expected value is pj* as well. The third stage is straightforward algebra. To minimize the resulting sum, it is clear that it is best to choose pj = pj*, so that the squared term disappears and all that remains is a term that is just the variance of the true distribution governing the actual class. Minimizing the squared error has a long history in prediction problems. In the present context, the quadratic loss function forces the predictor to be honest about choosing its best estimate of the probabilities—or, rather, it gives preference to predictors that are able to make the best guess at the true probabilities. Moreover, the quadratic loss function has some useful theoretical properties that we will not go into here. For all these reasons, it is frequently used as the criterion of success in probabilistic prediction situations. Informational Loss Function Another popular criterion used to evaluate probabilistic prediction is the informa- tional loss function, − log2 pi where the ith prediction is the correct one. This is in fact identical to the negative of the log-likelihood function that is optimized by logistic regression, described in Section 4.6 (modulo a constant factor, which is determined by the base of the loga- rithm). It represents the information (in bits) required to express the actual class i with respect to the probability distribution p1, p2, …, pk. In other words, if you were given the probability distribution and someone had to communicate to you which class was the one that actually occurred, this is the number of bits they would need to encode the information if they did it as effectively as possible. (Of course, it is 162 CHAPTER 5 Credibility: Evaluating What’s Been Learned always possible to use more bits.) Because probabilities are always less than 1, their logarithms are negative, and the minus sign makes the outcome positive. For example, in a two-class situation—heads or tails—with an equal probability of each class, the occurrence of a head would take 1 bit to transmit because −log2 1/2 is 1. The expected value of the informational loss function, if the true probabilities are p1*, p2*, …, pk*, is − − − −p p p p p pk k1 2 1 2 2 2 2***log log log… Like the quadratic loss function, this expression is minimized by choosing pj = pj*, in which case the expression becomes the entropy of the true distribution: − − − −p p p p p pk k1 2 1 2 2 2 2******log log log… Thus, the informational loss function also rewards honesty in predictors that know the true probabilities, and encourages predictors that do not to put forward their best guess. One problem with the informational loss function is that if you assign a probabil- ity of 0 to an event that actually occurs, the function’s value is infinity. This corre- sponds to losing your shirt when gambling. Prudent predictors operating under the informational loss function do not assign zero probability to any outcome. This does lead to a problem when no information is available about that outcome on which to base a prediction. This is called the zero-frequency problem, and various plausible solutions have been proposed, such as the Laplace estimator discussed for Naïve Bayes in Chapter 4 (page 93). Discussion If you are in the business of evaluating predictions of probabilities, which of the two loss functions should you use? That’s a good question, and there is no universally agreed-on answer—it’s really a matter of taste. They both do the fundamental job expected of a loss function: They give maximum reward to predictors that are capable of predicting the true probabilities accurately. However, there are some objective differences between the two that may help you form an opinion. The quadratic loss function takes into account not only the probability assigned to the event that actually occurred but also the other probabilities. For example, in a four-class situation, suppose you assigned 40% to the class that actually came up and distributed the remainder among the other three classes. The quadratic loss will depend on how you distributed it because of the sum of the pj 2 that occurs in the expression given earlier for the quadratic loss function. The loss will be smallest if the 60% was distributed evenly among the three classes: An uneven distribution will increase the sum of the squares. The informational loss function, on the other hand, depends solely on the probability assigned to the class that actually occurred. If 5.7 Counting the Cost 163 you’re gambling on a particular event coming up, and it does, who cares about potential winnings from other events? If you assign a very small probability to the class that actually occurs, the infor- mation loss function will penalize you massively. The maximum penalty, for a zero probability, is infinite. The quadratic loss function, on the other hand, is milder, being bounded by 1 2+ ∑ pjj which can never exceed 2. Finally, proponents of the informational loss function point to a general theory of performance assessment in learning called the minimum description length (MDL) principle. They argue that the size of the structures that a scheme learns can be measured in bits of information, and if the same units are used to measure the loss, the two can be combined in useful and powerful ways. We return to this in Section 5.9. 5.7 COUNTING THE COST The evaluations that have been discussed so far do not take into account the cost of making wrong decisions, wrong classifications. Optimizing the classification rate without considering the cost of the errors often leads to strange results. In one case, machine learning was being used to determine the exact day that each cow in a dairy herd was in estrus, or “in heat.” Cows were identified by electronic ear tags, and various attributes were used such as milk volume and chemical composition (recorded automatically by a high-tech milking machine) and milking order—for cows are regular beasts and generally arrive in the milking shed in the same order, except in unusual circumstances such as estrus. In a modern dairy operation it’s important to know when a cow is ready: Animals are fertilized by artificial insemination and missing a cycle will delay calving unnecessarily, causing complications down the line. In early experiments, machine learning schemes stubbornly predicted that each cow was never in estrus. Like humans, cows have a menstrual cycle of approxi- mately 30 days, so this “null” rule is correct about 97% of the time—an impressive degree of accuracy in any agricultural domain! What was wanted, of course, was rules that predicted the “in estrus” situation more accurately than the “not in estrus” one: The costs of the two kinds of error were different. Evaluation by classification accuracy tacitly assumes equal error costs. Other examples where errors cost different amounts include loan decisions: The cost of lending to a defaulter is far greater than the lost-business cost of refusing a loan to a nondefaulter. And oil-slick detection: The cost of failing to detect an environment-threatening real slick is far greater than the cost of a false alarm. And load forecasting: The cost of gearing up electricity generators for a storm that doesn’t hit is far less than the cost of being caught completely unprepared. And diagnosis: 164 CHAPTER 5 Credibility: Evaluating What’s Been Learned Table 5.3 Different Outcomes of a Two-Class Prediction Predicted Class yes no Actual Class yes true positive false negative no false positive true negative The cost of misidentifying problems with a machine that turns out to be free of faults is less than the cost of overlooking problems with one that is about to fail. And promotional mailing: The cost of sending junk mail to a household that doesn’t respond is far less than the lost-business cost of not sending it to a household that would have responded. Why—these are all the examples from Chapter 1! In truth, you’d be hard pressed to find an application in which the costs of different kinds of errors were the same. In the two-class case with classes yes and no—lend or not lend, mark a suspicious patch as an oil slick or not, and so on—a single prediction has the four different possible outcomes shown in Table 5.3. The true positives (TP) and true negatives (TN) are correct classifications. A false positive (FP) is when the outcome is incor- rectly predicted as yes (or positive) when it is actually no (negative). A false negative (FN) is when the outcome is incorrectly predicted as negative when it is actually positive. The true positive rate is TP divided by the total number of positives, which is TP + FN; the false positive rate is FP divided by the total number of negatives, which is FP + TN. The overall success rate is the number of correct classifications divided by the total number of classifications: TP TN TP TN FP FN + + + + Finally, the error rate is 1 minus this. In multiclass prediction, the result on a test set is often displayed as a two- dimensional confusion matrix with a row and column for each class. Each matrix element shows the number of test examples for which the actual class is the row and the predicted class is the column. Good results correspond to large numbers down the main diagonal and small, ideally zero, off-diagonal elements. Table 5.4(a) shows a numeric example with three classes. In this case, the test set has 200 instances (the sum of the nine numbers in the matrix), and 88 + 40 + 12 = 140 of them are predicted correctly, so the success rate is 70%. But is this a fair measure of overall success? How many agreements would you expect by chance? This predictor predicts a total of 120 a’s, 60 b’s, and 20 c’s; what if you had a random predictor that predicted the same total numbers of the three classes? The answer is shown in Table 5.4(b). Its first row divides the 100 a’s in the test set into these overall proportions, and the second and third rows do the same 165 Table 5.4 Different Outcomes of a Three-Class Prediction: (a) Actual and (b) Expected Predicted Class Predicted Class a b c Total a b c Total Actual Class a 88 10 2 100 Actual Class a 60 30 10 100 b 14 40 6 60 b 36 18 6 60 c 18 10 12 40 c 24 12 4 40 Total 120 60 20 Total 120 60 20 (a) (b) 166 CHAPTER 5 Credibility: Evaluating What’s Been Learned thing for the other two classes. Of course, the row and column totals for this matrix are the same as before—the number of instances hasn’t changed, and we have ensured that the random predictor predicts the same number of a’s, b’s, and c’s as the actual predictor. This random predictor gets 60 + 18 + 4 = 82 instances correct. A measure called the Kappa statistic takes this expected figure into account by deducting it from the predictor’s successes and expressing the result as a proportion of the total for a perfect predictor, to yield 140 – 82 = 58 extra successes out of a possible total of 200 – 82 = 118, or 49.2%. The maximum value of Kappa is 100%, and the expected value for a random predictor with the same column totals is 0. In summary, the Kappa statistic is used to measure the agreement between predicted and observed categorizations of a dataset, while correcting for an agreement that occurs by chance. However, like the plain success rate, it does not take costs into account. Cost-Sensitive Classification If the costs are known, they can be incorporated into a financial analysis of the decision-making process. In the two-class case, in which the confusion matrix is like that of Table 5.3, the two kinds of error—false positives and false negatives—will have different costs; likewise, the two types of correct classification may have different benefits. In the two-class case, costs can be summarized in the form of a 2 × 2 matrix in which the diagonal elements represent the two types of correct clas- sification and the off-diagonal elements represent the two types of error. In the multiclass case this generalizes to a square matrix whose size is the number of classes, and again the diagonal elements represent the cost of correct classification. Table 5.5(a) and (b) shows default cost matrixes for the two- and three-class cases, whose values simply give the number of errors: Misclassification costs are all 1. Taking the cost matrix into account replaces the success rate by the average cost (or, thinking more positively, profit) per decision. Although we will not do so here, a complete financial analysis of the decision-making process might also take into account the cost of using the machine learning tool—including the cost of gathering the training data—and the cost of using the model, or decision structure, that it Table 5.5 Default Cost Matrixes: (a) Two-Class Case and (b) Three-Class Case Predicted Class Predicted Class yes no a b c Actual Class yes 0 1 Actual Class a 0 1 1 no 1 0 b 1 0 1 c 1 1 0 (a) (b) 5.7 Counting the Cost 167 produces—including the cost of determining the attributes for the test instances. If all costs are known, and the projected number of the four different outcomes in the cost matrix can be estimated, say using cross-validation, it is straightforward to perform this kind of financial analysis. Given a cost matrix, you can calculate the cost of a particular learned model on a given test set just by summing the relevant elements of the cost matrix for the model’s prediction for each test instance. Here, costs are ignored when making predictions, but taken into account when evaluating them. If the model outputs the probability associated with each prediction, it can be adjusted to minimize the expected cost of the predictions. Given a set of predicted probabilities for each outcome on a certain test instance, one normally selects the most likely outcome. Instead, the model could predict the class with the smallest expected misclassification cost. For example, suppose in a three-class situation the model assigns the classes a, b, and c to a test instance with probabilities pa, pb, and pc, and the cost matrix is that in Table 5.5(b). If it predicts a, the expected cost of the prediction is obtained by multiplying the first column of the matrix, [0,1,1], by the probability vector, [pa, pb, pc], yielding pb + pc , or 1 – pa , because the three probabilities sum to 1. Similarly, the costs for predicting the other two classes are 1 – pb and 1 – pc. For this cost matrix, choosing the prediction with the lowest expected cost is the same as choosing the one with the greatest probability. For a different cost matrix it might be different. We have assumed that the learning scheme outputs probabilities, as Naïve Bayes does. Even if they do not normally output probabilities, most classifiers can easily be adapted to compute them. In a decision tree, for example, the probability distribu- tion for a test instance is just the distribution of classes at the corresponding leaf. Cost-Sensitive Learning We have seen how a classifier, built without taking costs into consideration, can be used to make predictions that are sensitive to the cost matrix. In this case, costs are ignored at training time but used at prediction time. An alternative is to do just the opposite: Take the cost matrix into account during the training process and ignore costs at prediction time. In principle, better performance might be obtained if the classifier were tailored by the learning algorithm to the cost matrix. In the two-class situation, there is a simple and general way to make any learning scheme cost sensitive. The idea is to generate training data with a different propor- tion of yes and no instances. Suppose you artificially increase the number of no instances by a factor of 10 and use the resulting dataset for training. If the learning scheme is striving to minimize the number of errors, it will come up with a decision structure that is biased toward avoiding errors on the no instances because such errors are effectively penalized tenfold. If data with the original proportion of no instances is used for testing, fewer errors will be made on these than on yes instances—that is, there will be fewer false positives than false negatives—because false positives have been weighted 10 times more heavily than false negatives. 168 CHAPTER 5 Credibility: Evaluating What’s Been Learned Varying the proportion of instances in the training set is a general technique for building cost-sensitive classifiers. One way to vary the proportion of training instances is to duplicate instances in the dataset. However, many learning schemes allow instances to be weighted. (As we mentioned in Section 3.2, this is a common technique for handling missing values.) Instance weights are normally initialized to 1. To build cost-sensitive clas- sifiers the weights can be initialized to the relative cost of the two kinds of error, false positives and false negatives. Lift Charts In practice, costs are rarely known with any degree of accuracy, and people will want to ponder various different scenarios. Imagine you’re in the direct-mailing business and are contemplating a mass mailout of a promotional offer to 1,000,000 households, most of whom won’t respond, of course. Let us say that, based on previ- ous experience, the proportion that normally respond is known to be 0.1% (1000 respondents). Suppose a data mining tool is available that, based on known informa- tion about the households, identifies a subset of 100,000 for which the response rate is 0.4% (400 respondents). It may well pay off to restrict the mailout to these 100,000 households; this, of course, depends on the mailing cost compared with the return gained for each response to the offer. In marketing terminology, the increase in response rate, a factor of 4 in this case, is known as the lift factor yielded by the learning tool. If you knew the costs, you could determine the payoff implied by a particular lift factor. But you probably want to evaluate other possibilities too. The same data mining scheme, with different parameter settings, may be able to identify 400,000 house- holds for which the response rate will be 0.2% (800 respondents), corresponding to a lift factor of 2. Again, whether this would be a more profitable target for the mailout can be calculated from the costs involved. It may be necessary to factor in the cost of creating and using the model, including collecting the infor- mation that is required to come up with the attribute values. After all, if developing the model is very expensive, a mass mailing may be more cost effective than a targeted one. Given a learning scheme that outputs probabilities for the predicted class of each member of the set of test instances (as Naïve Bayes does), your job is to find subsets of test instances that have a high proportion of positive instances, higher than in the test set as a whole. To do this, the instances should be sorted in descending order of predicted probability of yes. Then, to find a sample of a given size with the greatest possible proportion of positive instances, just read the requisite number of instances off the list, starting at the top. If each test instance’s class is known, you can calculate the lift factor by simply counting the number of positive instances that the sample includes, dividing by the sample size to obtain a success proportion, and dividing by the success proportion for the complete test set to determine the lift factor. 5.7 Counting the Cost 169 Table 5.6 Data for a Lift Chart Rank Predicted Actual Class 1 0.95 yes 2 0.93 yes 3 0.93 no 4 0.88 yes 5 0.86 yes 6 0.85 yes 7 0.82 yes 8 0.80 yes 9 0.80 no 10 0.79 yes 11 0.77 no 12 0.76 yes 13 0.73 yes 14 0.65 no 15 0.63 yes 16 0.58 no 17 0.56 yes 18 0.49 no 19 0.48 yes … … … and not the actual classes, your best bet would be the top 10 ranking instances. Eight of these are positive, so the success proportion for this sample is 80%, corresponding to a lift factor of about 2.4. If you knew the different costs involved, you could work them out for each sample size and choose the most profitable. But a graphical depiction of the various possibili- ties will often be far more revealing than presenting a single “optimal” decision. Repeating the operation for different-size samples allows you to plot a lift chart like that of Figure 5.1. The horizontal axis shows the sample size as a proportion of the total possible mailout. The vertical axis shows the number of responses obtained. The lower left and upper right points correspond to no mailout at all, with a response of 0, and a full mailout, with a response of 1000. The diagonal line gives the expected result for different-size random samples. But we do not choose random samples; we choose those instances that, according to the data mining tool, are most likely to generate a positive response. These correspond to the upper line, which is derived by summing the actual responses over the corresponding percentage of the instance list sorted in probability order. The two particular scenarios described previously are marked: a 10% mailout that yields 400 respondents and a 40% one that yields 800. Where you’d like to be in a lift chart is near the upper left corner: At the very best, 1000 responses from a mailout of just 1000, where you send only to those Table 5.6 shows an example, for a small dataset that has 150 instances, of which 50 are yes responses—an overall success proportion of 33%. The instances have been sorted in descending probability order according to the predicted probability of a yes response. The first instance is the one that the learning scheme thinks is the most likely to be positive, the second is the next most likely, and so on. The numeric values of the probabili- ties are unimportant: Rank is the only thing that matters. With each rank is given the actual class of the instance. Thus, the learning scheme was correct about items 1 and 2—they are indeed positives— but wrong about item 3, which turned out to be negative. Now, if you were seeking the most prom- ising sample of size 10, but only knew the predicted probabilities 170 CHAPTER 5 Credibility: Evaluating What’s Been Learned FIGURE 5.1 A hypothetical lift chart. 0 200 400 600 800 1000 0 20 40 60 80 100 Sample Size (%) Number of Respondents households that will respond and are rewarded with a 100% success rate. Any selec- tion procedure worthy of the name will keep you above the diagonal—otherwise, you’d be seeing a response that is worse than for random sampling. So the operating part of the diagram is the upper triangle, and the farther to the upper left the better. Figure 5.2(a) shows a visualization that allows various cost scenarios to be explored in an interactive fashion (called the cost–benefit analyzer, it forms part of the Weka workbench described in Part III). Here it is displaying results for predictions generated by the Naïve Bayes classifier on a real-world direct- mail data set. In this example, 47,706 instances were used for training and a further 47,706 for testing. The test instances were ranked according to the predicted probability of a response to the mailout. The graphs show a lift chart on the left and the total cost (or benefit), plotted against the sample size, on the right. At the lower left is a confusion matrix; at the lower right is a cost matrix. Cost or benefit values associated with incorrect or correct classifications can be entered into the matrix and affect the shape of the curve above. The horizontal slider in the middle allows users to vary the percentage of the population that is selected from the ranked list. Alternatively, one can determine the sample size by adjusting the recall level (the proportion of positives to be included in the sample) or by adjusting a threshold on the probability of the positive class, which here corresponds to a response to the mailout. When the slider is moved, a large cross shows the cor- responding point on both graphs. The total cost or benefit associated with the selected sample size is shown at the lower right, along with the expected response to a random mailout of the same size. 5.7 Counting the Cost 171 FIGURE 5.2 Analyzing the expected benefit of a mailing campaign when the cost of mailing is (a) $0.50 and (b) $0.80. (a) (b) 172 CHAPTER 5 Credibility: Evaluating What’s Been Learned In the cost matrix in Figure 5.2(a), a cost of $0.50—the cost of mailing—has been associated with nonrespondents and a benefit of $15.00 with respondents (after deducting the mailing cost). Under these conditions, and using the Naïve Bayes classifier, there is no subset from the ranked list of prospects that yields a greater profit than mailing to the entire population. However, a slightly higher mailing cost changes the situation dramatically, and Figure 5.2(b) shows what happens when it is increased to $0.80. Assuming the same profit of $15.00 per respondent, a maximum profit of $4,560.60 is achieved by mailing to the top 46.7% of the population. In this situation, a random sample of the same size achieves a loss of $99.59. ROC Curves Lift charts are a valuable tool, widely used in marketing. They are closely related to a graphical technique for evaluating data mining schemes known as ROC curves, which are used in just the same situation, where the learner is trying to select samples of test instances that have a high proportion of positives. The acronym stands for receiver operating characteristic, a term used in signal detection to characterize the tradeoff between hit rate and false-alarm rate over a noisy channel. ROC curves depict the performance of a classifier without regard to class distribution or error costs. They plot the true positive rate on the vertical axis against the true negative rate on the horizontal axis. The former is the number of positives included in the sample, expressed as a percentage of the total number of positives (TP Rate = 100 × TP/(TP + FN)); the latter is the number of negatives included in the sample, expressed as a percentage of the total number of negatives (FP Rate = 100 × FP/(FP + TN)). The vertical axis is the same as the lift chart’s except that it is expressed as a percentage. The horizontal axis is slightly different—it is the number of negatives rather than the sample size. However, in direct marketing situ- ations where the proportion of positives is very small anyway (like 0.1%), there is negligible difference between the size of a sample and the number of negatives it contains, so the ROC curve and lift chart look very similar. As with lift charts, the upper left corner is the place to be. Figure 5.3 shows an example ROC curve—the jagged line—for the sample of test data shown earlier in Table 5.6. You can follow it along with the table. From the origin: Go up two (two positives), along one (one negative), up five (five posi- tives), along two (two negatives), up one, along one, up two, and so on. Each point corresponds to drawing a line at a certain position on the ranked list, counting the yes’s and no’s above it, and plotting them vertically and horizontally, respectively. As you go farther down the list, corresponding to a larger sample, the number of positives and negatives both increase. The jagged ROC line in Figure 5.3 depends intimately on the details of the par- ticular sample of test data. This sample dependence can be reduced by applying cross-validation. For each different number of no’s—that is, each position along the horizontal axis—take just enough of the highest-ranked instances to include that number of no’s, and count the number of yes’s they contain. Finally, average that 5.7 Counting the Cost 173 FIGURE 5.3 A sample ROC curve. 0 20 40 60 80 100 0 20 40 60 80 100 False Positives (%) True Positives (%) number over different folds of the cross-validation. The result is a smooth curve like that in Figure 5.3—although in reality such curves do not generally look quite so smooth. This is just one way of using cross-validation to generate ROC curves. A simpler approach is to collect the predicted probabilities for all the various test sets (of which there are 10 in a tenfold cross-validation), along with the true class labels of the corresponding instances, and generate a single ranked list based on this data. This assumes that the probability estimates from the classifiers built from the different training sets are all based on equally sized random samples of the data. It is not clear which method is preferable. However, the latter method is easier to implement. If the learning scheme does not allow the instances to be ordered, you can first make it cost-sensitive as described earlier. For each fold of a tenfold cross-validation, weight the instances for a selection of different cost ratios, train the scheme on each weighted set, count the true positives and false positives in the test set, and plot the resulting point on the ROC axes. (It doesn’t matter whether the test set is weighted or not because the axes in the ROC diagram are expressed as the percentage of true and false positives.) However, for probabilistic classifiers such as Naïve Bayes it is far more costly than the method described previously because it involves a separate learning problem for every point on the curve. It is instructive to look at ROC curves obtained using different learning schemes. For example, in Figure 5.4, method A excels if a small, focused sample is sought— that is, if you are working toward the left side of the graph. Clearly, if you aim to cover just 40% of the true positives you should choose method A, which gives a false positive rate of around 5%, rather than method B, which gives more than 20% 174 CHAPTER 5 Credibility: Evaluating What’s Been Learned FIGURE 5.4 ROC curves for two learning schemes. 0 20 40 60 80 100 0 20 40 60 80 100 False Positives (%) True Positives (%) A B false positives. But method B excels if you are planning a large sample: If you are covering 80% of the true positives, B will give a false positive rate of 60% as com- pared with method A’s 80%. The shaded area is called the convex hull of the two curves, and you should always operate at a point that lies on the upper boundary of the convex hull. What about the region in the middle where neither method A nor method B lies on the convex hull? It is a remarkable fact that you can get anywhere in the shaded region by combining methods A and B and using them at random with appropriate probabilities. To see this, choose a particular probability cutoff for method A that gives true and false positive rates of tA and fA, respectively, and another cutoff for method B that gives tB and fB. If you use these two schemes at random with prob- abilities p and q, where p + q = 1, then you will get true and false positive rates of p . tA + q . tB and p . fA + q . fB. This represents a point lying on the straight line joining the points (tA, fA) and (tB, fB), and by varying p and q you can trace out the whole line between these two points. By this device, the entire shaded region can be reached. Only if a particular scheme generates a point that lies on the convex hull should it be used alone. Otherwise, it would always be better to use a combi- nation of classifiers corresponding to a point that lies on the convex hull. Recall–Precision Curves People have grappled with the fundamental tradeoff illustrated by lift charts and ROC curves in a wide variety of domains. Information retrieval is a good example. Given a query, a Web search engine produces a list of hits that represent documents supposedly relevant to the query. Compare one system that locates 100 documents, 40 of which are relevant, with another that locates 400 documents, 80 of which are relevant. Which is better? The answer should now be obvious: It depends on the relative cost of false positives, documents returned that aren’t relevant, and false negatives, documents that are relevant but aren’t returned. Information retrieval researchers define parameters called recall and precision: Recall number of documents retrieved that are relevant tota= ll number of documents that are relevant Precision number of documents retrieved that are relevant t= ootal number of documents that are retrieved For example, if the list of yes’s and no’s in Table 5.6 represented a ranked list of retrieved documents and whether they were relevant or not, and the entire col- lection contained a total of 40 relevant documents, then “recall at 10” would refer to the recall for the top 10 documents—that is, 8/40 = 20%—while “precision at 10” would be 8/10 = 80%. Information retrieval experts use recall–precision curves that plot one against the other, for different numbers of retrieved documents, in just the same way as ROC curves and lift charts—except that, because the axes are dif- ferent, the curves are hyperbolic in shape and the desired operating point is toward the upper right. Discussion Table 5.7 summarizes the three different ways introduced for evaluating the same basic tradeoff; TP, FP, TN, and FN are the numbers of true positives, false positives, true negatives, and false negatives, respectively. You want to choose a set of instances with a high proportion of yes instances and a high coverage of the yes instances: You can increase the proportion by (conservatively) using a smaller coverage, or (liberally) increase the coverage at the expense of the proportion. Different tech- niques give different tradeoffs, and can be plotted as different lines on any of these graphical charts. People also seek single measures that characterize performance. Two that are used in information retrieval are three-point average recall, which gives the average precision obtained at recall values of 20%, 50%, and 80%, and 11-point average recall, which gives the average precision obtained at recall values of 0%, 10%, 20%, 30%, 40%, 50%, 60%, 70%, 80%, 90%, and 100%. Also used in information retrieval is the F-measure, which is 2 2 2 × × + = × × + + recall precision recall precision TP TP FP FN Different terms are used in different domains. Physicians, for example, talk about the sensitivity and specificity of diagnostic tests. Sensitivity refers to the proportion 5.7 Counting the Cost 175 176 Table 5.7 Different Measures Used to Evaluate False Positive versus False Negative Tradeoff Domain Plot Axes Explanation of Axes Lift chart Marketing TP vs. subset size TP subset size number of true positives TP FP TP FP TN FN + + + + × 100% ROC curve Communications TP rate vs. FP rate TP rate FP rate tp = + × TP TP FN 100% fp = + × FP FP TN 100% Recall–precision curve Information retrieval Recall vs. precision Recall precision same as TP rate of tp above TP TP FP + × 100% of people with disease who have a positive test result—that is, tp. Specificity refers to the proportion of people without disease who have a negative test result, which is 1 – fp. Sometimes the product of these is used as an overall measure: sensitivity specificity TP TN TP FN FP TN× = − = × + × + tp fp()()()1 Finally, of course, there is our old friend the success rate: TP TN TP FP TN FN + + + + To summarize ROC curves in a single quantity, people sometimes use the area under the curve (AUC) because, roughly speaking, the larger the area the better the model. The area also has a nice interpretation as the probability that the classifier ranks a randomly chosen positive instance above a randomly chosen negative one. Although such measures may be useful if costs and class distributions are unknown and one scheme must be chosen to handle all situations, no single number is able to capture the tradeoff. That can only be done by two-dimensional depictions such as lift charts, ROC curves, and recall–precision diagrams. Several methods are commonly employed for computing the area under the ROC curve. One, corresponding to a geometric interpretation, is to approximate it by fitting several trapezoids under the curve and summing up their area. Another is to compute the probability that the classifier ranks a randomly chosen positive instance above a randomly chosen negative one. This can be accomplished by calculating the Mann–Whitney U statistic, or, more specifically, the ρ statistic from the U statistic. This value is easily obtained from a list of test instances sorted in descending order of predicted probability of the positive class. For each positive instance, count how many negative ones are ranked below it (increase the count by 12 if positive and negative instances tie in rank). The U statistic is simply the total of these counts. The ρ statistic is obtained by dividing U by the product of the number of positive and negative instances in the test set—in other words, the U value that would result if all positive instances were ranked above the negative ones. The area under the precision–recall curve (AUPRC) is an alternative summary statistic that is preferred by some practitioners, particularly in the information retrieval area. Cost Curves ROC curves and their relatives are very useful for exploring the tradeoffs among different classifiers over a range of scenarios. However, they are not ideal for evalu- ating machine learning models in situations with known error costs. For example, it is not easy to read off the expected cost of a classifier for a fixed cost matrix and class distribution. Neither can you easily determine the ranges of applicability of different classifiers. For example, from the crossover point between the two ROC 5.7 Counting the Cost 177 178 CHAPTER 5 Credibility: Evaluating What’s Been Learned (a) Probability, p [+] Expected Error Always wrong Always pick + Always pick – 1 0.5 fn fp 0 0 0.5 Always right 1 Probability Cost Function, pc [+] (b) 0.5 0.25 0 0 0.5 fn fp 1 A A B Normalized Expected Cost curves in Figure 5.4 it is hard to tell for what cost and class distributions classifier A outperforms classifier B. Cost curves are a different kind of display on which a single classifier corre- sponds to a straight line that shows how the performance varies as the class distribu- tion changes. Again, they work best in the two-class case, although you can always make a multiclass problem into a two-class one by singling out one class and evalu- ating it against the remaining ones. Figure 5.5(a) plots the expected error against the probability of one of the classes. You could imagine adjusting this probability by resampling the test set in a non uniform way. We denote the two classes by + and –. The diagonals show the per- formance of two extreme classifiers: One always predicts +, giving an expected error of 1 if the dataset contains no + instances and 0 if all its instances are +; the other always predicts –, giving the opposite performance. The dashed horizontal line shows the performance of the classifier that is always wrong, and the x-axis itself represents the classifier that is always correct. In practice, of course, neither of these is realizable. Good classifiers have low error rates, so where you want to be is as close to the bottom of the diagram as possible. The line marked A represents the error rate of a particular classifier. If you cal- culate its performance on a certain test set, its false positive rate, fp, is its expected error on a subsample of the test set that contains only examples that are negative (p[+] = 0), and its false negative rate, fn, is the error on a subsample that contains only positive examples, (p[+] = 1). These are the values of the intercepts at the left and right, respectively. You can see immediately from the plot that if p[+] is smaller than about 0.2, predictor A is outperformed by the extreme classifier that always predicts –, while if it is larger than about 0.65, the other extreme classifier is better. FIGURE 5.5 Effect of varying the probability threshold: (a) error curve and (b) cost curve. So far we have not taken costs into account, or rather we have used the default cost matrix in which all errors cost the same. Cost curves, which do take cost into account, look very similar—very similar indeed—but the axes are different. Figure 5.5(b) shows a cost curve for the same classifier A (note that the vertical scale has been enlarged, for convenience, and ignore the gray lines for now). It plots the expected cost of using A against the probability cost function, which is a distorted version of p[+] that retains the same extremes: 0 when p[+] = 0 and 1 when p[+] = 1. Denote by C[+ | –] the cost of predicting + when the instance is actually –, and the reverse by C[– | +]. Then the axes of Figure 5.5(b) are Normalized expected cost = × + + × − +fn p fp pC C[ ] ( [ ])1 Probability cost function p p C p C p CC [][ ] [ | ] [ ] [ | ] [ ] [+ = + − + + − + + − ++−| ] We are assuming here that correct predictions have no cost: C[+ | +] = C[– | –] = 0. If that is not the case, the formulas are a little more complex. The maximum value that the normalized expected cost can have is 1—that is why it is “normalized.” One nice thing about cost curves is that the extreme cost values at the left and right sides of the graph are fp and fn, just as they are for the error curve, so you can draw the cost curve for any classifier very easily. Figure 5.5(b) also shows classifier B, whose expected cost remains the same across the range—that is, its false positive and false negative rates are equal. As you can see, it outperforms classifier A if the probability cost function exceeds about 0.45, and knowing the costs we could easily work out what this corresponds to in terms of class distribution. In situations that involve different class distributions, cost curves make it easy to tell when one classifier will outperform another. In what circumstances might this be useful? To return to our example of predict- ing when cows will be in estrus, their 30-day cycle, or 1/30 prior probability, is unlikely to vary greatly (barring a genetic cataclysm!). But a particular herd may have different proportions of cows that are likely to reach estrus in any given week, perhaps synchronized with—who knows?—the phase of the moon. Then, different classifiers would be appropriate at different times. In the oil spill example, different batches of data may have different spill probabilities. In these situations cost curves can help to show which classifier to use when. Each point on a lift chart, ROC curve, or recall–precision curve represents a classifier, typically obtained by using different threshold values for a method such as Naïve Bayes. Cost curves represent each classifier by a straight line, and a suite of classifiers will sweep out a curved envelope whose lower limit shows how well that type of classifier can do if the parameter is well chosen. Figure 5.5(b) indicates this with a few gray lines. If the process were continued, it would sweep out the dotted parabolic curve. The operating region of classifier B ranges from a probability cost value of about 0.25 to a value of about 0.75. Outside this region, classifier B is outperformed by the trivial classifiers represented by dashed lines. Suppose we decide to use classifier 5.7 Counting the Cost 179 180 CHAPTER 5 Credibility: Evaluating What’s Been Learned B within this range and the appropriate trivial classifier below and above it. All points on the parabola are certainly better than this scheme. But how much better? It is hard to answer such questions from an ROC curve, but the cost curve makes them easy. The performance difference is negligible if the probability cost value is around 0.5, and below a value of about 0.2 and above 0.8 it is barely perceptible. The greatest difference occurs at probability cost values of 0.25 and 0.75 and is about 0.04, or 4% of the maximum possible cost figure. 5.8 EVALUATING NUMERIC PREDICTION All the evaluation measures we have described pertain to classification situations rather than numeric prediction situations. The basic principles—using an indepen- dent test set rather than the training set for performance evaluation, the holdout method, cross-validation—apply equally well to numeric prediction. But the basic quality measure offered by the error rate is no longer appropriate: Errors are not simply present or absent; they come in different sizes. Several alternative measures, some of which are summarized in Table 5.8, can be used to evaluate the success of numeric prediction. The predicted values on the test instances are p1, p2, …, pn; the actual values are a1, a2, …, an. Notice that pi means Mean-squared error ()()p a p a n n n1 1 2 2− + … + − Root mean-squared error ()()p a p a n n n1 1 2 2− + … + − Mean-absolute error p a p a n n n1 1− + … + − Relative-squared error* ()() ()() p a p a a a a a n n n 1 1 2 2 1 2 2 − + … + − − + … + − Root relative-squared error* ()() ()() p a p a a a a a n n n 1 1 2 2 1 2 2 − + … + − − + … + − Relative-absolute error* p a p a a a a a n n n 1 1 1 − + … + − − + … + − Correlation coefficient** S SS PA PA , where S p p a a nPA i ii= − − − ∑ ( )( ) 1 , S p p nP ii= − − ∑ ()2 1 , S a a nA ii= − − ∑ ()2 1 *Here, a is the mean value over the training data. **Here, a is the mean value over the test data. Table 5.8 Performance Measures for Numeric Prediction 5.8 Evaluating Numeric Prediction 181 something very different here from what it meant in the last section: There it was the probability that a particular prediction was in the ith class; here it refers to the numerical value of the prediction for the ith test instance. Mean-squared error is the principal and most commonly used measure; some- times the square root is taken to give it the same dimensions as the predicted value itself. Many mathematical techniques (such as linear regression, explained in Chapter 4) use the mean-squared error because it tends to be the easiest measure to manipu- late mathematically: It is, as mathematicians say, “well behaved.” However, here we are considering it as a performance measure: All the performance measures are easy to calculate, so mean-squared error has no particular advantage. The question is, is it an appropriate measure for the task at hand? Mean absolute error is an alternative: Just average the magnitude of the indi- vidual errors without taking account of their sign. Mean-squared error tends to exaggerate the effect of outliers—instances when the prediction error is larger than the others—but absolute error does not have this effect: All sizes of error are treated evenly according to their magnitude. Sometimes it is the relative rather than absolute error values that are of impor- tance. For example, if a 10% error is equally important whether it is an error of 50 in a prediction of 500 or an error of 0.2 in a prediction of 2, then averages of absolute error will be meaningless—relative errors are appropriate. This effect would be taken into account by using the relative errors in the mean-squared error calculation or the mean absolute error calculation. Relative squared error in Table 5.8 refers to something quite different. The error is made relative to what it would have been if a simple predictor had been used. The simple predictor in question is just the average of the actual values from the training data, denoted by a. Thus, relative squared error takes the total squared error and normalizes it by dividing by the total squared error of the default predictor. The root relative squared error is obtained in the obvious way. The next error measure goes by the glorious name of relative absolute error and is just the total absolute error, with the same kind of normalization. In these three relative error measures, the errors are normalized by the error of the simple predictor that predicts average values. The final measure in Table 5.8 is the correlation coefficient, which measures the statistical correlation between the a’s and the p’s. The correlation coefficient ranges from 1 for perfectly correlated results, through 0 when there is no correlation, to –1 when the results are perfectly correlated negatively. Of course, negative values should not occur for reasonable prediction methods. Correlation is slightly different from the other measures because it is scale independent in that, if you take a particu- lar set of predictions, the error is unchanged if all the predictions are multiplied by a constant factor and the actual values are left unchanged. This factor appears in every term of SPA in the numerator and in every term of SP in the denominator, thus canceling out. (This is not true for the relative error figures, despite normalization: If you multiply all the predictions by a large constant, then the difference between the predicted and actual values will change dramatically, as will the percentage 182 CHAPTER 5 Credibility: Evaluating What’s Been Learned Table 5.9 Performance Measures for Four Numeric Prediction Models A B C D Root mean-squared error 67.8 91.7 63.3 57.4 Mean absolute error 41.3 38.5 33.4 29.2 Root relative squared error 42.2% 57.2% 39.4% 35.8% Relative absolute error 43.1% 40.1% 34.8% 30.4% Correlation coefficient 0.88 0.88 0.89 0.91 errors.) It is also different in that good performance leads to a large value of the correlation coefficient, whereas because the other methods measure error, good performance is indicated by small values. Which of these measures is appropriate in any given situation is a matter that can only be determined by studying the application itself. What are we trying to minimize? What is the cost of different kinds of error? Often it is not easy to decide. The squared error measures and root-squared error measures weigh large discrepancies much more heavily than small ones, whereas the abso- lute error measures do not. Taking the square root (root mean-squared error) just reduces the figure to have the same dimensionality as the quantity being predicted. The relative error figures try to compensate for the basic predictability or unpre- dictability of the output variable: If it tends to lie fairly close to its average value, then you expect prediction to be good and the relative figure compensates for this. Otherwise, if the error figure in one situation is far greater than in another situation, it may be because the quantity in the first situation is inher- ently more variable and therefore harder to predict, not because the predictor is any worse. Fortunately, it turns out that in most practical situations the best numerical prediction method is still the best no matter which error measure is used. For example, Table 5.9 shows the result of four different numeric prediction techniques on a given dataset, measured using cross-validation. Method D is the best accord- ing to all five metrics: It has the smallest value for each error measure and the largest correlation coefficient. Method C is the second best by all five metrics. The performance of A and B is open to dispute: They have the same correlation coefficient; A is better than B according to mean-squared and relative squared errors, and the reverse is true for absolute and relative absolute error. It is likely that the extra emphasis that the squaring operation gives to outliers accounts for the differences in this case. When comparing two different learning schemes that involve numeric prediction, the methodology developed in Section 5.5 still applies. The only difference is that success rate is replaced by the appropriate performance measure (e.g., root mean- squared error) when performing the significance test. 5.9 Minimum Description Length Principle 183 5.9 MINIMUM DESCRIPTION LENGTH PRINCIPLE What is learned by a machine learning scheme is a kind of “theory” of the domain from which the examples are drawn, a theory that is predictive in that it is capable of generating new facts about the domain—in other words, the class of unseen instances. Theory is rather a grandiose term: We are using it here only in the sense of a predictive model. Thus, theories might comprise decision trees or sets of rules—they don’t have to be any more “theoretical” than that. There is a long-standing tradition in science that, other things being equal, simple theories are preferable to complex ones. This is known as Occam’s Razor after the medieval philosopher William of Occam (or Ockham). Occam’s Razor shaves philosophical hairs off a theory. The idea is that the best scientific theory is the smallest one that explains all the facts. As Einstein is reputed to have said, “Everything should be made as simple as possible, but no simpler.” Of course, quite a lot is hidden in the phrase “other things being equal,” and it can be hard to assess objectively whether a particular theory really does “explain” all the facts on which it is based—that’s what controversy in science is all about. In our case, in machine learning, most theories make errors. And if what is learned is a theory, then the errors it makes are like exceptions to the theory. One way to ensure that other things are equal is to insist that the information embodied in the exceptions is included as part of the theory when its “simplicity” is judged. Imagine an imperfect theory for which there are a few exceptions. Not all the data is explained by the theory, but most is. What we do is simply adjoin the exceptions to the theory, specifying them explicitly as exceptions. This new theory is larger: That is a price that, quite justifiably, has to be paid for its inability to explain all the data. However, it may be that the simplicity—is it too much to call it elegance?—of the original theory is sufficient to outweigh the fact that it does not quite explain everything compared with a large, baroque theory that is more comprehensive and accurate. For example, even though Kepler’s three laws of planetary motion did not at the time account for the known data quite so well as Copernicus’ latest refinement of the Ptolemaic theory of epicycles, they had the advantage of being far less complex, and that would have justified any slight apparent inaccuracy. Kepler was well aware of the benefits of having a theory that was compact, despite the fact that his theory violated his own aesthetic sense because it depended on “ovals” rather than pure circular motion. He expressed this in a forceful metaphor: “I have cleared the Augean stables of astronomy of cycles and spirals, and left behind me only a single cartload of dung.” The minimum description length, or MDL, principle takes the stance that the best theory for a body of data is one that minimizes the size of the theory plus the amount of information necessary to specify the exceptions relative to the theory—the small- est “cartload of dung.” In statistical estimation theory, this has been applied success- fully to various parameter-fitting problems. It applies to machine learning as follows: Given a set of instances, a learning scheme infers a theory—be it ever so simple; 184 CHAPTER 5 Credibility: Evaluating What’s Been Learned unworthy, perhaps, to be called a “theory”—from them. Using a metaphor of com- munication, imagine that the instances are to be transmitted through a noiseless channel. Any similarity that is detected among them can be exploited to give a more compact coding. According to the MDL principle, the best theory is the one that minimizes the number of bits required to communicate the theory, along with the labels of the examples from which it was made. Now the connection with the informational loss function introduced in Section 5.6 should be starting to emerge. That function measures the error in terms of the number of bits required to transmit the instances’ class labels, given the probabi- listic predictions made by the theory. According to the MDL principle, we need to add to this the “size” of the theory in bits, suitably encoded, to obtain an overall figure for complexity. However, the MDL principle refers to the information required to transmit the examples from which the theory was formed—that is, the training instances, not a test set. The overfitting problem is avoided because a complex theory that overfits will be penalized relative to a simple one by virtue of the fact that it takes more bits to encode. At one extreme is a very complex, highly over- fitted theory that makes no errors on the training set. At the other is a very simple theory—the null theory—which does not help at all when transmitting the training set. And in between are theories of intermediate complexity, which make proba- bilistic predictions that are imperfect and need to be corrected by transmitting some information about the training set. The MDL principle provides a means of comparing all these possibilities on an equal footing to see which is the best. We have found the holy grail: an evaluation scheme that works on the training set alone and does not need a separate test set. But the devil is in the details, as we will see. Suppose a learning scheme comes up with a theory T, based on a training set E of examples, that requires a certain number of bits L[T] to encode, where L is for length. We are only interested in predicting class labels correctly, so we assume that E stands for the collection of class labels in the training set. Given the theory, the training set itself can be encoded in a certain number of bits, L[E | T]. L[E | T] is in fact given by the informational loss function summed over all members of the training set. Then the total description length of theory plus training set is L L[] [ | ]TET+ and the MDL principle recommends choosing the theory T that minimizes this sum. There is a remarkable connection between the MDL principle and basic probabil- ity theory. Given a training set E, we seek the “most likely” theory T—that is, the theory for which the a posteriori probability Pr[T | E]—the probability after the examples have been seen—is maximized. Bayes’ rule of conditional probability (the very same rule that we encountered in Section 4.2) dictates that Pr[ | ] Pr[ | ]Pr[ ] Pr[ ]TEETT E= 5.9 Minimum Description Length Principle 185 Taking negative logarithms, − = − − +log Pr[ | ] log Pr[ | ] log Pr[ ] log Pr[ ]TEETTE Maximizing the probability is the same as minimizing its negative logarithm. Now (as we saw in Section 5.6) the number of bits required to code something is just the negative logarithm of its probability. Furthermore, the final term, log Pr[E], depends solely on the training set and not on the learning method. Thus, choosing the theory that maximizes the probability Pr[T | E] is tantamount to choosing the theory that minimizes L L[| ] [ ]ETT+ In other words, the MDL principle! This astonishing correspondence with the notion of maximizing the a posteriori probability of a theory after the training set has been taken into account gives cre- dence to the MDL principle. But it also points out where the problems will sprout when the principle is applied in practice. The difficulty with applying Bayes’ rule directly is in finding a suitable prior probability distribution Pr[T] for the theory. In the MDL formulation, that translates into finding how to code the theory T into bits in the most efficient way. There are many ways of coding things, and they all depend on presuppositions that must be shared by encoder and decoder. If you know in advance that the theory is going to take a certain form, you can use that information to encode it more efficiently. How are you going to actually encode T? The devil is in the details. Encoding E with respect to T to obtain L[E | T] seems a little more straightfor- ward: We have already met the informational loss function. But actually, when you encode one member of the training set after another, you are encoding a sequence rather than a set. It is not necessary to transmit the training set in any particular order, and it ought to be possible to use that fact to reduce the number of bits required. Often, this is simply approximated by subtracting log n! (where n is the number of elements in E), which is the number of bits needed to specify a particular permutation of the training set (and because this is the same for all theories, it doesn’t actually affect the comparison between them). But one can imagine using the fre- quency of the individual errors to reduce the number of bits needed to code them. Of course, the more sophisticated the method that is used to code the errors, the less the need for a theory in the first place—so whether a theory is justified or not depends to some extent on how the errors are coded. The details, the details. We end this section as we began, on a philosophical note. It is important to appreciate that Occam’s Razor, the preference of simple theories over complex ones, has the status of a philosophical position or “axiom” rather than something that can be proven from first principles. While it may seem self-evident to us, this is a func- tion of our education and the times we live in. A preference for simplicity is—or may be—culture specific rather than absolute. 186 CHAPTER 5 Credibility: Evaluating What’s Been Learned The Greek philosopher Epicurus (who enjoyed good food and wine and suppos- edly advocated sensual pleasure—in moderation—as the highest good) expressed almost the opposite sentiment. His principle of multiple explanations advises that “If more than one theory is consistent with the data, keep them all” on the basis that if several explanations are equally in agreement, it may be possible to achieve a higher degree of precision by using them together—and, anyway, it would be unscientific to discard some arbitrarily. This brings to mind instance-based learning, in which all the evidence is retained to provide robust predictions, and resonates strongly with deci- sion combination methods such as bagging and boosting (described in Chapter 8) that actually do gain predictive power by using multiple explanations together. 5.10 APPLYING THE MDL PRINCIPLE TO CLUSTERING One of the nice things about the minimum description length principle is that, unlike other evaluation criteria, it can be applied under widely different circumstances. Although in some sense equivalent to Bayes’ rule in that, as we have seen, devising a coding scheme for theories is tantamount to assigning them a prior probability distribution, schemes for coding are somehow far more tangible and easier to think about in concrete terms than intuitive prior probabilities. To illustrate this we will briefly describe—without entering into coding details—how you might go about applying the MDL principle to clustering. Clustering seems intrinsically difficult to evaluate. Whereas classification or association learning has an objective criterion of success—predictions made on test cases are either right or wrong—this is not so with clustering. It seems that the only realistic evaluation is whether the result of learning—the clustering—proves useful in the application context. (It is worth pointing out that really this is the case for all types of learning, not just clustering.) Despite this, clustering can be evaluated from a description-length perspective. Suppose a cluster-learning technique divides the training set E into k clusters. If these clusters are natural ones, it should be possible to use them to encode E more efficiently. The best clustering will support the most efficient encoding. One way of encoding the instances in E with respect to a given clustering is to start by encoding the cluster centers—the average value of each attribute over all instances in the cluster. Then, for each instance in E, transmit which cluster it belongs to (in log2 k bits) followed by its attribute values with respect to the cluster center— perhaps as the numeric difference of each attribute value from the center. Couched as it is in terms of averages and differences, this description presupposes numeric attributes and raises thorny questions of how to code numbers efficiently. Nominal attributes can be handled in a similar manner: For each cluster there is a probability distribution for the attribute values, and the distributions are different for different clusters. The coding issue becomes more straightforward: Attribute values are coded with respect to the relevant probability distribution, a standard operation in data compression. 5.11 Further Reading 187 If the data exhibits extremely strong clustering, this technique will result in a smaller description length than simply transmitting the elements of E without any clusters. However, if the clustering effect is not so strong, it will likely increase rather than decrease the description length. The overhead of transmitting cluster- specific distributions for attribute values will more than offset the advantage gained by encoding each training instance relative to the cluster it lies in. This is where more sophisticated coding techniques come in. Once the cluster centers have been communicated, it is possible to transmit cluster-specific probability distributions adaptively, in tandem with the relevant instances: The instances themselves help to define the probability distributions, and the probability distributions help to define the instances. We will not venture further into coding techniques here. The point is that the MDL formulation, properly applied, may be flexible enough to support the evaluation of clustering. But actually doing it satisfactorily in practice is not easy. 5.11 FURTHER READING The statistical basis of confidence tests is well covered in most statistics texts, which also give tables of the normal distribution and Student’s distribution. (We use an excellent course text by Wild and Seber (1995) that we recommend very strongly if you can get hold of it.) “Student” is the nom de plume of a statistician called William Gosset, who obtained a post as a chemist in the Guinness brewery in Dublin, Ireland, in 1899 and invented the t-test to handle small samples for quality control in brewing. The corrected resampled t-test was proposed by Nadeau and Bengio (2003). Cross-validation is a standard statistical technique, and its application in machine learning has been extensively investigated and compared with the bootstrap by Kohavi (1995a). The bootstrap technique itself is thoroughly covered by Efron and Tibshirani (1993). The Kappa statistic was introduced by Cohen (1960). Ting (2002) has investi- gated a heuristic way of generalizing to the multiclass case the algorithm given in Section 5.7 to make two-class learning schemes cost sensitive. Lift charts are described by Berry and Linoff (1997). The use of ROC analysis in signal detection theory is covered by Egan (1975); this work has been extended for visualizing and analyzing the behavior of diagnostic systems (Swets, 1988) and is also used in medicine (Beck and Schultz, 1986). Provost and Fawcett (1997) brought the idea of ROC analysis to the attention of the machine learning and data mining community. Witten et al. (1999b) explain the use of recall and precision in information retrieval systems; the F-measure is described by van Rijsbergen (1979). Drummond and Holte (2000) introduced cost curves and investigated their properties. The MDL principle was formulated by Rissanen (1985). Kepler’s discovery of his economical three laws of planetary motion, and his doubts about them, are recounted by Koestler (1964). Epicurus’ principle of multiple explanations is mentioned by Li and Vityani (1992), quoting from Asmis (1984). This page intentionally left blank PART IIAdvanced Data Mining This page intentionally left blank 191Data Mining: Practical Machine Learning Tools and Techniques Copyright © 2011 Elsevier Inc. All rights of reproduction in any form reserved. CHAPTER 6 Implementations: Real Machine Learning Schemes We have seen the basic ideas of several machine learning methods and studied in detail how to assess their performance on practical data mining problems. Now we are well prepared to look at real, industrial-strength, machine learning algorithms. Our aim is to explain these algorithms both at a conceptual level and with a fair amount of technical detail so that you can understand them fully and appreciate the key implementation issues that arise. In truth, there is a world of difference between the simplistic methods described in Chapter 4 and the actual algorithms that are widely used in practice. The principles are the same. So are the inputs and outputs—methods of knowledge representation. But the algorithms are far more complex, principally because they have to deal robustly and sensibly with real-world problems such as numeric attributes, missing values, and—most challenging of all—noisy data. To understand how the various schemes cope with noise, we will have to draw on some of the statistical knowledge that we learned in Chapter 5. Chapter 4 opened with an explanation of how to infer rudimentary rules and then examined statistical modeling and decision trees. Then we returned to rule induction and continued with association rules, linear models, the nearest-neighbor method of instance-based learning, and clustering. This chapter develops all these topics. We begin with decision tree induction and work up to a full description of the C4.5 system, a landmark decision tree program that is probably the machine learning workhorse most widely used in practice to date. Then we describe decision rule induction. Despite the simplicity of the idea, inducing decision rules that perform comparably with state-of-the-art decision trees turns out to be quite difficult in practice. Most high-performance rule inducers find an initial rule set and then refine it using a rather complex optimization stage that discards or adjusts individual rules to make them work better together. We describe the ideas that underlie rule learning in the presence of noise and then go on to cover a scheme that operates by forming partial decision trees, an approach that has been demonstrated to perform well while avoiding complex and ad hoc heuristics. Following this, we take a brief look at how to generate rules with exceptions, which were described in Section 3.4, and examine fast data structures for learning association rules. There has been a resurgence of interest in linear models with the introduction of support vector machines, a blend of linear modeling and instance-based learning. 192 CHAPTER 6 Implementations: Real Machine Learning Schemes Support vector machines select a small number of critical boundary instances called support vectors from each class and build a linear discriminant function that sepa- rates them as widely as possible. This instance-based approach transcends the limita- tions of linear boundaries by making it practical to include extra nonlinear terms in the function, making it possible to form quadratic, cubic, and higher-order decision boundaries. The same techniques can be applied to the perceptron described in Section 4.6 to implement complex decision boundaries, and also to least squares regression. An older technique for extending the perceptron is to connect units together into multilayer “neural networks.” All of these ideas are described in Section 6.4. Section 6.5 describes classic instance-based learners, developing the simple nearest-neighbor method introduced in Section 4.7 and showing some more pow- erful alternatives that perform explicit generalization. Following that we extend linear regression for numeric prediction to a more sophisticated procedure that comes up with the tree representation introduced in Section 3.3 and go on to describe locally weighted regression, an instance-based strategy for numeric pre- diction. Then we examine Bayesian networks, a potentially very powerful way of extending the Naïve Bayes method to make it less “naïve” by dealing with datasets that have internal dependencies. Next we return to clustering and review some methods that are more sophisticated than simple k-means, methods that produce hierarchical clusters and probabilistic clusters. We also look at semi-supervised learning, which can be viewed as combining clustering and classification. Finally, we discuss more advanced schemes for multi-instance learning than those covered in Section 4.9. Because of the nature of the material it contains, this chapter differs from the others in the book. Sections can be read independently, and each is self-contained, including the references to further reading, which are gathered together in Discus- sion sections. 6.1 DECISION TREES The first machine learning scheme that we will develop in detail, the C4.5 algorithm, derives from the simple divide-and-conquer algorithm for producing decision trees that was described in Section 4.3. It needs to be extended in several ways before it is ready for use on real-world problems. First, we consider how to deal with numeric attributes and, after that, missing values. Then we look at the all-important problem of pruning decision trees, because although trees constructed by the divide-and- conquer algorithm as described perform well on the training set, they are usually overfitted to the training data and do not generalize well to independent test sets. We then briefly consider how to convert decision trees to classification rules and examine the options provided by the C4.5 algorithm itself. Finally, we look at an alternative pruning strategy that is implemented in the famous CART system for learning classification and regression trees. 6.1 Decision Trees 193 Numeric Attributes The method we described in Section 4.3 only works when all the attributes are nominal, whereas, as we have seen, most real datasets contain some numeric attri- butes. It is not too difficult to extend the algorithm to deal with these. For a numeric attribute we will restrict the possibilities to a two-way, or binary, split. Suppose we use the version of the weather data that has some numeric features (see Table 1.3). Then, when temperature is being considered for the first split, the temperature values involved are 64 65 68 69 70 71 72 75 80 81 83 85 yes no yes yes yes no no yes yes yes no yes yes no Repeated values have been collapsed together, and there are only 11 possible posi- tions for the breakpoint—8 if the breakpoint is not allowed to separate items of the same class. The information gain for each can be calculated in the usual way. For example, the test temperature < 71.5 produces four yes’s and two no’s, whereas temperature > 71.5 produces five yes’s and three no’s, and so the information value of this test is info info info([,],[,]) ( ) ([,])( ) ([,]) .4 2 5 3 6 14 4 2 8 14 5 3 0 93= × + × = 99 bits It is common to place numeric thresholds halfway between the values that delimit the boundaries of a concept, although something might be gained by adopting a more sophisticated policy. For example, we will see in the following that although the simplest form of instance-based learning puts the dividing line between concepts in the middle of the space between them, other methods that involve more than just the two nearest examples have been suggested. When creating decision trees using the divide-and-conquer method, once the first attribute to split on has been selected, a top-level tree node is created that splits on that attribute, and the algorithm proceeds recursively on each of the child nodes. For each numeric attribute, it appears that the subset of instances at each child node must be re-sorted according to that attribute’s values—and, indeed, this is how programs for inducing decision trees are usually written. However, it is not actually necessary to re-sort because the sort order at a parent node can be used to derive the sort order for each child, leading to a speedier implementation. Consider the temperature attribute in the weather data, whose sort order (this time including duplicates) is 64 65 68 69 70 71 72 72 75 75 80 81 83 85 7 6 5 9 4 14 8 12 10 11 2 13 3 1 The italicized numbers below each temperature value give the number of the instance that has that value. Thus, instance number 7 has temperature value 64, instance 6 has temperature 65, and so on. Suppose we decide to split at the top level 194 CHAPTER 6 Implementations: Real Machine Learning Schemes 9 8 11 2 1 on the attribute outlook. Consider the child node for which outlook = sunny—in fact, the examples with this value of outlook are numbers 1, 2, 8, 9, and 11. If the italicized sequence is stored with the example set (and a different sequence must be stored for each numeric attribute)—that is, instance 7 contains a pointer to instance 6, instance 6 points to instance 5, instance 5 points to instance 9, and so on—then it is a simple matter to read off the examples for which outlook = sunny in order. All that is neces- sary is to scan through the instances in the indicated order, checking the outlook attribute for each and writing down the ones with the appropriate value: Thus, repeated sorting can be avoided by storing with each subset of instances the sort order for that subset according to each numeric attribute. The sort order must be determined for each numeric attribute at the beginning; no further sorting is necessary thereafter. When a decision tree tests a nominal attribute as described in Section 4.3, a branch is made for each possible value of the attribute. However, we have restricted splits on numeric attributes to be binary. This creates an important difference between numeric attributes and nominal ones: Once you have branched on a nominal attri- bute, you have used all the information that it offers; however, successive splits on a numeric attribute may continue to yield new information. Whereas a nominal attribute can only be tested once on any path from the root of a tree to the leaf, a numeric one can be tested many times. This can yield trees that are messy and dif- ficult to understand because the tests on any single numeric attribute are not located together but can be scattered along the path. An alternative, which is harder to accomplish but produces a more readable tree, is to allow a multiway test on a numeric attribute, testing against several different constants at a single node of the tree. A simpler but less powerful solution is to prediscretize the attribute as described in Section 7.2. Missing Values The next enhancement to the decision tree–building algorithm deals with the prob- lems of missing values. Missing values are endemic in real-world datasets. As explained in Chapter 2 (page 58), one way of handling them is to treat them as just another possible value of the attribute; this is appropriate if the fact that the attribute is missing is significant in some way. In that case, no further action need be taken. But if there is no particular significance in the fact that a certain instance has a missing attribute value, a more subtle solution is needed. It is tempting to simply ignore all instances in which some of the values are missing, but this solution is often too draconian to be viable. Instances with missing values often provide a good deal of information. Sometimes the attributes with values that are missing play no part in the decision, in which case these instances are as good as any other. One question is how to apply a given decision tree to an instance in which some of the attributes to be tested have missing values. We outlined a solution in Section 6.1 Decision Trees 195 3.3 that involves notionally splitting the instance into pieces, using a numeric weighting scheme, and sending part of it down each branch in proportion to the number of training instances going down that branch. Eventually, the various parts of the instance will each reach a leaf node, and the decisions at these leaf nodes must be recombined using the weights that have percolated to the leaves. The infor- mation gain and gain ratio calculations described in Section 4.3 can also be applied to partial instances. Instead of having integer counts, the weights are used when computing both gain figures. Another question is how to partition the training set once a splitting attribute has been chosen, to allow recursive application of the decision tree formation procedure on each of the daughter nodes. The same weighting procedure is used. Instances for which the relevant attribute value is missing are notionally split into pieces, one piece for each branch, in the same proportion as the known instances go down the various branches. Pieces of the instance contribute to decisions at lower nodes in the usual way through the information gain calculation, except that they are weighted accordingly. They may be further split at lower nodes, of course, if the values of other attributes are unknown as well. Pruning Fully expanded decision trees often contain unnecessary structure, and it is generally advisable to simplify them before they are deployed. Now it is time to learn how to prune decision trees. By building the complete tree and pruning it afterward we are adopting a strategy of postpruning (sometimes called backward pruning) rather than prepruning (or forward pruning). Prepruning would involve trying to decide during the tree- building process when to stop developing subtrees—quite an attractive prospect because that would avoid all the work of developing subtrees only to throw them away afterward. However, postpruning does seem to offer some advantages. For example, situations occur in which two attributes individually seem to have nothing to contribute but are powerful predictors when combined—a sort of combination- lock effect in which the correct combination of the two attribute values is very informative but the attributes taken individually are not. Most decision tree builders postprune; however, prepruning can be a viable alternative when runtime is of particular concern. Two rather different operations have been considered for postpruning: subtree replacement and subtree raising. At each node, a learning scheme might decide whether it should perform subtree replacement, subtree raising, or leave the subtree as it is, unpruned. Subtree replacement is the primary pruning operation, and we look at it first. The idea is to select some subtrees and replace them with single leaves. For example, the whole subtree in Figure 1.3(a), involving two internal nodes and four leaf nodes, has been replaced by the single leaf bad. This will certainly cause the accuracy on the training set to decrease if the original tree was produced by the decision tree algorithm described previously, because that continued to build 196 CHAPTER 6 Implementations: Real Machine Learning Schemes FIGURE 6.1 Example of subtree raising, where (a) node C is “raised” to subsume node B (b). A B C 4 5 1 2 3 A C 1′ 2′ 3′ (a) (b) the tree until all leaf nodes were pure (or until all attributes had been tested). However, it may increase the accuracy on an independently chosen test set. When subtree replacement is implemented, it proceeds from the leaves and works back up toward the root. In the Figure 1.3 example, the whole subtree in (a) would not be replaced at once. First, consideration would be given to replacing the three daughter nodes in the health plan contribution subtree with a single leaf node. Assume that a decision is made to perform this replacement—we will explain how this decision is made shortly. Then, continuing to work back from the leaves, con- sideration would be given to replacing the working hours per week subtree, which now has just two daughter nodes, by a single leaf node. In the Figure 1.3 example, this replacement was indeed made, which accounts for the entire subtree in (a) being replaced by a single leaf marked bad. Finally, consideration would be given to replacing the two daughter nodes in the wage increase 1st year subtree with a single leaf node. In this case, that decision was not made, so the tree remains as shown in Figure 1.3(a). Again, we will examine how these decisions are actually made shortly. The second pruning operation, subtree raising, is more complex, and it is not clear that it is necessarily always worthwhile. However, because it is used in the influential decision tree–building system C4.5, we describe it here. Subtree raising does not occur in the Figure 1.3 example, so we use the artificial example of Figure 6.1 for illustration. Here, consideration is given to pruning the tree in Figure 6.1(a), and the result is shown in Figure 6.1(b). The entire subtree from C downward has been “raised” to replace the B subtree. Note that although the daughters of B and C are shown as leaves, they can be entire subtrees. Of course, if we perform this raising operation, it is necessary to reclassify the examples at the nodes marked 4 and 5 into the new subtree headed by C. This is why the daughters of that node are marked with primes—1′, 2′, and 3′—to indicate that they are not the same as the original 6.1 Decision Trees 197 daughters 1, 2, and 3 but differ by the inclusion of the examples originally covered by 4 and 5. Subtree raising is a potentially time-consuming operation. In actual implementa- tions it is generally restricted to raising the subtree of the most popular branch. That is, we consider doing the raising illustrated in Figure 6.1 provided that the branch from B to C has more training examples than the branches from B to node 4 or from B to node 5. Otherwise, if (for example) node 4 were the majority daughter of B, we would consider raising node 4 to replace B and reclassifying all examples under C, as well as the examples from node 5, into the new node. Estimating Error Rates So much for the two pruning operations. Now we must address the question of how to decide whether to replace an internal node by a leaf (for subtree replacement) or whether to replace an internal node by one of the nodes below it (for subtree raising). To make this decision rationally, it is necessary to estimate the error rate that would be expected at a particular node given an independently chosen test set. We need to estimate the error at internal nodes as well as at leaf nodes. If we had such an esti- mate, it would be clear whether to replace, or raise, a particular subtree simply by comparing the estimated error of the subtree with that of its proposed replacement. Before estimating the error for a subtree proposed for raising, examples that lie under siblings of the current node—the examples at 4 and 5 of Figure 6.1—would have to be temporarily reclassified into the raised tree. It is no use taking the training set error as the error estimate: That would not lead to any pruning because the tree has been constructed expressly for that particular training set. One way of coming up with an error estimate is the standard verification technique: Hold back some of the data originally given and use it as an independent test set to estimate the error at each node. This is called reduced-error pruning. It suffers from the disadvantage that the actual tree is based on less data. The alternative is to try to make some estimate of error based on the training data itself. That is what C4.5 does, and we will describe its method here. It is a heuristic based on some statistical reasoning, but the statistical underpinning is rather weak. However, it seems to work well in practice. The idea is to consider the set of instances that reach each node and imagine that the majority class is chosen to represent that node. That gives us a certain number of “errors,” E, out of the total number of instances, N. Now imagine that the true probability of error at the node is q, and that the N instances are generated by a Bernoulli process with parameter q, of which E turn out to be errors. This is almost the same situation as we considered when looking at the holdout method in Section 5.2, where we calculated confidence intervals on the true success probability p given a certain observed success rate. There are two differences. One is trivial: Here we are looking at the error rate q rather than the success rate p; these are simply related by p + q = 1. The second is more serious: Here the figures E and N are measured from the training data, whereas in Section 5.2 we were considering 198 CHAPTER 6 Implementations: Real Machine Learning Schemes The mathematics involved is just the same as before. Given a particular confidence c (the default figure used by C4.5 is c = 25%), we find confidence limits z such that Pr ()/ f q q q N z c− − > =1 where N is the number of samples, f = E /N is the observed error rate, and q is the true error rate. As before, this leads to an upper confidence limit for q. Now we use that upper confidence limit as a (pessimistic) estimate for the error rate e at the node: e f z N z f N f N z N z N = + + − + + 2 2 2 2 2 2 4 1 Note the use of the + sign before the square root in the numerator to obtain the upper confidence limit. Here, z is the number of standard deviations corresponding to the confidence c, which for c = 25% is z = 0.69. To see how all this works in practice, let’s look again at the labor negotiations decision tree of Figure 1.3, salient parts of which are reproduced in Figure 6.2 with the number of training examples that reach the leaves added. We use the previous formula with a 25% confidence figure—that is, with z = 0.69. Consider the lower left leaf, for which E = 2, N = 6, and so f = 0.33. Plugging these figures into the formula, the upper confidence limit is calculated as e = 0.47. That means that instead of using the training set error rate for this leaf, which is 33%, we will use the pessimistic estimate of 47%. This is pessimistic indeed, considering that it would be a bad mistake to let the error rate exceed 50% for a two-class problem. But things are worse for the neighboring leaf, where E = 1 and N = 2, because the upper confidence limit becomes e = 0.72. The third leaf has the same value of e as the first. The next step is to combine the error estimates for these three leaves in the ratio of the number of examples they cover, 6 : 2 : 6, which leads to a combined error estimate of 0.51. Now we consider the error estimate for the parent node, health plan contribution. This covers nine bad examples and five good ones, so the training set error rate is f = 5/14. For these values, the previous formula yields a pessimistic error estimate of e = 0.46. Because this is less than the combined error estimate of the three children, they are pruned away. The next step is to consider the working hours per week node, which now has two children that are both leaves. The error estimate for the first, with E = 1 and N = 2, is e = 0.72, while for the second it is e = 0.46, as we have just seen. Combining these in the appropriate ratio of 2 : 14 leads to a value that is higher than the error estimate for the working hours node, so the subtree is pruned away and replaced by a leaf node. The estimated error figures obtained in these examples should be taken with a grain of salt because the estimate is only a heuristic one and is based on a number of shaky assumptions: the use of the upper confidence limit; the assumption of a normal distribution; and the fact that statistics from the training set are used. However, the qualitative behavior of the error formula is correct and the method seems to work reasonably well in practice. If necessary, the underlying confidence level, which we have taken to be 25%, can be tweaked to produce more satisfactory results. independent test data. Because of this difference we make a pessimistic estimate of the error rate by using the upper confidence limit rather than stating the estimate as a confidence range. 6.1 Decision Trees 199 Complexity of Decision Tree Induction Now that we have learned how to accomplish the pruning operations, we have finally covered all the central aspects of decision tree induction. Let’s take stock and examine the computational complexity of inducing decision trees. We will use the standard order notation: O(n) stands for a quantity that grows at most linearly with n, O(n2) grows at most quadratically with n, and so on. Suppose the training data contains n instances and m attributes. We need to make some assumption about the size of the tree, and we will assume that its depth is on the order of log n, that is O(log n). This is the standard rate of growth of a tree with n leaves, provided that it remains “bushy” and doesn’t degenerate into a few very long, stringy branches. Note that we are tacitly assuming that most of the instances are different from each other and—this is almost the same thing—that the m attri- butes provide enough tests to allow the instances to be differentiated. For example, if there were only a few binary attributes, they would allow only so many instances to be differentiated and the tree could not grow past a certain point, rendering an “in the limit” analysis meaningless. The computational cost of building the tree in the first place is O(mnlog n). Consider the amount of work done for one attribute over all nodes of the tree. Not all the examples need to be considered at each node, of course. But at each possible tree depth, the entire set of n instances must be considered in the worst case. And because there are log n different depths in the tree, the amount of work for this one attribute is O(n log n). At each node all attributes are considered, so the total amount of work is O(mn log n). FIGURE 6.2 Pruning the labor negotiations decision tree. wage increase 1st year working hours per week <= 2.5 > 2.5 1 bad 1 good <= 36 health plan contribution > 36 4 bad 2 good none 1 bad 1 good half 4 bad 2 good full 200 CHAPTER 6 Implementations: Real Machine Learning Schemes This reasoning makes some assumptions. If some attributes are numeric, they must be sorted, but once the initial sort has been done there is no need to re-sort at each tree depth if the appropriate algorithm is used (described previously—see page 193). The initial sort takes O(n log n) operations for each of up to m attributes; thus, the above complexity figure is unchanged. If the attributes are nominal, all attributes do not have to be considered at each tree node because attributes that are used further up the tree cannot be reused. However, if attributes are numeric, they can be reused and so they have to be considered at every tree level. Next, consider pruning by subtree replacement. First an error estimate must be made for every tree node. Provided that counts are maintained appropriately, this is linear in the number of nodes in the tree. Then each node needs to be considered for replacement. The tree has at most n leaves, one for each instance. If it were a binary tree, each attribute being numeric or two-valued, that would give it 2n – 1 nodes; multiway branches would only serve to decrease the number of internal nodes. Thus, the complexity of subtree replacement is O(n). Finally, subtree lifting has a basic complexity equal to subtree replacement. But there is an added cost because instances need to be reclassified during the lifting operation. During the whole process, each instance may have to be reclassified at every node between its leaf and the root—that is, as many as O(log n) times. That makes the total number of reclassifications O(n log n). And reclassification is not a single operation: One that occurs near the root will take O(log n) operations, and one of average depth will take half of this. Thus, the total complexity of subtree lifting is as follows: O(n(log n)2). Taking into account all these operations, the full complexity of decision tree induction is O( log ) O( (log ) )mn n n n+ 2 From Trees to Rules It is possible to read a set of rules directly off a decision tree, as noted in Section 3.4, by generating a rule for each leaf and making a conjunction of all the tests encountered on the path from the root to that leaf. This produces rules that are unambiguous in that it doesn’t matter in what order they are executed. However, the rules are more complex than necessary. The estimated error rate described previously provides exactly the mechanism necessary to prune the rules. Given a particular rule, each condition in it is considered for deletion by tentatively removing it, working out which of the training examples are now covered by the rule, calculating from this a pessimistic estimate of the error rate of the new rule, and comparing this with the pessimistic estimate for the original rule. If the new rule is better, delete that condition and carry on, looking for other conditions to delete. Leave the rule when there are no remaining conditions that will improve it if they are removed. Once all rules have been pruned in this way, it is necessary to see if there are any duplicates and remove them from the rule set. 6.1 Decision Trees 201 This is a greedy approach to detecting redundant conditions in a rule, and there is no guarantee that the best set of conditions will be removed. An improvement would be to consider all subsets of conditions, but this is usually prohibitively expensive. Another solution might be to use an optimization technique such as simulated annealing or a genetic algorithm to select the best version of this rule. However, the simple greedy solution seems to produce quite good rule sets. The problem, even with the greedy method, is computational cost. For every condition that is a candidate for deletion, the effect of the rule must be reevalu- ated on all the training instances. This means that rule generation from trees tends to be very slow. The next section describes much faster methods that generate classification rules directly without forming a decision tree first. C4.5: Choices and Options The decision tree program C4.5 and its successor C5.0 were devised by Ross Quinlan over a 20-year period beginning in the late 1970s. A complete description of C4.5, the early 1990s version, appears as an excellent and readable book (Quinlan, 1993), along with the full source code. The more recent version, C5.0, is available com- mercially. Its decision tree induction seems to be essentially the same as that used by C4.5, and tests show some differences but negligible improvements. However, its rule generation is greatly sped up and clearly uses a different technique, although this has not been described in the open literature. C4.5 works essentially as described in the previous sections. The default confi- dence value is set at 25% and works reasonably well in most cases; possibly it should be altered to a lower value, which causes more drastic pruning, if the actual error rate of pruned trees on test sets is found to be much higher than the estimated error rate. There is one other important parameter whose effect it is to eliminate tests for which almost all of the training examples have the same outcome. Such tests are often of little use. Consequently, tests are not incorporated into the decision tree unless they have at least two outcomes that have at least a minimum number of instances. The default value for this minimum is 2, but it is controllable and should perhaps be increased for tasks that have a lot of noisy data. Another heuristic in C4.5 is that candidate splits on numeric attributes are only considered if they cut off a certain minimum number of instances: at least 10% of the average number of instances per class at the current node, or 25 instances—whichever value is smaller (but the minimum just mentioned, 2 by default, is also enforced). C4.5 Release 8, the last noncommercial version of C4.5, includes an MDL-based adjustment to the information gain for splits on numeric attributes. More specifically, if there are S candidate splits on a certain numeric attribute at the node currently considered for splitting, log2(S)/N is subtracted from the information gain, where N is the number of instances at the node. This heuristic, described by Quinlan (1986), is designed to prevent overfitting. The information gain may be negative after sub- traction, and tree growing will stop if there are no attributes with positive informa- tion gain—a form of prepruning. We mention this here because it can be surprising 202 CHAPTER 6 Implementations: Real Machine Learning Schemes to obtain a pruned tree even if postpruning has been turned off! This heuristic is also implemented in the software described in Part 3 of this book. Cost-Complexity Pruning As mentioned, the postpruning method in C4.5 is based on shaky statistical assump- tions, and it turns out that it often does not prune enough. On the other hand, it is very fast and thus popular in practice. However, in many applications it is worth- while expending more computational effort to obtain a more compact decision tree. Experiments have shown that C4.5’s pruning method can yield unnecessary addi- tional structure in the final tree: Tree size continues to grow when more instances are added to the training data even when this does not further increase performance on independent test data. In that case, the more conservative cost-complexity pruning method from the Classification and Regression Trees (CART) learning system may be more appropriate. Cost-complexity pruning is based on the idea of first pruning those subtrees that, relative to their size, lead to the smallest increase in error on the training data. The increase in error is measured by a quantity α that is defined to be the average error increase per leaf of the subtree concerned. By monitoring this quantity as pruning progresses, the algorithm generates a sequence of successively smaller pruned trees. In each iteration it prunes all subtrees that exhibit the smallest value of α among the remaining subtrees in the current version of the tree. Each candidate tree in the resulting sequence of pruned trees corresponds to one particular threshold value, αi. The question becomes, which tree should be chosen as the final classification model? To determine the most predictive tree, cost- complexity pruning either uses a holdout set to estimate the error rate of each tree, or, if data is limited, employs cross-validation. Using a holdout set is straightforward. However, cross-validation poses the problem of relating the α values observed in the sequence of pruned trees for train- ing fold k of the cross-validation to the α values from the sequence of trees for the full dataset: These values are usually different. This problem is solved by first com- puting the geometric average of αi and αi+1 for tree i from the full dataset. Then, for each fold k of the cross-validation, the tree that exhibits the largest α value smaller than this average is picked. The average of the error estimates for these trees from the k folds, estimated from the corresponding test datasets, is the cross-validation error for tree i from the full dataset. Discussion Top-down induction of decision trees is probably the most extensively researched method of machine learning used in data mining. Researchers have investigated a panoply of variations for almost every conceivable aspect of the learning process— for example, different criteria for attribute selection or modified pruning methods. However, they are rarely rewarded by substantial improvements in accuracy over a spectrum of diverse datasets. As discussed, the pruning method used by the CART system for learning decision trees (Breiman et al., 1984) can often produce smaller 6.2 Classification Rules 203 trees than C4.5’s pruning method. This has been investigated empirically by Oates and Jensen (1997). In our description of decision trees, we have assumed that only one attribute is used to split the data into subsets at each node of the tree. However, it is possible to allow tests that involve several attributes at a time. For example, with numeric attributes each test can be on a linear combination of attribute values. Then the final tree consists of a hierarchy of linear models of the kind we described in Section 4.6, and the splits are no longer restricted to being axis-parallel. Trees with tests involv- ing more than one attribute are called multivariate decision trees, in contrast to the simple univariate trees that we normally use. The CART system has the option of generating multivariate tests. They are often more accurate and smaller than univari- ate trees but take much longer to generate and are also more difficult to interpret. We briefly mention one way of generating them in the Principal Components Analy- sis section in Section 7.3. 6.2 CLASSIFICATION RULES We call the basic covering algorithm for generating rules that was described in Section 4.4 a separate-and-conquer technique because it identifies a rule that covers instances in a class (and excludes ones not in the class), separates them out, and continues on those that are left. Such algorithms have been used as the basis of many systems that generate rules. There, we described a simple correctness-based measure for choosing what test to add to the rule at each stage. However, there are many other possibilities, and the particular criterion that is used has a significant effect on the rules produced. We examine different criteria for choosing tests in this section. We also look at how the basic rule-generation algorithm can be extended to more practical situations by accommodating missing values and numeric attributes. But the real problem with all these rule-generation schemes is that they tend to overfit the training data and do not generalize well to independent test sets, particularly on noisy data. To be able to generate good rule sets for noisy data, it is necessary to have some way of measuring the real worth of individual rules. The standard approach to assessing the worth of rules is to evaluate their error rate on an independent set of instances, held back from the training set, and we explain this next. After that, we describe two industrial-strength rule learners: one that combines the simple separate- and-conquer technique with a global optimization step, and another that works by repeatedly building partial decision trees and extracting rules from them. Finally, we consider how to generate rules with exceptions, and exceptions to the exceptions. Criteria for Choosing Tests When we introduced the basic rule learner in Section 4.4, we had to figure out a way of deciding which of many possible tests to add to a rule to prevent it from covering any negative examples. For this we used the test that maximizes the ratio p/t, where t is the total number of instances that the new rule will cover, and p is 204 CHAPTER 6 Implementations: Real Machine Learning Schemes the number of these that are positive—that is, belong to the class in question. This attempts to maximize the “correctness” of the rule on the basis that the higher the proportion of positive examples it covers, the more correct a rule is. One alternative is to calculate an information gain: p p t P Tlog log− where p and t are the number of positive instances and the total number of instances covered by the new rule, as before, and P and T are the corresponding number of instances that satisfied the rule before the new test was added. The rationale for this is that it represents the total information gained regarding the current positive examples, which is given by the number of them that satisfy the new test, multiplied by the information gained regarding each one. The basic criterion for choosing a test to add to a rule is to find one that covers as many positive examples as possible while covering as few negative examples as possible. The original correctness-based heuristic, which is just the percentage of positive examples among all examples covered by the rule, attains a maximum when no negative examples are covered regardless of the number of positive examples covered by the rule. Thus, a test that makes the rule exact will be preferred to one that makes it inexact, no matter how few positive examples the former rule covers nor how many positive examples the latter covers. For example, if we consider a test that covers one example that is positive, this criterion will prefer it over a test that covers 1000 positive examples along with one negative one. The information-based heuristic, on the other hand, places far more emphasis on covering a large number of positive examples regardless of whether the rule so created is exact. Of course, both algorithms continue adding tests until the final rule produced is exact, which means that the rule will be finished earlier using the cor- rectness measure whereas more terms will have to be added if the information-based measure is used. Thus, the correctness-based measure might find special cases and eliminate them completely, saving the larger picture for later (when the more general rule might be simpler because awkward special cases have already been dealt with), whereas the information-based one will try to generate high-coverage rules first and leave the special cases until later. It is by no means obvious that either strategy is superior to the other at producing an exact rule set. Moreover, the whole situation is complicated by the fact that, as described in the following, rules may be pruned and inexact ones tolerated. Missing Values, Numeric Attributes As with divide-and-conquer decision tree algorithms, the nasty practical consider- ations of missing values and numeric attributes need to be addressed. In fact, there is not much more to say. Now that we know how these problems can be solved for decision tree induction, appropriate solutions for rule induction are easily given. 6.2 Classification Rules 205 When producing rules using covering algorithms, missing values can be best treated as though they don’t match any of the tests. This is particularly suitable when a decision list is being produced, because it encourages the learning algorithm to separate out positive instances using tests that are known to succeed. It has the effect either that instances with missing values are dealt with by rules involving other attributes that are not missing, or that any decisions about them are deferred until most of the other instances have been taken care of, at which time tests will probably emerge that involve other attributes. Covering algorithms for decision lists have a decided advantage over decision tree algorithms in this respect: Tricky examples can be left until late in the process, at which time they will appear less tricky because most of the other examples have already been classified and removed from the instance set. Numeric attributes can be dealt with in exactly the same way as they are dealt with for trees. For each numeric attribute, instances are sorted according to the attribute’s value and, for each possible threshold, a binary less-than/greater-than test is considered and evaluated in exactly the same way that a binary attribute would be. Generating Good Rules Suppose you don’t want to generate perfect rules that guarantee to give the correct classification on all instances in the training set, but would rather generate “sensible” ones that avoid overfitting the training set and thereby stand a better chance of performing well on new test instances. How do you decide which rules are worth- while? How do you tell when it becomes counterproductive to continue adding terms to a rule to exclude a few pesky instances of the wrong type, all the while excluding more and more instances of the correct type? Let’s look at a few examples of possible rules—some good and some bad—for the contact lens problem in Table 1.1. Consider first the rule If astigmatism = yes and tear production rate = normal then recommendation = hard This gives a correct result for four out of the six cases that it covers; thus, its success fraction is 4/6. Suppose we add a further term to make the rule a “perfect” one: If astigmatism = yes and tear production rate = normal and age = young then recommendation = hard This improves accuracy to 2/2. Which rule is better? The second one is more accurate on the training data but covers only two cases, whereas the first one covers six. It may be that the second version is just overfitting the training data. For a practical rule learner we need a principled way of choosing the appropriate version of a rule, preferably one that maximizes accuracy on future test data. Suppose we split the training data into two parts that we will call a growing set and a pruning set. The growing set is used to form a rule using the basic covering 206 CHAPTER 6 Implementations: Real Machine Learning Schemes algorithm. Then a test is deleted from the rule, and the effect is evaluated by trying out the truncated rule on the pruning set and seeing whether it performs better than the original rule. This pruning process repeats until the rule cannot be improved by deleting any further tests. The whole procedure is repeated for each class, obtaining one best rule for each class, and the overall best rule is established by evaluating the rules on the pruning set. This rule is then added to the rule set, the instances it covers are removed from the training data—from both growing and pruning sets—and the process is repeated. Why not do the pruning as we build up the rule, rather than building up the whole thing and then throwing parts away? That is, why not preprune rather than post- prune? Just as when pruning decision trees it is often best to grow the tree to its maximum size and then prune back, so with rules it is often best to make a perfect rule and then prune it. Who knows?—adding that last term may make a really good rule, a situation that we might never have noticed had we adopted an aggressive prepruning strategy. It is essential that the growing and pruning sets are separate because it is mis leading to evaluate a rule on the very data that was used to form it: That would lead to serious errors by preferring rules that were overfitted. Usually the training set is split so that two-thirds of instances are used for growing and one-third for pruning. A disadvantage, of course, is that learning occurs from instances in the growing set only, so the algorithm might miss important rules because some key instances had been assigned to the pruning set. Moreover, the wrong rule might be preferred because the pruning set contains only one-third of the data and may not be com- pletely representative. These effects can be ameliorated by resplitting the training data into growing and pruning sets at each cycle of the algorithm—that is, after each rule is finally chosen. The idea of using a separate pruning set for pruning—which is applicable to decision trees as well as rule sets—is called reduced-error pruning. The variant previously described prunes a rule immediately after it has been grown; it is called incremental reduced-error pruning. Another possibility is to build a full, unpruned, rule set first, pruning it afterwards by discarding individual tests. However, this method is much slower. Of course, there are many different ways to assess the worth of a rule based on the pruning set. A simple measure is to consider how well the rule would do at discriminating the predicted class from other classes if it were the only rule in the theory, operating under the closed-world assumption. Suppose it gets p instances right out of the t instances that it covers, and there are P instances of this class out a total of T instances altogether. The instances that it does not cover include N – n negative ones, where n = t – p is the number of negative instances that the rule covers and N = T – P is the total number of negative instances. Thus, in total the rule makes correct decisions on p + (N – n) instances, and so has an overall success ratio of [ ( )]p N n T+ − 6.2 Classification Rules 207 FIGURE 6.3 Algorithm for forming rules by incremental reduced-error pruning. Initialize E to the instance set Split E into Grow and Prune in the ratio 2:1 For each class C for which Grow and Prune both contain an instance Use the basic covering algorithm to create the best perfect rule for class C Calculate the worth w(R) for the rule on Prune, and for the rule with the final condition omitted w(R-) While w(R-) > w(R), remove the final condition from the rule and repeat the previous step From the rules generated, select the one with the largest w(R) Print the rule Remove the instances covered by the rule from E Continue This quantity, evaluated on the test set, has been used to evaluate the success of a rule when using reduced-error pruning. This measure is open to criticism because it treats noncoverage of negative examples as being as important as coverage of positive ones, which is unrealistic in a situation where what is being evaluated is one rule that will eventually serve alongside many others. For example, a rule that gets p = 2000 instances right out of a total coverage of 3000 (i.e., it gets n = 1000 wrong) is judged as more successful than one that gets p = 1000 out of a total coverage of 1001 (i.e., n = 1 wrong), because [p + (N – n)]/T is [1000 + N]/T in the first case but only [999 + N]/T in the second. This is counterintuitive: The first rule is clearly less predictive than the second because it has a 33.3% as opposed to only a 0.1% chance of being incorrect. Using the success rate p/t as a measure, as was done in the original formulation of the covering algorithm (Figure 4.8), is not the perfect solution either because it would prefer a rule that got a single instance right (p = 1) out of a total coverage of 1 (so n = 0) to the far more useful rule that got 1000 right out of 1001. Another heuristic that has been used is (p – n)/t, but that suffers from exactly the same problem because (p – n)/t = 2p/t – 1 and so the result, when comparing one rule with another, is just the same as with the success rate. It seems hard to find a simple measure of the worth of a rule that corresponds with intuition in all cases. Whatever heuristic is used to measure the worth of a rule, the incremental reduced-error pruning algorithm is the same. A possible rule-learning algorithm based on this idea is given in Figure 6.3. It generates a decision list, creating rules for each class in turn and choosing at each stage the best version of the rule accord- ing to its worth on the pruning data. The basic covering algorithm for rule generation (Figure 4.8) is used to come up with good rules for each class, choosing conditions to add to the rule using the accuracy measure p/t that we described earlier. 208 CHAPTER 6 Implementations: Real Machine Learning Schemes This method has been used to produce rule-induction schemes that can process vast amounts of data and operate very quickly. It can be accelerated by generating rules for the classes in order rather than generating a rule for each class at every stage and choosing the best. A suitable ordering is the increasing order in which they occur in the training set so that the rarest class is processed first and the most common ones are processed later. Another significant speedup is obtained by stop- ping the whole process when a rule of sufficiently low accuracy is generated, so as not to spend time generating a lot of rules at the end with very small coverage. However, very simple terminating conditions (such as stopping when the accuracy for a rule is lower than the default accuracy for the class it predicts) do not give the best performance. One criterion that seems to work well is a rather complicated one based on the MDL principle, described later. Using Global Optimization In general, rules generated using incremental reduced-error pruning in this manner seem to perform quite well, particularly on large datasets. However, it has been found that a worthwhile performance advantage can be obtained by performing a global optimization step on the set of rules induced. The motivation is to increase the accuracy of the rule set by revising or replacing individual rules. Experiments show that both the size and the performance of rule sets are significantly improved by postinduction optimization. On the other hand, the process itself is rather complex. To give an idea of how elaborate—and heuristic—industrial-strength rule learn- ers become, Figure 6.4 shows an algorithm called RIPPER, an acronym for repeated incremental pruning to produce error reduction. Classes are examined in increasing size and an initial set of rules for a class is generated using incremental reduced-error pruning. An extra stopping condition is introduced that depends on the description length of the examples and rule set. The description-length DL is a complex formula that takes into account the number of bits needed to send a set of examples with respect to a set of rules, the number of bits required to send a rule with k conditions, and the number of bits needed to send the integer k—times an arbitrary factor of 50% to compensate for possible redundancy in the attributes. Having produced a rule set for the class, each rule is reconsidered and two variants produced, again using reduced-error pruning—but at this stage, instances covered by other rules for the class are removed from the pruning set, and success rate on the remaining instances is used as the pruning criterion. If one of the two variants yields a better description length, it replaces the rule. Next we reactivate the original building phase to mop up any newly uncovered instances of the class. A final check is made, to ensure that each rule contributes to the reduction of description length, before proceeding to generate rules for the next class. Obtaining Rules from Partial Decision Trees There is an alternative approach to rule induction that avoids global optimization but nevertheless produces accurate, compact rule sets. The method combines the divide-and-conquer strategy for decision tree learning with the separate-and-conquer Initialize E to the instance set For each class C, from smallest to largest BUILD: Split E into Growing and Pruning sets in the ratio 2:1 Repeat until (a) there are no more uncovered examples of C; or (b) the description length (DL) of ruleset and examples is 64 bits greater than the smallest DL found so far, or (c) GROW phase: Grow a rule by greedily adding conditions until the rule is 100% accurate by testing every possible value of each attribute and selecting the condition with greatest PRUNE phase: Prune conditions in last-to-first order. Continue as long as the worth W of the rule increases OPTIMIZE: GENERATE VARIANTS: For each rule R for class C, Split E afresh into Growing and Pruning sets Remove all instances from the Pruning set that are covered by other rules for C Use GROW and PRUNE to generate and prune two competing rules from the newly split data: R1 is a new rule, rebuilt from scratch; R2 is generated by greedily adding antecedents to R. Prune using the metric A (instead of W) on this reduced data SELECT REPRESENTATIVE: Replace R by whichever of R, R1 and R2 has the smallest DL. MOP UP: If there are residual uncovered instances of class C, return to the BUILD stage to generate more rules based on these CLEAN UP: Calculate DL for the whole ruleset and for the ruleset with each rule in turn omitted; delete any rule that increases the DL the error rate exceeds 50%: information gain G instances. Remove instances covered by the rules just generated (a) Continue FIGURE 6.4 RIPPER: (a) algorithm for rule learning and (b) meaning of symbols. p = number of positive examples covered by this rule (true n = number of negative examples covered by this rule (false t = p + n; total number of examples covered by this rule n′ = N – n; number of negative examples not covered by this rule (true negatives) P = number of positive examples of this class N = number of negative examples of this class T = P + N; total number of examples of this class G = p[log(p/t)–log(P/T)] W = A = ; accuracy for this rule p+1 t+2 p+n′ T positives) negatives) (b) 210 CHAPTER 6 Implementations: Real Machine Learning Schemes FIGURE 6.5 Algorithm for expanding examples into a partial tree. Expand-subset (S): Choose a test T and use it to split the set of examples into subsets Sort subsets into increasing order of average entropy while (there is a subset X that has not yet been expanded AND all subsets expanded so far are leaves) expand-subset(X) if (all the subsets expanded are leaves AND estimated error for subtree ≥ estimated error for node) undo expansion into subsets and make node a leaf one for rule learning. It adopts the separate-and-conquer strategy in that it builds a rule, removes the instances it covers, and continues creating rules recursively for the remaining instances until none are left. However, it differs from the standard approach in the way that each rule is created. In essence, to make a single rule a pruned decision tree is built for the current set of instances, the leaf with the largest coverage is made into a rule, and the tree is discarded. The prospect of repeatedly building decision trees only to discard most of them is not as bizarre as it first seems. Using a pruned tree to obtain a rule instead of pruning a rule incrementally by adding conjunctions one at a time avoids a tendency to overprune, which is a characteristic problem of the basic separate-and-conquer rule learner. Using the separate-and-conquer methodology in conjunction with deci- sion trees adds flexibility and speed. It is indeed wasteful to build a full decision tree just to obtain a single rule, but the process can be accelerated significantly without sacrificing the advantages. The key idea is to build a partial decision tree instead of a fully explored one. A partial decision tree is an ordinary decision tree that contains branches to undefined subtrees. To generate such a tree, the construction and pruning operations are inte- grated in order to find a “stable” subtree that can be simplified no further. Once this subtree has been found, tree building ceases and a single rule is read off. The tree-building algorithm is summarized in Figure 6.5: It splits a set of instances recursively into a partial tree. The first step chooses a test and divides the instances into subsets accordingly. The choice is made using the same information-gain heu- ristic that is normally used for building decision trees (Section 4.3). Then the subsets are expanded in increasing order of their average entropy. The reason for this is that the later subsets will most likely not end up being expanded, and a subset with low- average entropy is more likely to result in a small subtree and therefore produce a more general rule. This proceeds recursively until a subset is expanded into a leaf, and then continues further by backtracking. But as soon as an internal node appears that has all its children expanded into leaves, the algorithm checks whether that node is better replaced by a single leaf. This is just the standard subtree replacement 6.2 Classification Rules 211 FIGURE 6.6 Example of building a partial tree. 1 2 3 4 1 2 3 4 5 1 2 3 4 5 1 2 3 4 1 2 4 (a) (b) (d) (e) (c) operation of decision tree pruning (see Section 6.1). If replacement is performed the algorithm backtracks in the standard way, exploring siblings of the newly replaced node. However, if during backtracking a node is encountered all of whose children expanded so far are not leaves—and this will happen as soon as a potential subtree replacement is not performed—then the remaining subsets are left unexplored and the corresponding subtrees are left undefined. Due to the recursive structure of the algorithm, this event automatically terminates tree generation. Figure 6.6 shows a step-by-step example. During the stages in Figure 6.6(a–c), tree building continues recursively in the normal way—except that at each point the lowest-entropy sibling is chosen for expansion: node 3 between stages (a) and (b). Gray elliptical nodes are as yet unexpanded; rectangular ones are leaves. Between stages (b) and (c), the rectangular node will have lower entropy than its sibling, node 5, but cannot be expanded further because it is a leaf. Backtracking occurs and node 5 is chosen for expansion. Once stage of Figure 6.6(c) is reached, there is a node— node 5—that has all its children expanded into leaves, and this triggers pruning. Subtree replacement for node 5 is considered and accepted, leading to stage (d). Next node 3 is considered for subtree replacement, and this operation is again accepted. Backtracking continues, and node 4, having lower entropy than node 2, is expanded into two leaves. Now subtree replacement is considered for node 4, but suppose that node 4 is not replaced. At this point, the process terminates with the three-leaf partial tree of stage (e). 212 CHAPTER 6 Implementations: Real Machine Learning Schemes If the data is noise-free and contains enough instances to prevent the algorithm from doing any pruning, just one path of the full decision tree has to be explored. This achieves the greatest possible performance gain over the naïve method that builds a full decision tree each time. The gain decreases as more pruning takes place. For datasets with numeric attributes, the asymptotic time complexity of the algo- rithm is the same as building the full decision tree because in this case the complexity is dominated by the time required to sort the attribute values in the first place. Once a partial tree has been built, a single rule is extracted from it. Each leaf corresponds to a possible rule, and we seek the “best” leaf of those subtrees (typi- cally a small minority) that have been expanded into leaves. Experiments show that it is best to aim at the most general rule by choosing the leaf that covers the greatest number of instances. When a dataset contains missing values, they can be dealt with exactly as they are when building decision trees. If an instance cannot be assigned to any given branch because of a missing attribute value, it is assigned to each of the branches with a weight proportional to the number of training instances going down that branch, normalized by the total number of training instances with known values at the node. During testing, the same procedure is applied separately to each rule, thus associating a weight with the application of each rule to the test instance. That weight is deducted from the instance’s total weight before it is passed to the next rule in the list. Once the weight has reduced to 0, the predicted class probabilities are com- bined into a final classification according to the weights. This yields a simple but surprisingly effective method for learning decision lists for noisy data. Its main advantage over other comprehensive rule-generation schemes is simplicity, because other methods appear to require a complex global optimization stage to achieve the same level of performance. Rules with Exceptions In Section 3.4 (page 73) we learned that a natural extension of rules is to allow them to have exceptions, and exceptions to the exceptions, and so on—indeed, the whole rule set can be considered as exceptions to a default classification rule that is used when no other rules apply. The method of generating a “good” rule, using one of the measures described previously, provides exactly the mechanism needed to gener- ate rules with exceptions. First, a default class is selected for the top-level rule: It is natural to use the class that occurs most frequently in the training data. Then, a rule is found pertaining to any class other than the default one. Of all such rules it is natural to seek the one with the most discriminatory power—for example, the one with the best evaluation on a test set. Suppose this rule has the form if then class = It is used to split the training data into two subsets: one containing instances for which the rule’s condition is true and the other containing those for which it is false. 6.2 Classification Rules 213 FIGURE 6.7 Rules with exceptions for the iris data. --> Iris setosa 50/150 petal length >= 2.45 petal width < 1.75 petal length < 5.35 --> Iris versicolor 49/52 petal length >= 4.95 petal width < 1.55 --> Iris virginica 2/2 petal length >= 3.35 --> Iris virginica 47/48 sepal length < 4.95 sepal width >= 2.45 --> Iris virginica 1/1 petal length < 4.85 sepal length < 5.95 --> Iris versicolor 1/1 If either subset contains instances of more than one class, the algorithm is invoked recursively on that subset. For the subset for which the condition is true, the “default class” is the new class as specified by the rule; for the subset where the condition is false, the default class remains as it was before. Let’s examine how this algorithm would work for the rules with exceptions that were given in Section 3.4 for the iris data of Table 1.4. We will represent the rules in the graphical form shown in Figure 6.7, which is in fact equivalent to the textual rules noted in Figure 3.8. The default of Iris setosa is the entry node at the top left. Hori- zontal, dotted paths show exceptions, so the next box, which contains a rule that concludes Iris versicolor, is an exception to the default. Below this is an alternative, a second exception—alternatives are shown by vertical, solid lines—leading to the conclusion Iris virginica. Following the upper path horizontally leads to an exception to the Iris versicolor rule that overrides it whenever the condition in the top right box holds, with the conclusion Iris virginica. Below this is an alternative, leading (as it happens) to the same conclusion. Returning to the box at bottom center, this has its own exception, the lower right box, which gives the conclusion Iris versicolor. The numbers at the lower right of each box give the “coverage” of the rule, expressed as the number of examples that satisfy it divided by the number that satisfy its condition but not its conclusion. For example, the condition in the top center box applies to 52 of the examples, and 49 of them are Iris versicolor. The strength of this representation is that you can get a very good feeling for the effect of the rules from the boxes toward the left side; the boxes at the right cover just a few exceptional cases. To create these rules, the default is first set to Iris setosa by taking the most frequently occurring class in the dataset. This is an arbitrary choice because, for this 214 CHAPTER 6 Implementations: Real Machine Learning Schemes dataset, all classes occur exactly 50 times; as shown in Figure 6.7 this default “rule” is correct in 50 out of 150 cases. Then the best rule that predicts another class is sought. In this case it is if petal-length ≥ 2.45 and petal-length < 5.355 and petal-width < 1.75 then Iris-versicolor This rule covers 52 instances, of which 49 are Iris versicolor. It divides the dataset into two subsets: the 52 instances that satisfy the condition of the rule and the remaining 98 that do not. We work on the former subset first. The default class for these instances is Iris versicolor: There are only three exceptions, all of which happen to be Iris virginica. The best rule for this subset that does not predict Iris versicolor is if petal-length ≥ 4.95 and petal-width < 1.55 then Iris-virginica It covers two of the three Iris virginicas and nothing else. Again, it divides the subset into two: those instances that satisfy its condition and those that do not. Fortunately, in this case, all those instances that satisfy the condition do indeed have class Iris virginica, so there is no need for a further exception. However, the remaining instances still include the third Iris virginica, along with 49 Iris versicolors, which are the default at this point. Again the best rule is sought: if sepal-length < 4.95 and sepal-width ≥ 2.45 then Iris-virginica This rule covers the remaining Iris virginica and nothing else, so it also has no exceptions. Furthermore, all remaining instances in the subset that do not satisfy its condition have the class Iris versicolor, which is the default, so no more needs to be done. Return now to the second subset created by the initial rule, the instances that do not satisfy the condition petal-length ≥ 2.45 and petal-length < 5.355 and petal-width < 1.75 Of the rules for these instances that do not predict the default class Iris setosa, the best is if petal-length ≥ 3.35 then Iris-virginica It covers all 47 Iris virginicas that are in the example set (3 were removed by the first rule, as explained previously). It also covers 1 Iris versicolor. This needs to be taken care of as an exception, by the final rule: if petal-length < 4.85 and sepal-length < 5.95 then Iris-versicolor Fortunately, the set of instances that do not satisfy its condition are all the default, Iris setosa. Thus, the procedure is finished. The rules that are produced have the property that most of the examples are covered by the high-level rules and the lower-level ones really do represent excep- tions. For example, the last exception clause and the deeply nested else clause both 6.2 Classification Rules 215 cover a solitary example, and removing them would have little effect. Even the remaining nested exception rule covers only two examples. Thus, one can get an excellent feeling for what the rules do by ignoring all the deeper structure and looking only at the first level or two. That is the attraction of rules with exceptions. Discussion All algorithms for producing classification rules that we have described use the basic covering or separate-and-conquer approach. For the simple, noise-free case this produces PRISM (Cendrowska, 1987), an algorithm that is simple and easy to understand. When applied to two-class problems with the closed-world assumption, it is only necessary to produce rules for one class: Then the rules are in disjunctive normal form and can be executed on test instances without any ambiguity arising. When applied to multiclass problems, a separate rule set is produced for each class; thus, a test instance may be assigned to more than one class, or to no class, and further heuristics are necessary if a unique prediction is sought. To reduce overfitting in noisy situations, it is necessary to produce rules that are not “perfect” even on the training set. To do this it is necessary to have a measure for the “goodness,” or worth, of a rule. With such a measure it is then possible to abandon the class-by-class approach of the basic covering algorithm and start by generating the very best rule, regardless of which class it predicts, and then remove all examples covered by this rule and continue the process. This yields a method for producing a decision list rather than a set of independent classification rules, and decision lists have the important advantage that they do not generate ambiguities when interpreted. The idea of incremental reduced-error pruning is from Fürnkranz and Widmer (1994) and forms the basis for fast and effective rule induction. The RIPPER rule learner is from Cohen (1995), although the published description appears to differ from the implementation in precisely how the description length (DL) affects the stopping condition. What we have presented here is the basic idea of the algorithm; there are many more details in the implementation. The whole question of measuring the value of a rule has not yet been satis- factorily resolved. Many different measures have been proposed, some blatantly heuristic and others based on information-theoretical or probabilistic grounds. However, there seems to be no consensus on the best measure to use. An exten- sive theoretical study of various criteria has been performed by Fürnkranz and Flach (2005). The rule-learning scheme based on partial decision trees was developed by Frank and Witten (1998). On standard benchmark datasets it produces rule sets that are as accurate as rules generated by C4.5 and more accurate than those of RIPPER; however, it produces larger rule sets than RIPPER. Its main advantage over other schemes is not performance but simplicity: By combining top-down decision tree induction with separate-and-conquer rule learning, it produces good rule sets without any need for global optimization. 216 CHAPTER 6 Implementations: Real Machine Learning Schemes The procedure for generating rules with exceptions was developed as an option in the Induct system by Gaines and Compton (1995), who called them ripple-down rules. In an experiment with a large medical dataset (22,000 instances, 32 attributes, and 60 classes), they found that people can understand large systems of rules with exceptions more readily than equivalent systems of regular rules because that is the way they think about the complex medical diagnoses that are involved. Richards and Compton (1998) describe their role as an alternative to classic knowledge engineering. 6.3 ASSOCIATION RULES In Section 4.5 we studied the Apriori algorithm for generating association rules that meet minimum support and confidence thresholds. Apriori follows a generate-and- test methodology for finding frequent item sets, generating successively longer candidate item sets from shorter ones that are known to be frequent. Each different size of candidate item set requires a scan through the dataset to determine whether its frequency exceeds the minimum support threshold. Although some improvements to the algorithm have been suggested to reduce the number of scans of the dataset, the combinatorial nature of this generation process can prove costly, particularly if there are many item sets or item sets are large. Both conditions readily occur even for modest datasets when low support thresholds are used. Moreover, no matter how high the threshold, if the data is too large to fit in main memory, it is undesir- able to have to scan it repeatedly—and many association rule applications involve truly massive datasets. These effects can be ameliorated by using appropriate data structures. We describe a method called FP-growth that uses an extended prefix tree—a frequent- pattern tree, or FP-tree—to store a compressed version of the dataset in main memory. Only two passes are needed to map a dataset into an FP-tree. The algorithm then processes the tree in a recursive fashion to grow large item sets directly, instead of generating candidate item sets and then having to test them against the entire database. Building a Frequent-Pattern Tree Like Apriori, the FP-growth algorithm begins by counting the number of times individual items (i.e., attribute–value pairs) occur in the dataset. After this initial pass, a tree structure is created in a second pass. Initially, the tree is empty and the structure emerges as each instance in the dataset is inserted into it. The key to obtaining a compact tree structure that can be quickly processed to find large item sets is to sort the items in each instance in descending order of their frequency of occurrence in the dataset, which has already been recorded in the first pass, before inserting them into the tree. Individual items in each instance that do not meet the minimum support threshold are not inserted into the tree, effectively removing them from the dataset. The hope is that many instances will share those 6.3 Association Rules 217 items that occur most frequently individually, resulting in a high degree of compres- sion close to the tree’s root. We illustrate the process with the weather data, reproduced in Table 6.1(a), using a minimum support threshold of 6. The algorithm is complex, and its complexity far exceeds what would be reasonable for such a trivial example, but a small illus- tration is the best way of explaining it. Table 6.1(b) shows the individual items, with their frequencies, that are collected in the first pass. They are sorted into descending order and ones whose frequency exceeds the minimum threshold are bolded. Table 6.1(c) shows the original instances, numbered as in Table 6.1(a), with the items in each instance sorted into descending frequency order. Finally, to give Table 6.1 Preparing Weather Data for Insertion into an FP-Tree (a) Outlook Temperature Humidity Windy Play 1 sunny hot high false no 2 sunny hot high true no 3 overcast hot high false yes 4 rainy mild high false yes 5 rainy cool normal false yes 6 rainy cool normal true no 7 overcast cool normal true yes 8 sunny mild high false no 9 sunny cool normal false yes 10 rainy mild normal false yes 11 sunny mild normal true yes 12 overcast mild high true yes 13 overcast hot normal false yes 14 rainy mild high true no (b) play = yes 9 windy = false 8 humidity = normal 7 humidity = high 7 windy = true 6 temperature = mild 6 play = no 5 outlook = sunny 5 outlook = rainy 5 temperature = hot 4 temperature = cool 4 outlook = overcast 4 Continued 218 CHAPTER 6 Implementations: Real Machine Learning Schemes (c) 1 windy = false, humidity = high, play = no, outlook = sunny, temperature = hot 2 humidity = high, windy = true, play = no, outlook = sunny, temperature = hot 3 play = yes, windy = false, humidity = high, temperature = hot, outlook = overcast 4 play = yes, windy = false, humidity = high, temperature = mild, outlook = rainy 5 play = yes, windy = false, humidity = normal, outlook = rainy, temperature = cool 6 humidity = normal, windy = true, play = no, outlook = rainy, temperature = cool 7 play = yes, humidity = normal, windy = true, temperature = cool, outlook = overcast 8 windy = false, humidity = high, temperature = mild, play = no, outlook = sunny 9 play = yes, windy = false, humidity = normal, outlook = sunny, temperature = cool 10 play = yes, windy = false, humidity = normal, temperature = mild, outlook = rainy 11 play = yes, humidity = normal, windy = true, temperature = mild, outlook = sunny 12 play = yes, humidity = high, windy = true, temperature = mild, outlook = overcast 13 play = yes, windy = false, humidity = normal, temperature = hot, outlook = overcast 14 humidity = high, windy = true, temperature = mild, play = no, outlook = rainy (d) play = yes and windy = false 6 play = yes and humidity = normal 6 (a) The original data, (b) frequency ordering of items with frequent item sets in bold, (c) the data with each instance sorted into frequency order, and (d) the two multiple-item frequent item sets. Table 6.1 Preparing Weather Data for Insertion into an FP-Tree Continued an advance peek at the final outcome, Table 6.1(d) shows the only two multiple- item sets whose frequency satisfies the minimum support threshold. Along with the six single-item sets shown in bold in Table 6.1(b), these form the final answer: a total of eight item sets. We are going to have to do a lot of work to find the two multiple-item sets in Table 6.1(d) using the FP-tree method. 6.3 Association Rules 219 Figure 6.8(a) shows the FP-tree structure that results from this data with a minimum support threshold of 6. The tree itself is shown with solid arrows. The numbers at each node show how many times the sorted prefix of items, up to and including the item at that node, occur in the dataset. For example, following the third branch from the left in the tree we can see that, after sorting, two instances begin with the prefix humidity = high—that is, the second and last instances of Table 6.1(c). Continuing down that branch, the next node records that the same two instances also have windy = true as their next most frequent item. The lowest node in the branch shows that one of these two instances— that is, the last in Table 6.1(c)—contains temperature = mild as well. The other instance—that is, the second in Table 6.1(c)—drops out at this stage because its next most frequent item does not meet the minimum support constraint and is therefore omitted from the tree. On the left side of the diagram a “header table” shows the frequencies of the individual items in the dataset (Table 6.1(b)). These items appear in descending frequency order, and only those with at least minimum support are included. Each item in the header table points to its first occurrence in the tree, and subsequent items in the tree with the same name are linked together to form a list. These lists, emanating from the header table, are shown in Figure 6.8(a) by dashed arrows. It is apparent from the tree that only two nodes have counts that satisfy the minimum support threshold, corresponding to the item sets play = yes (count of 9) and play = yes and windy = false (count of 6) in the leftmost branch. Each entry in the header table is itself a single-item set that also satisfies the threshold. This identifies as part of the final answer all the bold items in Table 6.1(b) and the first item set in Table 6.1(d). Since we know the outcome in advance we can see that there is only one more item set to go—the second in Table 6.1(d). But there is no hint of it in the data structure of Figure 6.8(a), and we will have to do a lot of work to discover it! Finding Large Item Sets The purpose of the links from the header table into the tree structure is to facili- tate traversal of the tree to find other large item sets, apart from the two that are already in the tree. This is accomplished by a divide-and-conquer approach that recursively processes the tree to grow large item sets. Each header table list is followed in turn, starting from the bottom of the table and working upward. Actually, the header table can be processed in any order, but it is easier to think about processing the longest paths in the tree first, and these correspond to the lower-frequency items. Starting from the bottom of the header table, we can immediately add tempera- ture = mild to the list of large item sets. Figure 6.8(b) shows the result of the next stage, which is an FP-tree for just those instances in the dataset that include 220 (a) Root play = yes (9) windy = false (6) humidity = normal (2) humidity = normal (4) temperature = mild (1) temperature = mild (6) windy = true (6) humidity = normal (7) humidity = high (7) windy = false (8) play = yes (9) temperature = mild (1) temperature = mild (1) temperature = mild (1) temperature = mild (1) temperature = mild (1) humidity = normal (1) humidity = high (1) humidity = high (2) humidity = high (2) humidity = high (2) windy = false (2) windy = true (2) windy = true (1) windy = true (2) windy = true (1) FIGURE 6.8 Extended prefix trees for the weather data: (a) the full data, (b) the data conditional on temperature = mild, and (c) the data conditional on humidity = normal. 221 (b) windy = true (3) humidity = normal (2) humidity = high (4) windy = false (3) windy = false (2) play = yes (4) windy = true (1) windy = true (1) windy = true (1) windy = false (1) humidity = normal (1) humidity = normal (1) humidity = high (1) humidity = high (1) humidity = high (1) humidity = high (1) play = yes (4) Root (c) windy = false (4) windy = false (4) play = yes (6) play = yes (6) Root FIGURE 6.8, cont’d 222 CHAPTER 6 Implementations: Real Machine Learning Schemes temperature = mild. This tree was not created by rescanning the dataset but by further processing of the tree in Figure 6.8(a), as follows. To see if a larger item set containing temperature = mild can be grown, we follow its link from the header table. This allows us to find all instances that contain tempera- ture = mild. From here the new tree in Figure 6.8(b) is created, with counts projected from the original tree corresponding to the set of instances that are conditional on the presence of temperature = mild. This is done by propagating the counts from the tem- perature = mild nodes up the tree, each node receiving the sum of its children’s counts. A quick glance at the header table for this new FP-tree shows that the temperature = mild pattern cannot be grown any larger because there are no individual items, con- ditional on temperature = mild, that meet the minimum support threshold. Note, however, that it is necessary to create the whole Figure 6.8(b) tree in order to discover this because it is effectively being created bottom up and the counts in the header table to the left are computed from the numbers in the tree. The recursion exits at this point, and processing continues on the remaining header table items in the original FP-tree. Figure 6.8(c) shows a second example, the FP-tree that results from following the header table link for humidity = normal. Here the windy = false node has a count of 4, corresponding to the four instances that had humidity = normal in the node’s left branch in the original tree. Similarly, play = yes has a count of 6, corresponding to the four instances from windy = false and the two instances that contain humidity = normal from the middle branch of the subtree rooted at play = yes in Figure 6.8(a). Processing the header list for this FP-tree shows that the humidity = normal item set can be grown to include play = yes since these two occur together six times, which meets the minimum support constraint. This corresponds to the second item set in Table 6.1(d), which in fact completes the output. However, in order to be sure that there are no other eligible item sets it is necessary to continue processing the entire header link table in Figure 6.8(a). Once the recursive tree mining process is complete all large item sets that meet the minimum support threshold have been found. Then association rules are created using the approach explained in Section 4.5. Studies have claimed that the FP-growth algorithm is up to an order of magnitude faster than Apriori at finding large item sets, although this depends on the details of the implementation and the nature of the dataset. Discussion The process of recursively creating projected FP-trees can be efficiently implemented within a single prefix tree structure by having a list of frequencies, indexed by recur- sion depth, at each node in the tree and each element of the header table. The tree structure itself is usually far smaller than the original dataset, and if the dataset is dense it achieves a high level of compression. This outweighs the overhead imposed by the pointers and counters that must be maintained at each node. Only when the support threshold is set very low does the FP-tree’s ability to compress the dataset degrade. Under these conditions, the tree becomes bushy, with little node sharing. On massive datasets for which the frequent-pattern tree exceeds main memory, 6.4 Extending Linear Models 223 disk-resident trees can be constructed using indexing techniques that have been developed for relational database systems. The FP-tree data structure and FP-growth algorithm for finding large item sets without candidate generation were introduced by Han et al. (2000) following pio- neering work by Zaki et al. (1997); Han et al. (2004) give a more comprehensive description. It has been extended in various ways. Wang et al. (2003) develop an algorithm called CLOSET+ to mine closed item sets—that is, sets for which there is no proper superset that has the same support. Finding large closed item sets pro- vides essentially the same information as finding the complete set of large item sets, but produces few redundant rules and thus eases the task that users face when exam- ining the output of the mining process. GSP (Generalized Sequential Patterns) is a method based on the Apriori algorithm for mining patterns in databases of event sequences (Srikant and Agrawal, 1996). A similar approach to FP-growth is used for event sequences by algorithms called PrefixSpan (Pei et al., 2004) and CloSpan (Yan et al., 2003), and for graph patterns by algorithms called gSpan (Yan and Han, 2002) and CloseGraph (Yan and Han, 2003). Ceglar and Roddick (2006) provide a comprehensive survey of association rule mining. Some authors have worked on integrating association rule mining with classification. For example, Liu et al. (1998) mine a kind of association rule that they call a “class association rule,” and build a classifier on the rules that are found using a technique they call CBA (Classification Based on Associations). Mutter et al. (2004) use classification to evaluate the output of confidence-based association rule mining, and find that standard learners for classification rules are generally preferable to CBA when runtime and size of the rule sets is of concern. 6.4 EXTENDING LINEAR MODELS Section 4.6 described how simple linear models can be used for classification in situations where all attributes are numeric. Their biggest disadvantage is that they can only represent linear boundaries between classes, which makes them too simple for many practical applications. Support vector machines use linear models to imple- ment nonlinear class boundaries. (Although it is a widely used term, support vector machines is something of a misnomer: These are algorithms, not machines.) How can this be possible? The trick is easy: Transform the input using a nonlinear mapping. In other words, transform the instance space into a new space. With a nonlinear mapping, a straight line in the new space doesn’t look straight in the original instance space. A linear model constructed in the new space can represent a nonlinear decision boundary in the original space. Imagine applying this idea directly to the ordinary linear models in Section 4.6. For example, the original set of attributes could be replaced by one giving all prod- ucts of n factors that can be constructed from these attributes. An example for two attributes, including all products with three factors, is x wa waa waa wa= + + +1 1 3 2 1 2 2 3 1 2 2 4 2 3 224 CHAPTER 6 Implementations: Real Machine Learning Schemes Here, x is the outcome, a1 and a2 are the two attribute values, and there are four weights wi to be learned. As described in Section 4.6, the result can be used for classification by training one linear system for each class and assigning an unknown instance to the class that gives the greatest output x—the standard technique of multiresponse linear regression. Then, a1 and a2 will be the attribute values for the test instance. To generate a linear model in the space that is spanned by these products, each training instance is mapped into the new space by computing all possible three-factor products of its two attribute values. The learning algorithm is then applied to the transformed instances. To classify an instance, it is processed by the same transformation prior to classification. There is nothing to stop us from adding in more synthetic attributes. For example, if a constant term were included, the original attributes and all two-factor products of them would yield a total of 10 weights to be learned. (Alternatively, adding an additional attribute with a value that was always a constant would have the same effect.) Indeed, polynomi- als of sufficiently high degree can approximate arbitrary decision boundaries to any required accuracy. It seems too good to be true—and it is. As you will probably have guessed, problems arise with this procedure due to the large number of coefficients introduced by the transformation in any realistic setting. The first snag is computational com- plexity. With 10 attributes in the original dataset, suppose we want to include all products with five factors: then the learning algorithm will have to determine more than 2000 coefficients. If its runtime is cubic in the number of attributes, as it is for linear regression, training will be infeasible. That is a problem of practicality. The second problem is one of principle: overfitting. If the number of coefficients is large relative to the number of training instances, the resulting model will be “too nonlinear”—it will overfit the training data. There are just too many parameters in the model. Maximum-Margin Hyperplane Support vector machines address both problems. They are based on an algorithm that finds a special kind of linear model: the maximum-margin hyperplane. We already know what a hyperplane is—it’s just another term for a linear model. To visualize a maximum-margin hyperplane, imagine a two-class dataset whose classes are linearly separable—that is, there is a hyperplane in instance space that classifies all training instances correctly. The maximum-margin hyperplane is the one that gives the greatest separation between the classes—it comes no closer to either than it has to. An example is shown in Figure 6.9, where the classes are represented by open and filled circles, respectively. Technically, the convex hull of a set of points is the tightest enclosing convex polygon: Its outline emerges when you connect every point of the set to every other point. Because we have supposed that the two classes are linearly separable, their convex hulls cannot overlap. Among all hyperplanes that separate the classes, the maximum-margin hyperplane is the one that is as far as 6.4 Extending Linear Models 225 can easily construct the maximum-margin hyperplane. All other training instances are irrelevant—they can be deleted without changing the position and orientation of the hyperplane. FIGURE 6.9 A maximum-margin hyperplane. maximum margin hyperplane support vectors A hyperplane separating the two classes might be written as x w w a w a= + +0 1 1 2 2 in the two-attribute case, where a1 and a2 are the attribute values and there are three weights wi to be learned. However, the equation defining the maximum-margin hyperplane can be written in another form, in terms of the support vectors. Write the class value y of a training instance as either 1 (for yes, it is in this class) or –1 (for no, it is not). Then the maximum-margin hyperplane can be written as x b yi i i = + •∑ α a(i) a is support vector Here, yi is the class value of training instance a(i), while b and αi are numeric parameters that have to be determined by the learning algorithm. Note that a(i) and a are vectors. The vector a represents a test instance—just as the vector [a1, a2] represented a test instance in the earlier formulation. The vectors a(i) are the support vectors, those circled in Figure 6.9; they are selected members of the training set. The term a(i) • a represents the dot product of the test instance with one of the support vectors: a(i) • a = Σja(i)jaj. If you are not familiar with dot product notation, you should still be able to understand the gist of what follows: Just think of a(i) as the whole set of attribute values for the ith support vector. Finally, b and αi are parameters that determine the hyperplane, just as the weights w0, w1, and w2 are parameters that determine the hyperplane in the earlier formulation. It turns out that finding the support vectors for the training instances and determining the parameters b and αi belongs to a standard class of optimization problems known as constrained quadratic optimization. There are off-the-shelf software packages for solving these problems (see Fletcher, 1987, for a comprehensive and practical account of solution methods). However, the computational complexity can be reduced, and learning accelerated, if special-purpose algorithms for training support vector machines are applied—but the details of these algorithms lie beyond the scope of this book (see Platt, 1998). possible from both convex hulls—it is the perpendicular bisector of the short- est line connecting the hulls (shown dashed in the figure). The instances that are closest to the maximum-margin hyperplane— the ones with the minimum distance to it—are called support vectors. There is always at least one support vector for each class, and often there are more. The important thing is that the set of support vectors uniquely defines the maximum-margin hyperplane for the learning problem. Given the sup port vectors for the two classes, we 226 CHAPTER 6 Implementations: Real Machine Learning Schemes Nonlinear Class Boundaries We motivated the introduction of support vector machines by claiming that they can be used to model nonlinear class boundaries. However, so far we have only described the linear case. Consider what happens when an attribute transformation, as described before, is applied to the training data before determining the maximum-margin hyperplane. Recall that there are two problems with the straightforward application of such transformations to linear models: computational complexity on the one hand and overfitting on the other. With support vectors, overfitting is unlikely to occur. The reason is that it is inevitably associated with instability: With an algorithm that overfits, changing one or two instance vectors will make sweeping changes to large sections of the decision boundary. But the maximum-margin hyperplane is relatively stable: It only moves if training instances are added or deleted that are support vectors—and this is true even in the high-dimensional space spanned by the nonlinear transformation. Over- fitting is caused by too much flexibility in the decision boundary. The support vectors are global representatives of the whole set of training points, and there are usually few of them, which gives little flexibility. Thus, overfitting is less likely to occur. What about computational complexity? This is still a problem. Suppose that the transformed space is a high-dimensional one so that the transformed support vectors and test instance have many components. According to the preceding equation, every time an instance is classified its dot product with all support vectors must be calcu- lated. In the high-dimensional space produced by the nonlinear mapping this is rather expensive. Obtaining the dot product involves one multiplication and one addition for each attribute, and the number of attributes in the new space can be huge. This problem occurs not only during classification but also during training because the optimization algorithms have to calculate the same dot products very frequently. Fortunately, it turns out that it is possible to calculate the dot product before the nonlinear mapping is performed, on the original attribute set, using a so-called kernel function based on the dot product. A high-dimensional version of the preceding equation is simply x b yi i n= + •∑α ()a(i) a where n is chosen as the number of factors in the transformation (three in the example we used earlier). If you expand the term (a(i) • a)n, you will find that it contains all the high-dimensional terms that would have been involved if the test and training vectors were first transformed by including all products of n factors and the dot product of the result was taken. (If you actually do the calculation, you will notice that some constant factors—binomial coefficients—are introduced. However, these do not matter: It is the dimensionality of the space that concerns us; the constants merely scale the axes.) Because of this mathematical equivalence, the dot products can be computed in the original low-dimensional space, and the problem becomes feasible. In implementation terms, you take a software package for constrained quadratic optimization and every time 6.4 Extending Linear Models 227 The function (x • y)n, which computes the dot product of two vectors x and y and raises the result to the power n, is called a polynomial kernel. A good way of choosing the value of n is to start with 1 (a linear model) and increment it until the estimated error ceases to improve. Usually, quite small values suffice. To include lower-order terms, we can use the kernel (x • y + 1)n. Other kernel functions can be used instead to implement different nonlinear mappings. Two that are often suggested are the radial basis function (RBF) kernel and the sigmoid kernel. Which one produces the best results depends on the applica- tion, although the differences are rarely large in practice. It is interesting to note that a support vector machine with the RBF kernel is simply a type of neural network called an RBF network (which we describe later), and one with the sigmoid kernel implements another type of neural network, a multilayer perceptron with one hidden layer (also described later). Mathematically, any function K(x, y) is a kernel function if it can be written as K(x, y) = Φ(x) • Φ(y), where Φ is a function that maps an instance into a (potentially high-dimensional) feature space. In other words, the kernel function represents a dot product in the feature space created by Φ. Practitioners sometimes apply functions that are not proper kernel functions (the sigmoid kernel with certain parameter set- tings is an example). Despite the lack of theoretical guarantees, this can nevertheless produce accurate classifiers. Throughout this section, we have assumed that the training data is linearly separable—either in the instance space or in the new space spanned by the nonlinear mapping. It turns out that support vector machines can be generalized to the case where the training data is not separable. This is accomplished by placing an upper bound on the coefficients αi. Unfortunately, this parameter must be chosen by the user, and the best setting can only be determined by experimentation. Also, except in trivial cases it is not possible to determine a priori whether the data is linearly separable or not. Finally, we should mention that compared with other methods such as decision tree learners, even the fastest training algorithms for support vector machines are slow when applied in the nonlinear setting. However, they often produce very accurate classifiers because subtle and complex decision boundaries can be obtained. Support Vector Regression The maximum-margin hyperplane concept only applies to classification. However, support vector machine algorithms have been developed for numeric prediction that share many of the properties encountered in the classification case: They a(i) • a is evaluated you evaluate (a(i) • a)n instead. It’s as simple as that because in both the optimization and the classification algorithms these vectors are only used in this dot product form. The training vectors, including the support vectors, and the test instance all remain in the original low-dimensional space throughout the calculations. 228 CHAPTER 6 Implementations: Real Machine Learning Schemes FIGURE 6.10 Support vector regression: (a) ε = 1, (b) ε = 2, and (c) ε = 0.5. 0 2 4 6 8 10 0 2 4 6 8 10 Class Attribute (c) 0 2 4 6 8 10 0 2 4 6 8 10 Class Attribute 0 2 4 6 8 10 0 2 4 6 8 10 Class Attribute (a) (b) produce a model that can usually be expressed in terms of a few support vectors and can be applied to nonlinear problems using kernel functions. As with regular support vector machines, we will describe the concepts involved, but will not attempt to describe the algorithms that actually perform the work. As with linear regression, covered in Section 4.6, the basic idea is to find a function that approximates the training points well by minimizing the prediction error. The crucial difference is that all deviations up to a user-specified parameter ε are simply discarded. Also, when minimizing the error, the risk of overfitting is reduced by simultaneously trying to maximize the flatness of the function. Another difference is that what is minimized is normally the predictions’ absolute error instead of the squared error used in linear regression. (There are, however, versions of the algorithm that use the squared error instead.) A user-specified parameter ε defines a tube around the regression function in which errors are ignored: For linear support vector regression, the tube is a cylinder. If all training points can fit within a tube of width 2ε, the algorithm outputs the function in the middle of the flattest tube that encloses them. In this case the total perceived error is 0. Figure 6.10(a) shows a regression problem with one attribute, 6.4 Extending Linear Models 229 a numeric class, and eight instances. In this case ε was set to 1, so the width of the tube around the regression function (indicated by dotted lines) is 2. Figure 6.10(b) shows the outcome of the learning process when ε is set to 2. As you can see, the wider tube makes it possible to learn a flatter function. The value of ε controls how closely the function will fit the training data. Too large a value will produce a meaningless predictor—in the extreme case, when 2ε exceeds the range of class values in the training data, the regression line is horizontal and the algorithm just predicts the mean class value. On the other hand, for small values of ε there may be no tube that encloses all the data. In that case, some train- ing points will have nonzero error, and there will be a tradeoff between the prediction error and the tube’s flatness. In Figure 6.10(c), ε was set to 0.5 and there is no tube of width 1 that encloses all the data. For the linear case, the support vector regression function can be written as x b i i = + •∑ α a(i) a is support vector As with classification, the dot product can be replaced by a kernel function for nonlinear problems. The support vectors are all those points that do not fall strictly within the tube—that is, the points outside the tube and on its border. As with classification, all other points have coefficient 0 and can be deleted from the train- ing data without changing the outcome of the learning process. In contrast to the classification case, the αi may be negative. We have mentioned that as well as minimizing the error, the algorithm simultane- ously tries to maximize the flatness of the regression function. In Figures 6.10(a) and (b), where there is a tube that encloses all the training data, the algorithm simply outputs the flattest tube that does so. However, in Figure 6.10(c), there is no tube with error 0, and a tradeoff is struck between the prediction error and the tube’s flatness. This tradeoff is controlled by enforcing an upper limit C on the absolute value of the coefficients αi. The upper limit restricts the influence of the support vectors on the shape of the regression function and is a parameter that the user must specify in addition to ε. The larger C is, the more closely the function can fit the data. In the degenerate case ε = 0, the algorithm simply performs least-absolute-error regression under the coefficient size constraint and all training instances become support vectors. Conversely, if ε is large enough that the tube can enclose all the data, the error becomes 0, there is no tradeoff to make, and the algorithm outputs the flattest tube that encloses the data irrespective of the value of C. Kernel Ridge Regression Chapter 4 introduced classic least-squares linear regression as a technique for pre- dicting numeric quantities. In the previous section we saw how the powerful idea of support vector machines can be applied to regression and, furthermore, how nonlinear problems can be tackled by replacing the dot product in the support vector formulation by a kernel function—this is often known as the “kernel trick.” For 230 CHAPTER 6 Implementations: Real Machine Learning Schemes classic linear regression using squared loss, only simple matrix operations are needed to find the model, but this is not the case for support vector regression with the user-specified loss parameter ε. It would be nice to combine the power of the kernel trick with the simplicity of standard least-squares regression. Kernel ridge regression does just that. In contrast to support vector regression, it does not ignore errors smaller than ε, and the squared error is used instead of the absolute error. Instead of expressing the linear regression model’s predicted class value for a given test instance a as a weighted sum of the attribute values, as in Chapter 4, it can be expressed as a weighted sum over the dot products of each training instance aj and the test instance in question: αj j j n a a• = ∑0 where we assume that the function goes through the origin and an intercept is not required. This involves a coefficient αj for each training instance, which resembles the situation with support vector machines—except that here j ranges over all instances in the training data, not just the support vectors. Again, the dot product can be replaced by a kernel function to yield a nonlinear model. The sum of the squared errors of the model’s predictions on the training data is given by yi j j i j n i n − • == ∑∑ α a a 0 2 1 This is the squared loss, just as in Chapter 4, and again we seek to minimize it by choosing appropriate αj’s. But now there is a coefficient for each training instance, not just for each attribute, and most data sets have far more instances than attributes. This means that there is a serious risk of overfitting the training data when a kernel function is used instead of the dot product to obtain a nonlinear model. That is where the ridge part of kernel ridge regression comes in. Instead of minimizing the squared loss, we trade closeness of fit against model complexity by introducing a penalty term: yi j j i j n i n i j j i i j n − ⋅ + • == = ∑∑ ∑α λ α αa a a a 0 2 1 1, The second sum penalizes large coefficients. This prevents the model from placing too much emphasis on individual training instances by giving them large coefficients, unless this yields a correspondingly large drop in error. The parameter λ controls the tradeoff between closeness of fit and model complexity. When matrix operations are used to solve for the coefficients of the model, the ridge penalty also has the added benefit of stabilizing degenerate cases. For this reason, it is often applied in standard least-squares linear regression as well. Although kernel ridge regression has the advantage over support vector machines of computational simplicity, one disadvantage is that there is no sparseness in the vector of coefficients—in other words, no concept of “support vectors.” This makes a difference at prediction time because support vector machines have to sum only over the set of support vectors, not the entire training set. 6.4 Extending Linear Models 231 In a typical situation with more instances than attributes, kernel ridge regression is more computationally expensive than standard linear regression, even when using the dot product rather than a kernel. This is because of the complexity of the matrix inversion operation used to find the model’s coefficient vector. Standard linear regres- sion requires inverting an m × m matrix, which has complexity O(m3), where m is the number of attributes in the data. Kernel ridge regression, on the other hand, involves an n × n matrix, with complexity O(n3) where n is the number of instances in the train- ing data. Nevertheless, it is advantageous to use kernel ridge regression in cases where a nonlinear fit is desired, or where there are more attributes than training instances. Kernel Perceptron In Section 4.6 we introduced the perceptron algorithm for learning a linear classifier. It turns out that the kernel trick can also be used to upgrade this algorithm to learn nonlinear decision boundaries. To see this, we first revisit the linear case. The perceptron algorithm repeatedly iterates through the training data instance by instance and updates the weight vector every time one of these instances is misclassified based on the weights learned so far. The weight vector is updated simply by adding or subtracting the instance’s attribute values to or from it. This means that the final weight vector is just the sum of the instances that have been misclassified. The perceptron makes its predictions based on whether w aii i∑ is greater or less than 0, where wi is the weight for the ith attribute and ai the corresponding attribute value of the instance that we wish to classify. Instead, we could use y j a j ai iji ()()′∑∑ Here, a′ (j ) is the jth misclassified training instance, a′(j )i its ith attribute value, and y (j ) its class value (either +1 or –1). To implement this we no longer keep track of an explicit weight vector: We simply store the instances that have been misclassified so far and use the previous expression to make a prediction. It looks like we’ve gained nothing—in fact, the algorithm is much slower because it iterates through all misclassified training instances every time a prediction is made. However, closer inspection of this formula reveals that it can be expressed in terms of dot products between instances. First, swap the summation signs to yield y j a j aj i ii()()∑ ∑ ′ The second sum is just a dot product between two instances and can be written as y jj ()′ •∑ a ( ) aj This rings a bell! A similar expression for support vector machines enabled the use of kernels. Indeed, we can apply exactly the same trick here and use a kernel function instead of the dot product. Writing this function as K(…) gives y j Kj ()(,)′∑ a ( ) aj In this way the perceptron algorithm can learn a nonlinear classifier simply by keeping track of the instances that have been misclassified during the training process and using this expression to form each prediction. 232 CHAPTER 6 Implementations: Real Machine Learning Schemes If a separating hyperplane exists in the high-dimensional space implicitly created by the kernel function, this algorithm will learn one. However, it won’t learn the maximum-margin hyperplane found by a support vector machine classifier. This means that classification performance is usually worse. On the plus side, the algorithm is easy to implement and supports incremental learning. This classifier is called the kernel perceptron. It turns out that all sorts of algo- rithms for learning linear models can be upgraded by applying the kernel trick in a similar fashion. For example, logistic regression can be turned into kernel logistic regression. As we saw before, the same applies to regression problems: Linear regression can also be upgraded using kernels. Again, a drawback of these advanced methods for linear and logistic regression (if they are done in a straightforward manner) is that the solution is not “sparse”: Every training instance contributes to the solution vector. In support vector machines and the kernel perceptron, only some of the training instances affect the solution, and this can make a big difference in computational efficiency. The solution vector found by the perceptron algorithm depends greatly on the order in which the instances are encountered. One way to make the algorithm more stable is to use all the weight vectors encountered during learning, not just the final one, letting them vote on a prediction. Each weight vector contributes a certain number of votes. Intuitively, the “correctness” of a weight vector can be measured roughly as the number of successive trials after its inception in which it correctly classified subsequent instances and thus didn’t have to be changed. This measure can be used as the number of votes given to the weight vector, giving an algorithm known as the voted perceptron that performs almost as well as a support vector machine. (Note that, as mentioned earlier, the various weight vectors in the voted perceptron don’t need to be stored explicitly, and the kernel trick can be applied here too.) Multilayer Perceptrons Using a kernel is not the only way to create a nonlinear classifier based on the per- ceptron. In fact, kernel functions are a recent development in machine learning. Previously, neural network proponents used a different approach for nonlinear clas- sification: They connected many simple perceptron-like models in a hierarchical structure. This can represent nonlinear decision boundaries. Section 4.6 explained that a perceptron represents a hyperplane in instance space. We mentioned there that it is sometimes described as an artificial “neuron.” Of course, human and animal brains successfully undertake very complex clas- sification tasks—for example, image recognition. The functionality of each indi- vidual neuron that is in a brain is certainly not sufficient to perform these feats. How can they be solved by brainlike structures? The answer must lie in the fact that the neurons in the brain are massively interconnected, allowing a problem to be decomposed into subproblems that can be solved at the neuron level. 6.4 Extending Linear Models 233 This observation inspired the development of artificial networks of neurons— neural nets. Consider the simple dataset in Figure 6.11. Part (a) shows a two-dimensional instance space with four instances having classes 0 and 1, represented by white and black dots, respectively. No matter how you draw a straight line through this space, you will not be able to find one that separates all the black points from all the white ones. In other words, the problem is not linearly separable, and the simple percep- tron algorithm will fail to generate a separating hyperplane (in this two-dimensional instance space a hyperplane is just a straight line). The situation is different in Figure 6.11(b) and Figure 6.11(c): Both these problems are linearly separable. The same holds for Figure 6.11(d), which shows two points in a one-dimensional instance space (in the case of one dimension the separating hyperplane degenerates to a separating point). If you are familiar with propositional logic, you may have noticed that the four situations in Figure 6.11 correspond to four types of logical connectives. Figure 6.11(a) represents a logical XOR (exclusive-OR), where the class is 1 if and only if exactly one of the attributes has value 1. Figure 6.11(b) represents logical AND, where the class is 1 if and only if both attributes have value 1. Figure 6.11(c) represents OR, where the class is 0 only if both attributes have value 0. Figure 6.11(d) represents NOT, where the class is 0 if and only if the attribute has value 1. Because the last three are linearly separable, a perceptron can represent AND, OR, and NOT. Indeed, perceptrons for the corresponding datasets are shown in Figures 6.11(f–h), respectively. However, a simple perceptron cannot represent XOR because that is not linearly separable. To build a classifier for this type of problem a single perceptron is not sufficient—we need several of them. Figure 6.11(e) shows a network with three perceptrons, or units, labeled A, B, and C. The first two are connected to what is sometimes called the input layer of the network, representing the attributes in the data. As in a simple perceptron, the input layer has an additional constant input called the bias. However, the third unit does not have any connections to the input layer. Its input consists of the output of units A and B (either 0 or 1) and another constant bias unit. These three units make up the hidden layer of the multilayer perceptron. They are called “hidden” because the units have no direct connection to the environment. This layer is what enables the system to represent XOR. You can verify this by trying all four possible combinations of input signals. For example, if attribute a1 has value 1 and a2 has value 1, then unit A will output 1 (because 1 × 1 + 1 × 1 + −0.5 × 1 > 0), unit B will output 0 (because –1 × 1 + –1 × 1 + –1.5 × 1 < 0), and unit C will output 0 (because 1 × 1 + 1 × 0 + –1.5 × 1 < 0). This is the correct answer. Closer inspection of the behavior of the three units reveals that the first one represents OR, the second represents NAND (NOT com- bined with AND), and the third represents AND. Together they represent the expres- sion (a1 OR a2) AND (a1 NAND a2), which is precisely the definition of XOR. As this example illustrates, any expression from propositional calculus can be converted into a multilayer perceptron, because the three connectives AND, OR, and 234 CHAPTER 6 Implementations: Real Machine Learning Schemes 0.5 attribute a1 1 (“bias”) –1–0.5 attribute a1 attribute a2 1 (“bias”) 11 (g) (h) (a) 1 0 1 0 0 1 1 –1.5 attribute a1 attribute a2 1 (“bias”) C A B 1 1 –1.5 –0.5 1.5 11 attribute a1 attribute a2 1 (“bias”) 1 (“bias”) –1 –1 1 1 0 0 1 1 0 0 1 (b) (d) (e) (f) (c) FIGURE 6.11 Example datasets and corresponding perceptrons. 6.4 Extending Linear Models 235 NOT are sufficient for this and we have seen how each can be represented using a perceptron. Individual units can be connected together to form arbitrarily complex expressions. Hence, a multilayer perceptron has the same expressive power as, say, a decision tree. In fact, it turns out that a two-layer perceptron (not counting the input layer) is sufficient. In this case, each unit in the hidden layer corresponds to a variant of AND—because we assume that it may negate some of the inputs before forming the conjunction—joined by an OR that is represented by a single unit in the output layer. In other words, each node in the hidden layer has the same role as a leaf in a decision tree or a single rule in a set of decision rules. The big question is how to learn a multilayer perceptron. There are two aspects to the problem: learning the structure of the network and learning the connection weights. It turns out that there is a relatively simple algorithm for determining the weights given a fixed network structure. This algorithm is called backpropagation and is described in the next section. However, although there are many algorithms that attempt to identify network structure, this aspect of the problem is commonly solved by experimentation—perhaps combined with a healthy dose of expert knowledge. Sometimes the network can be separated into distinct modules that represent identifi- able subtasks (e.g., recognizing different components of an object in an image recogni- tion problem), which opens up a way of incorporating domain knowledge into the learning process. Often a single hidden layer is all that is necessary, and an appropriate number of units for that layer is determined by maximizing the estimated accuracy. Backpropagation Suppose we have some data and seek a multilayer perceptron that is an accurate predictor for the underlying classification problem. Given a fixed network structure, we must determine appropriate weights for the connections in the network. In the absence of hidden layers, the perceptron learning rule from Section 4.6 can be used to find suitable values. But suppose there are hidden units. We know what the output unit should predict and could adjust the weights of the connections leading to that unit based on the perceptron rule. But the correct outputs for the hidden units are unknown, so the rule cannot be applied there. It turns out that, roughly speaking, the solution is to modify the weights of the connections leading to the hidden units based on the strength of each unit’s contribu- tion to the final prediction. There is a standard mathematical optimization algorithm, called gradient descent, which achieves exactly that. Unfortunately, it requires taking derivatives, and the step function that the simple perceptron uses to convert the weighted sum of the inputs into a 0/1 prediction is not differentiable. We need to see whether the step function can be replaced by something else. Figure 6.12(a) shows the step function: If the input is smaller than 0, it outputs 0; otherwise, it outputs 1. We want a function that is similar in shape but differentiable. A commonly used replacement is shown in Figure 6.12(b). In neural networks terminol- ogy it is called the sigmoid function, and the version we consider here is defined by f x e x() = + − 1 1 236 CHAPTER 6 Implementations: Real Machine Learning Schemes FIGURE 6.12 Step versus sigmoid: (a) step function and (b) sigmoid function. 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 –10 –5 0 5 10–10 –5 0 5 10 (a) (b) We encountered it in Section 4.6 when we described the logit transform used in logistic regression. In fact, learning a multilayer perceptron is closely related to logistic regression. To apply the gradient descent procedure, the error function—the thing that is to be minimized by adjusting the weights—must also be differentiable. The number of misclassifications—measured by the discrete 0 – 1 loss mentioned in Section 5.6— does not fulfill this criterion. Instead, multilayer perceptrons are usually trained by minimizing the squared error of the network’s output, essentially treating it as an estimate of the class probability. (Other loss functions are also applicable. For example, if the negative log-likelihood is used instead of the squared error, learning a sigmoid-based perceptron is identical to logistic regression.) We work with the squared-error loss function because it is most widely used. For a single training instance, it is E y f x= −1 2 2( ( )) where f(x) is the network’s prediction obtained from the output unit and y is the instance’s class label (in this case, it is assumed to be either 0 or 1). The factor 12 is included just for convenience and will drop out when we start taking derivatives. Gradient descent exploits information given by the derivative of the function that is to be minimized—in this case, the error function. As an example, consider a hypothetical error function that happens to be identical to w 2 + 1, shown in Figure 6.13. The x-axis represents a hypothetical parameter w that is to be optimized. The derivative of w 2 + 1 is simply 2w. The crucial observation is that, based on the derivative, we can figure out the slope of the function at any particular point. If the derivative is negative, the function slopes downward to the right; if it is positive, 6.4 Extending Linear Models 237 FIGURE 6.13 Gradient descent using the error function w 2 + 1. 0 5 10 15 20 –4 –2 0 2 4 it slopes downward to the left; and the size of the derivative determines how steep the decline is. Gradient descent is an iterative optimization procedure that uses this information to adjust a function’s parameters. It takes the value of the derivative, multiplies it by a small constant called the learning rate, and subtracts the result from the current parameter value. This is repeated for the new parameter value, and so on, until a minimum is reached. Returning to the example, assume that the learning rate is set to 0.1 and the current parameter value w is 4. The derivative is double this—8 at this point. Multi plying by the learning rate yields 0.8, and subtracting this from 4 gives 3.2, which becomes the new parameter value. Repeating the process for 3.2, we get 2.56, then 2.048, and so on. The little crosses in Figure 6.13 show the values encountered in this process. The process stops once the change in parameter value becomes too small. In the example this happens when the value approaches 0, the value corre- sponding to the location on the x-axis where the minimum of the hypothetical error function is located. The learning rate determines the step size and hence how quickly the search converges. If it is too large and the error function has several minima, the search may overshoot and miss a minimum entirely, or it may oscillate wildly. If it is too small, progress toward the minimum may be slow. Note that gradient descent can only find a local minimum. If the function has several minima—and error functions for multilayer perceptrons usually have many—it may not find the best one. This is a significant drawback of standard multilayer perceptrons compared with, for example, support vector machines. 238 CHAPTER 6 Implementations: Real Machine Learning Schemes To use gradient descent to find the weights of a multilayer perceptron, the derivative of the squared error must be determined with respect to each parameter—that is, each weight in the network. Let’s start with a simple perceptron without a hidden layer. Differentiating the error function with respect to a particular weight wi yields dE dw f x y f x dwi i = −(())() Here, f(x) is the perceptron’s output and x is the weighted sum of the inputs. To compute the second factor on the right side, the derivative of the sigmoid function f(x) is needed. It turns out that this has a particularly simple form that can be written in terms of f(x) itself: df x dx f x f x()( )( ( ))= −1 We use f′(x) to denote this derivative. But we seek the derivative with respect to wi, not x. Because x w ai ii= ∑ the derivative of f(x) with respect to wi is df x dw f x a i i ()()= ′ Plugging this back into the derivative of the error function yields dE dw f x y f x a i i= − ′(())() This expression gives all that is needed to calculate the change of weight wi caused by a particular example vector a (extended by 1 to represent the bias, as explained previously). Having repeated this computation for each training instance, we add up the changes associated with a particular weight wi, multiply by the learning rate, and subtract the result from wi’s current value. So far so good. But all this assumes that there is no hidden layer. With a hidden layer, things get a little trickier. Suppose f(xi) is the output of the ith hidden unit, wij is the weight of the connection from input j to the ith hidden unit, and wi is the weight of the ith hidden unit to the output unit. The situation is depicted in Figure 6.14. As before, f(x) is the output of the single unit in the output layer. The update rule for the weights wi is essentially the same as above, except that ai is replaced by the output of the ith hidden unit: dE dw f x y f x f x i i= − ′(())()() However, to update the weights wij the corresponding derivatives must be calculated. Applying the chain rule gives dE dw dE dx dx dw f x y f x dx dwij ij ij = = − ′(())() The first two factors are the same as in the previous equation. To compute the third factor, differentiate further. Because x w f xi ii= ∑ () then dx dw w df x dwij i i ij = () 6.4 Extending Linear Models 239 Furthermore, x w ai ij jj= ∑ so df x dw f x dx dw f x ai ij i i ij i j ()()()= ′ = ′ This means that we are finished. Putting everything together yields an equation for the derivative of the error function with respect to the weights wij: dE dw f x y f x w f x a ij i i j= − ′′(())()() As before, we calculate this value for every training instance, add up the changes associated with a particular weight wij, multiply by the learning rate, and subtract the outcome from the current value of wij. This derivation applies to a perceptron with one hidden layer. If there are two hidden layers, the same strategy can be applied a second time to update the weights pertaining to the input connections of the first hidden layer, propagating the error from the output unit through the second hidden layer to the first one. Because of this error propagation mechanism, this version of the generic gradient descent strategy is called backpropagation. FIGURE 6.14 Multilayer perceptron with a hidden layer. hidden unit 0 hidden unit 1 hidden unit l output unit f(x) f(x0) w 0 w 10 w l0 w 01 w l1 w0k w 1k w lkw 11 w 00 w l w 1 f(x1) f(x l) input a0 input a1 input ak We have tacitly assumed that the network’s output layer has just one unit, which is appropriate for two-class problems. For more than two classes, a separate network could be learned for each class that distinguishes it from the remaining classes. A more compact classifier can be obtained from a single network by creating an output unit for each class, connecting every unit in the hidden layer to every output unit. 240 CHAPTER 6 Implementations: Real Machine Learning Schemes The squared error for a particular training instance is the sum of squared errors taken over all output units. The same technique can be applied to predict several targets, or attribute values, simultaneously by creating a separate output unit for each one. Intuitively, this may give better predictive accuracy than building a separate classi- fier for each class attribute if the underlying learning tasks are in some way related. We have assumed that weights are only updated after all training instances have been fed through the network and all the corresponding weight changes have been accumulated. This is batch learning because all the training data is processed together. But exactly the same formulas can be used to update the weights incre- mentally after each training instance has been processed. This is called stochastic backpropagation because the overall error does not necessarily decrease after every update. It can be used for online learning, in which new data arrives in a continuous stream and every training instance is processed just once. In both variants of back- propagation, it is often helpful to standardize the attributes to have zero mean and unit standard deviation. Before learning starts, each weight is initialized to a small, randomly chosen value based on a normal distribution with zero mean. Like any other learning scheme, multilayer perceptrons trained with backpropa- gation may suffer from overfitting, especially if the network is much larger than what is actually necessary to represent the structure of the underlying learning problem. Many modifications have been proposed to alleviate this. A very simple one, called early stopping, works like reduced-error pruning in rule learners: A holdout set is used to decide when to stop performing further iterations of the backpropagation algorithm. The error on the holdout set is measured and the algo- rithm is terminated once the error begins to increase because that indicates overfit- ting to the training data. Another method, called weight decay, adds to the error function a penalty term that consists of the squared sum of all weights in the network, as in ridge regression. This attempts to limit the influence of irrelevant connections on the network’s predictions by penalizing large weights that do not contribute a correspondingly large reduction in the error. Although standard gradient descent is the simplest technique for learning the weights in a multilayer perceptron, it is by no means the most efficient one. In practice, it tends to be rather slow. A trick that often improves performance is to include a momentum term when updating weights: Add to the new weight change a small proportion of the update value from the previous iteration. This smoothes the search process by making changes in direction less abrupt. More sophisticated methods make use of information obtained from the second derivative of the error function as well; they can converge much more quickly. However, even those algo- rithms can be very slow compared with other methods of classification learning. A serious disadvantage of multilayer perceptrons that contain hidden units is that they are essentially opaque. There are several techniques that attempt to extract rules from trained neural networks. However, it is unclear whether they offer any advan- tages over standard rule learners that induce rule sets directly from data, especially considering that this can generally be done much more quickly than learning a multilayer perceptron in the first place. 6.4 Extending Linear Models 241 Although multilayer perceptrons are the most prominent type of neural network, many others have been proposed. Multilayer perceptrons belong to a class of net- works called feed-forward networks because they do not contain any cycles and the network’s output depends only on the current input instance. Recurrent neural net- works do have cycles. Computations derived from earlier input are fed back into the network, which gives them a kind of memory. Radial Basis Function Networks Another popular type of feed-forward network is the radial basis function (RBF) network. It has two layers, not counting the input layer, and differs from a multilayer perceptron in the way that the hidden units perform computations. Each hidden unit essentially represents a particular point in input space, and its output, or activation, for a given instance depends on the distance between its point and the instance, which is just another point. Intuitively, the closer these two points, the stronger the activation. This is achieved by using a nonlinear transformation function to convert the distance into a similarity measure. A bell-shaped Gaussian activation function, of which the width may be different for each hidden unit, is commonly used for this purpose. The hidden units are called RBFs because the points in instance space for which a given hidden unit produces the same activation form a hypersphere or hyperellipsoid. (In a multilayer perceptron, this is a hyperplane.) The output layer of an RBF network is the same as that of a multilayer perceptron: It takes a linear combination of the outputs of the hidden units and—in classification problems—pipes it through the sigmoid function (or something with a similar shape). The parameters that such a network learns are (a) the centers and widths of the RBFs and (b) the weights used to form the linear combination of the outputs obtained from the hidden layer. A significant advantage over multilayer perceptrons is that the first set of parameters can be determined independently of the second set and still produce accurate classifiers. One way to determine the first set of parameters is to use clustering. The simple k-means clustering algorithm described in Section 4.8 can be applied, clustering each class independently to obtain k-basis functions for each class. Intuitively, the resulting RBFs represent prototype instances. The second set of parameters is then learned by keeping the first parameters fixed. This involves learning a simple linear classifier using one of the techniques we have discussed (e.g., linear or logistic regression). If there are far fewer hidden units than training instances, this can be done very quickly. A disadvantage of RBF networks is that they give every attribute the same weight because all are treated equally in the distance computation, unless attribute weight parameters are included in the overall optimization process. Thus, they cannot deal effectively with irrelevant attributes, in contrast to multilayer perceptrons. Support vector machines share the same problem. In fact, support vector machines with Gaussian kernels (i.e., “RBF kernels”) are a particular type of RBF network, in which one basis function is centered on every training instance, all basis functions 242 CHAPTER 6 Implementations: Real Machine Learning Schemes have the same width, and the outputs are combined linearly by computing the maximum-margin hyperplane. This has the effect that only some of the RBFs have a nonzero weight—the ones that represent the support vectors. Stochastic Gradient Descent We have introduced gradient descent and stochastic backpropagation as optimization methods for learning the weights in a neural network. Gradient descent is, in fact, a general-purpose optimization technique that can be applied whenever the objective function is differentiable. Actually, it turns out that it can even be applied in cases where the objective function is not completely differentiable through use of a device called subgradients. One application is the use of gradient descent to learn linear models such as linear support vector machines or logistic regression. Learning such models using gradient descent is easier than optimizing nonlinear neural networks because the objective function has a global minimum rather than many local minima, which is usually the case for nonlinear networks. For linear problems, a stochastic gradient descent procedure can be designed that is computationally simple and converges very rapidly, allowing models such as linear support vector machines and logistic regression to be learned from large datasets. Moreover, stochastic gradient descent allows models to be learned incrementally, in an online setting. For support vector machines, the error function—the thing that is to be minimized—is called the hinge loss. Illustrated in Figure 6.15, this is so named because it comprises a downwards sloping linear segment joined to a horizontal part at z = 1—more formally, E(z) = max{0, 1 – z}. For comparison, the figure also shows the 0 – 1 loss, which is discontinuous, and the squared loss, which is both continuous FIGURE 6.15 Hinge, squared, and 0 – 1 loss functions. 0 1 2 3 4 5 6 7 8 9 –2 –1.5 –1 –0.5 0 0.5 1 1.5 2 Loss (z) z hinge loss squared loss 0 – 1 loss 6.4 Extending Linear Models 243 The hinge loss is continuous, unlike the 0 – 1 loss, but is not differentiable at z = 1, unlike the squared loss, which is differentiable everywhere. This lack of differentiability presents a problem if gradient descent is used to update the model’s weights after a training example has been processed, because the loss function’s derivative is needed for this. That is where subgradients come in. The basic idea is that even though the gradient cannot be computed, the minimum will still be found if something resembling a gradient can be substituted. In the case of the hinge loss, the gradient is taken to be 0 at the point of nondifferentiability. In fact, since the hinge loss is 0 for z ≥ 1, we can focus on that part of the function that is differentiable (z < 1) and proceed as usual. Ignoring the weight decay necessary to find the smallest weight vector, the weight update for a linear support vector machine using the hinge loss is Δwi = ηai y, where η is the learning rate. For stochastic gradient descent, all that is needed to compute z for each training instance is to take the dot product between the current weight vector and the instance, multiply the result by the instance’s class value, and check to see if the resulting value is less than 1. If so, the weights are updated accordingly. As with perceptrons, a bias term can be included by extending the weight vector by one element and including an additional attribute with each training instance that always has the value 1. and differentiable. These functions are plotted as a function of the margin z = y f(x), where the class y is either –1 or +1 and f(x) is the output of the linear model. Mis- classification occurs when z < 0, so all loss functions incur their most serious penal- ties in the negative region. In the linearly separable case, the hinge loss is 0 for a function that successfully separates the data. The maximum-margin hyperplane is given by the smallest weight vector that achieves a zero hinge loss. Discussion Support vector machines originated from research in statistical learning theory (Vapnik, 1999), and a good starting point for exploration is a tutorial by Burges (1998). A general description, including generalization to the case in which the data is not linearly separable, has been published by Cortes and Vapnik (1995). We have introduced the standard version of support vector regression; Schölkopf et al. (1999) present a different version that has one parameter instead of two. Smola and Schölkopf (2004) provide an extensive tutorial on support vector regression. Ridge regression was introduced in statistics by Hoerl and Kennard (1970) and can now be found in standard statistics texts. Hastie et al. (2009) give a good description of kernel ridge regression. Kernel ridge regression is equivalent to a technique called Gaussian process regression in terms of point estimates produced, but a discussion of Gaussian processes is beyond the scope of this book. The com- plexity of the most efficient general matrix inversion algorithm is in fact O(n2.807) rather than O(n3). The (voted) kernel perceptron is due to Freund and Schapire (1999). Cristianini and Shawe-Taylor (2000) provide a nice introduction to support vector machines 244 CHAPTER 6 Implementations: Real Machine Learning Schemes and other kernel-based methods, including the optimization theory underlying the support vector learning algorithms. We have barely skimmed the surface of these learning schemes, mainly because advanced mathematics lies just beneath. The idea of using kernels to solve nonlinear problems has been applied to many algorithms, for example, principal components analysis (described in Section 7.3). A kernel is essentially a similarity function with certain mathematical properties, and it is pos- sible to define kernel functions over all sorts of structures—for example, sets, strings, trees, and probability distributions. Shawe-Taylor and Cristianini (2004) and Schölkopf and Smola (2002) cover kernel-based learning in detail. There is extensive literature on neural networks, and Bishop (1995) provides an excellent introduction to both multilayer perceptrons and RBF networks. Interest in neural networks appears to have declined since the arrival of support vector machines, perhaps because the latter generally require fewer parameters to be tuned to achieve the same (or greater) accuracy. However, multilayer perceptrons have the advantage that they can learn to ignore irrelevant attributes, and RBF networks trained using k-means can be viewed as a quick-and-dirty method for finding a nonlinear classifier. Recent studies have shown that multilayer perceptrons achieve performance competitive with more modern learning techniques on many practical datasets. Recently there has been renewed interest in gradient methods for learning clas- sifiers. In particular, stochastic gradient methods have been explored because they are applicable to large data sets and online learning scenarios. Kivinen et al. (2002), Zhang (2004), and Shalev-Shwartz et al. (2007) explore such methods when applied to learning support vector machines. Kivinen et al. and Shalev-Shwartz et al. provide heuristics for setting the learning rate for gradient descent based on the current itera- tion, which only require the user to provide a value for a single parameter that determines the closeness of fit to the training data (a so-called regularization param- eter). In the vanilla approach, regularization is performed by limiting the number of updates that can be performed. 6.5 INSTANCE-BASED LEARNING In Section 4.7 we saw how the nearest-neighbor rule can be used to implement a basic form of instance-based learning. There are several practical problems with this simple scheme. First, it tends to be slow for large training sets because the entire set must be searched for each test instance—unless sophisticated data structures such as kD-trees or ball trees are used. Second, it performs badly with noisy data because the class of a test instance is determined by its single nearest neighbor without any “averaging” to help eliminate noise. Third, it performs badly when different attri- butes affect the outcome to different extents—in the extreme case, when some attributes are completely irrelevant—because all attributes contribute equally to the distance formula. Fourth, it does not perform explicit generalization, although we 6.5 Instance-Based Learning 245 intimated in Section 3.5 (and illustrated in Figure 3.10) that some instance-based learning systems do indeed perform explicit generalization. Reducing the Number of Exemplars The plain nearest-neighbor rule stores a lot of redundant exemplars. Yet it is almost always completely unnecessary to save all the examples seen so far. A simple variant is to classify each example with respect to the examples already seen and to save only ones that are misclassified. We use the term exemplars to refer to the already-seen instances that are used for classification. Discarding correctly classified instances reduces the number of exemplars and proves to be an effective way to prune the exemplar database. Ideally, only a single exemplar is stored for each important region of the instance space. However, early in the learning process examples may be discarded that later turn out to be important, possibly leading to some decrease in predictive accuracy. As the number of stored instances increases, the accuracy of the model improves, and so the system makes fewer mistakes. Unfortunately, the strategy of only storing misclassified instances does not work well in the face of noise. Noisy examples are very likely to be misclassified, and so the set of stored exemplars tends to accumulate those that are least useful. This effect is easily observed experimentally. Thus, this strategy is only a stepping-stone on the way toward more effective instance-based learners. Pruning Noisy Exemplars Noisy exemplars inevitably lower the performance of any nearest-neighbor scheme that does not suppress them, because they have the effect of repeatedly misclassify- ing new instances. There are two ways of dealing with this. One is to locate, instead of the single nearest neighbor, the k-nearest neighbors for some predetermined con- stant k, and assign the majority class to the unknown instance. The only problem here is determining a suitable value of k. Plain nearest-neighbor learning corresponds to k = 1. The more noise, the greater the optimal value of k. One way to proceed is to perform cross-validation tests with several different values and choose the best. Although this is expensive in computation time, it often yields excellent predictive performance. A second solution is to monitor the performance of each exemplar that is stored and discard ones that do not perform well. This can be done by keeping a record of the number of correct and incorrect classification decisions that each exemplar makes. Two predetermined thresholds are set on the success ratio. When an exem- plar’s performance drops below the lower one, it is deleted from the exemplar set. If its performance exceeds the upper threshold, it is used for predicting the class of new instances. If its performance lies between the two, it is not used for prediction but, whenever it is the closest exemplar to the new instance (and thus would have 246 CHAPTER 6 Implementations: Real Machine Learning Schemes been used for prediction if its performance record had been good enough), its success statistics are updated as though it had been used to classify that new instance. To accomplish this, we use the confidence limits on the success probability of a Bernoulli process that we derived in Section 5.2. Recall that we took a certain number of successes S out of a total number of trials N as evidence on which to base confidence limits on the true underlying success rate p. Given a certain confidence level of, say, 5%, we can calculate upper and lower bounds and be 95% sure that p lies between them. To apply this to the problem of deciding when to accept a particular exemplar, suppose that it has been used n times to classify other instances and that s of these have been successes. That allows us to estimate bounds, at a particular confidence level, on the true success rate of this exemplar. Now suppose that the exemplar’s class has occurred c times out of a total number N of training instances. This allows us to estimate bounds on the default success rate—that is, the probability of suc- cessfully classifying an instance of this class without any information about other instances. We insist that the lower confidence bound on an exemplar’s success rate exceeds the upper confidence bound on the default success rate. We use the same method to devise a criterion for rejecting a poorly performing exemplar, requiring that the upper confidence bound on its success rate lies below the lower confidence bound on the default success rate. With suitable choices of thresholds, this scheme works well. In a particular implementation, called IB3 for Instance-Based Learner version 3, a confidence level of 5% is used to determine acceptance whereas a level of 12.5% is used for rejec- tion. The lower percentage figure produces a wider confidence interval, which makes for a more stringent criterion because it is harder for the lower bound of one interval to lie above the upper bound of the other. The criterion for acceptance is more stringent than for rejection, making it more difficult for an instance to be accepted. The reason for a less stringent rejection criterion is that there is little to be lost by dropping instances with only moderately poor classification accuracies: They will probably be replaced by similar instances later. Using these thresholds has been found to improve the performance of instance-based learning and, at the same time, dramatically reduce the number of exemplars—particularly noisy exemplars—that are stored. Weighting Attributes The Euclidean distance function, modified to scale all attribute values to between 0 and 1, works well in domains in which the attributes are equally relevant to the outcome. Such domains, however, are the exception rather than the rule. In most domains some attributes are irrelevant and some relevant ones are less important than others. The next improvement in instance-based learning is to learn the rele- vance of each attribute incrementally by dynamically updating feature weights. In some schemes, the weights are class specific in that an attribute may be more important to one class than to another. To cater for this, a description is produced for each class that distinguishes its members from members of all other classes. This leads to the problem that an unknown test instance may be assigned to several dif- ferent classes, or no classes at all—a problem that is all too familiar from our description of rule induction. Heuristic solutions are applied to resolve these situations. The distance metric incorporates the feature weights w1, w2, …, wm on each dimension: w x y w x y w x ym m m1 2 1 1 2 2 2 2 2 2 2 2()()()− + − + + −… In the case of class-specific feature weights, there will be a separate set of weights for each class. All attribute weights are updated after each training instance is classified, and the most similar exemplar (or the most similar exemplar of each class) is used as the basis for updating. Call the training instance x and the most similar exemplar y. For each attribute i, the difference xi − yi is a measure of the contribution of that attribute to the decision. If this difference is small then the attribute contributes positively, whereas if it is large it may contribute negatively. The basic idea is to update the ith weight on the basis of the size of this difference and whether the classification was indeed correct. If the classification is correct the associated weight is increased, and if it is incorrect it is decreased, the amount of increase or decrease being governed by the size of the difference: large if the difference is small and vice versa. The weight change is generally followed by a renormalization step. A simpler strategy, which may be equally effective, is to leave the weights alone if the decision is correct, and if it is incorrect to increase the weights for those attri- butes that differ most greatly, accentuating the difference. Details of these weight adaptation algorithms are described by Aha (1992). A good test of whether an attribute weighting scheme works is to add irrelevant attributes to all examples in a dataset. Ideally, the introduction of irrelevant attributes should not affect either the quality of predictions or the number of exemplars stored. Generalizing Exemplars Generalized exemplars are rectangular regions of instance space, called hyperrect- angles because they are high-dimensional. When classifying new instances it is necessary to modify the distance function as described below to allow the distance to a hyperrectangle to be computed. When a new exemplar is classified correctly, it is generalized by simply merging it with the nearest exemplar of the same class. The nearest exemplar may be either a single instance or a hyperrectangle. In the former case, a new hyperrectangle is created that covers the old and the new instance. In the latter, the hyperrectangle is enlarged to encompass the new instance. Finally, if the prediction is incorrect and it was a hyperrectangle that was responsible for the incorrect prediction, the hyperrectangle’s boundaries are altered so that it shrinks away from the new instance. 6.5 Instance-Based Learning 247 248 CHAPTER 6 Implementations: Real Machine Learning Schemes It is necessary to decide at the outset whether overgeneralization caused by nesting or overlapping hyperrectangles is to be permitted or not. If it is to be avoided, a check is made before generalizing a new example to see whether any regions of feature space conflict with the proposed new hyperrectangle. If they do, the gener- alization is aborted and the example is stored verbatim. Note that overlapping hyper- rectangles are precisely analogous to situations in which the same example is covered by two or more rules in a rule set. In some schemes, generalized exemplars can be nested in that they may be completely contained within one another, in the same way that in some representa- tions rules may have exceptions. To do this, whenever an example is incorrectly classified, a fallback heuristic is tried using the second nearest neighbor if it pro- duces a correct prediction in a further attempt to perform generalization. This second-chance mechanism promotes nesting of hyperrectangles. If an example falls within a rectangle of the wrong class that already contains an exemplar of the same class, the two are generalized into a new “exception” hyperrectangle nested within the original one. For nested generalized exemplars, the learning process frequently begins with a small number of seed instances to prevent all examples of the same class from being generalized into a single rectangle that covers most of the problem space. Distance Functions for Generalized Exemplars With generalized exemplars it is necessary to generalize the distance function to compute the distance from an instance to a generalized exemplar, as well as to another instance. The distance from an instance to a hyperrectangle is defined to be zero if the point lies within the hyperrectangle. The simplest way to generalize the distance function to compute the distance from an exterior point to a hyper- rectangle is to choose the closest instance within it and to measure the distance to that. However, this reduces the benefit of generalization because it reintroduces dependence on a particular single example. More precisely, whereas new instances that happen to lie within a hyperrectangle continue to benefit from generalizations, ones that lie outside do not. It might be better to use the distance from the nearest part of the hyperrectangle instead. Figure 6.16 shows the implicit boundaries that are formed between two rectan- gular classes if the distance metric is adjusted to measure distance to the nearest point of a rectangle. Even in two dimensions the boundary contains a total of nine regions (they are numbered for easy identification); the situation will be more complex for higher-dimensional hyperrectangles. Proceeding from the lower left, the first region, in which the boundary is linear, lies outside the extent of both rectangles—to the left of both borders of the larger one and below both borders of the smaller one. The second is within the extent of one rectangle—to the right of the leftmost border of the larger rectangle—but outside that of the other—below both borders of the smaller one. In this region the boundary is parabolic because the locus of a point that is the same distance from a given line FIGURE 6.16 A boundary between two rectangular classes. 1 2 3 4 5 6 7 8 9 as from a given point is a parabola. The third region is where the boundary meets the lower border of the larger rectangle when projected upward and the left border of the smaller one when projected to the right. The boundary is linear in this region because it is equidistant from these two borders. The fourth is where the boundary lies to the right of the larger rectangle but below the bottom of that rectangle. In this case the boundary is parabolic because it is the locus of points equidistant from the lower right corner of the larger rectangle and the left side of the smaller one. The fifth region lies between the two rectangles: Here the boundary is ver- tical. The pattern is repeated in the upper right part of the diagram: first parabolic, then linear, then parabolic (although this particular parabola is almost indistinguish- able from a straight line), and finally linear as the boundary escapes from the scope of both rectangles. This simple situation certainly defines a complex boundary! Of course, it is not necessary to represent the boundary explicitly; it is generated implicitly by the nearest-neighbor calculation. Nevertheless, the solution is still not a very good one. Whereas taking the distance from the nearest instance within a hyperrect- angle is overly dependent on the position of that particular instance, taking the distance to the nearest point of the hyperrectangle is overly dependent on that corner of the rectangle—the nearest example might be far from the corner. A final problem concerns measuring the distance to hyperrectangles that overlap or are nested. This complicates the situation because an instance may fall within more than one hyperrectangle. A suitable heuristic for use in this case is to choose the class of the most specific hyperrectangle containing the instance—that is, the one covering the smallest area of instance space. Whether or not overlap or nesting is permitted, the distance function should be modified to take account of both the observed prediction accuracy of exemplars and the relative importance of different features, as described in the sections above on pruning noisy exemplars and attribute weighting. Generalized Distance Functions There are many different ways of defining a distance function, and it is hard to find rational grounds for any particular choice. An elegant solution is to consider one 6.5 Instance-Based Learning 249 250 CHAPTER 6 Implementations: Real Machine Learning Schemes instance being transformed into another through a sequence of predefined elemen- tary operations and to calculate the probability of such a sequence occurring if operations are chosen randomly. Robustness is improved if all possible transforma- tion paths are considered, weighted by their probabilities, and the scheme generalizes naturally to the problem of calculating the distance between an instance and a set of other instances by considering transformations to all instances in the set. Through such a technique it is possible to consider each instance as exerting a “sphere of influence,” but a sphere with soft boundaries rather than the hard-edged cutoff implied by the k-nearest-neighbor rule, in which any particular example is either “in” or “out” of the decision. With such a measure, given a test instance that has a class that is unknown, its distance to the set of all training instances in each class in turn is calculated and the closest class is chosen. It turns out that nominal and numeric attributes can be treated in a uniform manner within this transformation-based approach by defining different transformation sets, and it is even possible to take account of unusual attribute types—such as degrees of arc or days of the week, which are measured on a circular scale. Discussion Nearest-neighbor methods gained popularity in machine learning through the work of Aha (1992), who showed that, when combined with noisy exemplar pruning and attribute weighting, instance-based learning performs well in comparison with other methods. It is worth noting that although we have described it solely in the context of classification rather than numeric prediction problems, it applies to these equally well: Predictions can be obtained by combining the predicted values of the k-nearest neighbors and weighting them by distance. Viewed in instance space, the standard rule- and tree-based representations are only capable of representing class boundaries that are parallel to the axes defined by the attributes. This is not a handicap for nominal attributes, but it is for numeric ones. Non-axis-parallel class boundaries can only be approximated by covering the region above or below the boundary with several axis-parallel rectangles, the number of rectangles determining the degree of approximation. In contrast, the instance- based method can easily represent arbitrary linear boundaries. Even with just one example of each of two classes, the boundary implied by the nearest-neighbor rule is a straight line of arbitrary orientation, namely the perpendicular bisector of the line joining the examples. Plain instance-based learning does not produce explicit knowledge representa- tions except by selecting representative exemplars. However, when combined with exemplar generalization, a set of rules can be obtained that may be compared with those produced by other machine learning schemes. The rules tend to be more con- servative because the distance metric, modified to incorporate generalized exem- plars, can be used to process examples that do not fall within the rules. This reduces the pressure to produce rules that cover the whole example space or even all of the 6.6 Numeric Prediction With Local Linear Models 251 training examples. On the other hand, the incremental nature of most instance-based learning schemes means that rules are formed eagerly, after only part of the training set has been seen; this inevitably reduces their quality. We have not given precise algorithms for variants of instance-based learning that involve generalization, because it is not clear what the best way to do gener- alization is. Salzberg (1991) suggested that generalization with nested exemplars can achieve a high degree of classification of accuracy on a variety of different problems, a conclusion disputed by Wettschereck and Dietterich (1995), who argued that these results were fortuitous and did not hold in other domains. Martin (1995) explored the idea that it is not generalization but the overgeneralization that occurs when hyperrectangles nest or overlap that is responsible for poor performance, and demonstrated that if nesting and overlapping are avoided, excellent results are achieved in a large number of domains. The generalized distance function based on transformations is described by Cleary and Trigg (1995). Exemplar generalization is a rare example of a learning strategy in which the search proceeds from specific to general rather than from general to specific as in the case of tree or rule induction. There is no particular reason why specific-to- general searching should necessarily be handicapped by forcing the examples to be considered in a strictly incremental fashion, and batch-oriented methods exist that generate rules using a basic instance-based approach. Moreover, it seems that the idea of producing conservative generalizations and coping with instances that are not covered by choosing the “closest” generalization may be generally useful for tree and rule inducers. 6.6 NUMERIC PREDICTION WITH LOCAL LINEAR MODELS Trees that are used for numeric prediction are just like ordinary decision trees, except that at each leaf they store either a class value that represents the average value of instances that reach the leaf, in which case the tree is called a regression tree, or a linear regression model that predicts the class value of instances that reach the leaf, in which case it is called a model tree. In what follows we will talk about model trees because regression trees are really a special case. Regression and model trees are constructed by first using a decision tree induc- tion algorithm to build an initial tree. However, whereas most decision tree algo- rithms choose the splitting attribute to maximize the information gain, it is appropriate for numeric prediction to instead minimize the intrasubset variation in the class values down each branch. Once the basic tree has been formed, consideration is given to pruning the tree back from each leaf, just as with ordinary decision trees. The only difference between regression tree and model tree induction is that, for the latter, each node is replaced by a regression plane instead of a constant value. The attributes that serve to define that plane are generally those that participate in deci- sions in the subtree that will be pruned—that is, in nodes beneath the current one and perhaps those that occur on the path to the root node. 252 CHAPTER 6 Implementations: Real Machine Learning Schemes Following an extensive description of model trees, we briefly explain how to generate rules from model trees, and then describe another approach to numeric prediction based on generating local linear models: locally weighted linear regres- sion. Whereas model trees derive from the basic divide-and-conquer decision tree methodology, locally weighted regression is inspired by the instance-based methods for classification that we described in the previous section. Like instance-based learning, it performs all “learning” at prediction time. Although locally weighted regression resembles model trees in that it uses linear regression to fit models locally to particular areas of instance space, it does so in quite a different way. Model Trees When a model tree is used to predict the value for a test instance, the tree is followed down to a leaf in the normal way, using the instance’s attribute values to make routing decisions at each node. The leaf will contain a linear model based on some of the attribute values, and this is evaluated for the test instance to yield a raw pre- dicted value. Instead of using this raw value directly, however, it turns out to be beneficial to use a smoothing process to reduce the sharp discontinuities that will inevitably occur between adjacent linear models at the leaves of the pruned tree. This is a particular problem for models constructed from a small number of training instances. Smooth- ing can be accomplished by producing linear models for each internal node, as well as for the leaves, at the time the tree is built. Then, once the leaf model has been used to obtain the raw predicted value for a test instance, that value is filtered along the path back to the root, smoothing it at each node by combining it with the value predicted by the linear model for that node. An appropriate smoothing calculation is ′ = + + p np kq n k where p′ is the prediction passed up to the next higher node, p is the prediction passed to this node from below, q is the value predicted by the model at this node, n is the number of training instances that reach the node below, and k is a smoothing constant. Experiments show that smoothing substantially increases the accuracy of predictions. However, discontinuities remain and the resulting function is not smooth. In fact, exactly the same smoothing process can be accomplished by incorporating the inte- rior models into each leaf model after the tree has been built. Then, during the classification process, only the leaf models are used. The disadvantage is that the leaf models tend to be larger and more difficult to comprehend because many coef- ficients that were previously zero become nonzero when the interior nodes’ models are incorporated. 6.6 Numeric Prediction With Local Linear Models 253 Building the Tree The splitting criterion is used to determine which attribute is the best to split that portion T of the training data that reaches a particular node. It is based on treating the standard deviation of the class values in T as a measure of the error at that node, and calculating the expected reduction in error as a result of testing each attribute at that node. The attribute that maximizes the expected error reduction is chosen for splitting at the node. The expected error reduction, which we call SDR for standard deviation reduction, is calculated by SDR = − ×∑sd T T T sd Ti i i() | | | | () where T1, T2, … are the sets that result from splitting the node according to the chosen attribute, and sd(T ) is the standard deviation of the class values. The splitting process terminates when the class values of the instances that reach a node vary just slightly—that is, when their standard deviation is only a small frac- tion (say less than 5%) of the standard deviation of the original instance set. Splitting also terminates when just a few instances remain (say four or fewer). Experiments show that the results obtained are not very sensitive to the exact choice of these parameters. Pruning the Tree As noted earlier, a linear model is needed for each interior node of the tree, not just at the leaves, for use in the smoothing process. Before pruning, a model is calculated for each node of the unpruned tree. The model takes the form w w a w a w ak k0 1 1 2 2+ + + +… where a1, a2, …, ak are attribute values. The weights w1, w2, …, wk are calculated using standard regression. However, only a subset of the attributes are generally used here—for example, those that are tested in the subtree below this node and perhaps those occurring along the path to the root node. Note that we have tacitly assumed that attributes are numeric; we describe the handling of nominal attributes in the next section. The pruning procedure makes use of an estimate, at each node, of the expected error for test data. First, the absolute difference between the predicted value and the actual class value is averaged over each of the training instances that reach that node. Because the tree has been built expressly for this dataset, this average will underes- timate the expected error for unseen cases. To compensate, it is multiplied by the factor (n + ν)/(n – ν), where n is the number of training instances that reach the node and ν is the number of parameters in the linear model that gives the class value at that node. 254 CHAPTER 6 Implementations: Real Machine Learning Schemes The expected error for test data at a node is calculated as described previously, using the linear model for prediction. Because of the compensation factor (n + ν)/ (n – ν), it may be that the linear model can be further simplified by dropping terms to minimize the estimated error. Dropping a term decreases the multiplication factor, which may be enough to offset the inevitable increase in average error over the training instances. Terms are dropped one by one, greedily, as long as the error estimate decreases. Finally, once a linear model is in place for each interior node, the tree is pruned back from the leaves as long as the expected estimated error decreases. The expected error for the linear model at that node is compared with the expected error from the subtree below. To calculate the latter, the error from each branch is combined into a single, overall value for the node by weighting the branch by the proportion of the training instances that go down it and combining the error estimates linearly using those weights. Alternatively, one can calculate the training error of the subtree and multiply it by the above modification factor based on an ad hoc estimate of the number of parameters in the tree—perhaps adding one for each split point. Nominal Attributes Before constructing a model tree, all nominal attributes are transformed into binary variables that are then treated as numeric. For each nominal attribute, the average class value corresponding to each possible value in the set is calculated from the training instances, and the values are sorted according to these averages. Then, if the nominal attribute has k possible values, it is replaced by k – 1 synthetic binary attributes, the ith being 0 if the value is one of the first i in the ordering and 1 other wise. Thus, all splits are binary: They involve either a numeric attribute or a synthetic binary attribute that is treated as numeric. It is possible to prove analytically that the best split at a node for a nominal variable with k values is one of the k – 1 positions obtained by ordering the average class values for each value of the attribute. This sorting operation should really be repeated at each node; however, there is an inevitable increase in noise due to small numbers of instances at lower nodes in the tree (and in some cases nodes may not represent all values for some attributes), and not much is lost by performing the sorting just once before starting to build a model tree. Missing Values To take account of missing values, a modification is made to the SDR formula. The final formula, including the missing value compensation, is SDR T= × − × ∈ ∑m sd T T T sd Tj j L R j| | () | | | | () {,} where m is the number of instances without missing values for that attribute, and T is the set of instances that reach this node. TL, TR are sets that result from splitting on this attribute because all tests on attributes are now binary. 6.6 Numeric Prediction With Local Linear Models 255 When processing both training and test instances, once an attribute is selected for splitting it is necessary to divide the instances into subsets according to their value for this attribute. An obvious problem arises when the value is missing. An interesting technique called surrogate splitting has been developed to handle this situation. It involves finding another attribute to split on in place of the original one and using it instead. The attribute is chosen as the one most highly correlated with the original attribute. However, this technique is both complex to implement and time consuming to execute. A simpler heuristic is to use the class value as the surrogate attribute, in the belief that, a priori, this is the attribute most likely to be correlated with the one being used for splitting. Of course, this is only possible when processing the training set because for test examples the class is not known. A simple solution for test examples is simply to replace the unknown attribute value by the average value of that attribute for the training examples that reach the node, which has the effect, for a binary attribute, of choosing the most populous subnode. This simple approach seems to work well in practice. Let’s consider in more detail how to use the class value as a surrogate attribute during the training process. We first deal with all instances for which the value of the splitting attribute is known. We determine a threshold for splitting in the usual way, by sorting the instances according to the splitting attribute’s value and, for each pos- sible split point, calculating the SDR according to the preceding formula, choosing the split point that yields the greatest reduction in error. Only the instances for which the value of the splitting attribute is known are used to determine the split point. Next we divide these instances into the two sets L and R according to the test. We determine whether the instances in L or R have the greater average class value, and we calculate the average of these two averages. Then an instance for which this attribute value is unknown is placed into L or R according to whether its class value exceeds this overall average or not. If it does, it goes into whichever of L and R has the greater average class value; otherwise, it goes into the one with the smaller average class value. When the splitting stops, all the missing values will be replaced by the average values of the corresponding attributes of the training instances reaching the leaves. Pseudocode for Model Tree Induction Figure 6.17 gives pseudocode for the model tree algorithm we have described. The two main parts are creating a tree by successively splitting nodes, performed by split, and pruning it from the leaves upward, performed by prune. The node data structure contains a type flag indicating whether it is an internal node or a leaf, pointers to the left and right child, the set of instances that reach that node, the attribute that is used for splitting at that node, and a structure representing the linear model for the node. The sd function called at the beginning of the main program and again at the beginning of split calculates the standard deviation of the class values of a set of instances. This is followed by the procedure for obtaining synthetic binary attributes that was described previously. Standard procedures for creating new nodes and print- ing the final tree are not shown. In split, sizeof returns the number of elements in a 256 CHAPTER 6 Implementations: Real Machine Learning Schemes FIGURE 6.17 Pseudocode for model tree induction. MakeModelTree (instances) { SD = sd(instances) for each k-valued nominal attribute convert into k-1 synthetic binary attributes root = newNode root.instances = instances split(root) prune(root) printTree(root) } split(node) { if sizeof(node.instances) < 4 or sd(node.instances) < 0.05*SD node.type = LEAF else node.type = INTERIOR for each attribute for all possible split positions of the attribute calculate the attribute's SDR node.attribute = attribute with maximum SDR split(node.left) split(node.right) } prune(node) { if node = INTERIOR then prune(node.leftChild) prune(node.rightChild) node.model = linearRegression(node) if subtreeError(node) > error(node) then node.type = LEAF } subtreeError(node) { l = node.left; r = node.right if node = INTERIOR then return (sizeof(l.instances)*subtreeError(l) + sizeof(r.instances)*subtreeError(r))/sizeof(node. else return error(node) } instances) set. Missing attribute values are dealt with as described earlier. The SDR is calculated according to the equation at the beginning of the previous section. Although not shown in the code, it is set to infinity if splitting on the attribute would create a leaf with less than two instances. In prune, the linearRegression routine recursively descends the subtree collecting attributes, performs a linear regression on the instances 6.6 Numeric Prediction With Local Linear Models 257 at that node as a function of those attributes, and then greedily drops terms if doing so improves the error estimate, as described earlier. Finally, the error function returns n n n instances+ − × ∑ν ν deviation from predicted class value where n is the number of instances at the node and ν is the number of parameters in the node’s linear model. Figure 6.18 gives an example of a model tree formed by this algorithm for a problem with two numeric and two nominal attributes. What is to be predicted is the rise time of a simulated servo system involving a servo amplifier, motor, lead screw, and sliding carriage. The nominal attributes play important roles. Four synthetic binary attributes have been created for each of the five-valued nominal attributes motor and screw and are shown in Table 6.2 in terms of the two sets of values to which they correspond. The ordering of these values—D, E, C, B, A for motor and coinciden- tally D, E, C, B, A for screw also—is determined from the training data: the rise time averaged over all examples for which motor = D is less than that averaged over exam- ples for which motor = E, which is less than when motor = C, and so on. It is apparent from the magnitude of the coefficients in Table 6.2 that motor = D versus E, C, B, A and screw = D, E, C, B versus A play leading roles in the LM2, LM3, and LM4 models (among others). Both motor and screw also play a minor role in several of the models. FIGURE 6.18 Model tree for a dataset with nominal attributes. pgain pgainmotor screw screw LM1 LM5 LM6 LM7 LM10 LM8LM2 LM3 LM4 D,E,C,B A E,C,B,AD D,E,C LM9 LM11motormotor screw vgain screw D,E,C B,A D,E,C,B A D,E C,B,A D,E,C B,A <=4.5 >4.5 <=2.5 >2.5 >3.5<=3.5 B,A 258 Table 6.2 Linear Models in the Model Tree Model LM1 LM2 LM3 LM4 LM5 LM6 LM7 LM8 LM9 LM10 LM 11 constant term 0.96 1.14 1.43 1.52 2.69 2.91 0.88 0.98 1.11 1.06 0.97 pgain –0.38 –0.38 –0.38 –0.38 –0.38 –0.38 –0.24 –0.24 –0.24 –0.25 –0.25 vgain 0.71 0.49 0.49 0.49 0.56 0.45 0.13 0.15 0.15 0.10 0.14 motor = D vs. E, C, B, A 0.66 1.14 1.06 1.06 0.50 0.50 0.30 0.40 0.30 0.14 0.14 motor = D, E vs. C, B, A 0.97 0.61 0.65 0.59 0.42 0.42 –0.02 0.06 0.06 0.17 0.22 motor = D, E, C vs. B, A 0.32 0.32 0.32 0.32 0.41 0.41 0.05 motor = D, E, C, B vs. A 0.08 0.05 screw = D vs. E, C, B, A screw = D, E vs. C, B, A 0.13 screw = D, E, C vs. B, A 0.49 0.54 0.54 0.54 0.39 0.40 0.30 0.20 0.16 0.08 0.08 screw = D, E, C, B vs. A 1.73 1.79 1.79 0.96 1.13 0.22 0.15 0.15 0.16 0.19 6.6 Numeric Prediction With Local Linear Models 259 Rules from Model Trees Model trees are essentially decision trees with linear models at the leaves. Like decision trees, they may suffer from the replicated subtree problem explained in Section 3.4, and sometimes the structure can be expressed much more concisely using a set of rules instead of a tree. Can we generate rules for numeric predic- tion? Recall the rule learner described in Section 6.2 that uses separate-and-conquer in conjunction with partial decision trees to extract decision rules from trees. The same strategy can be applied to model trees to generate decision lists for numeric prediction. First build a partial model tree from all the data. Pick one of the leaves and make it into a rule. Remove the data covered by that leaf, then repeat the process with the remaining data. The question is, how do we build the partial model tree—that is, a tree with unexpanded nodes? This boils down to the question of how to pick which node to expand next. The algorithm of Figure 6.5 (Section 6.2) picks the node of which the entropy for the class attribute is smallest. For model trees, the predictions of which are numeric, simply use the standard deviation instead. This is based on the same rationale: The lower the standard deviation, the shallower the subtree and the shorter the rule. The rest of the algorithm stays the same, with the model tree learner’s split selection method and pruning strategy replacing the decision tree learner’s. Because the model tree’s leaves are linear models, the corresponding rules will have linear models on the right side. There is one caveat when using model trees in this fashion to generate rule sets. It turns out that using smoothed model trees does not reduce the error in the final rule set’s predictions. This may be because smoothing works best for contiguous data, but the separate-and-conquer scheme removes data covered by previous rules, leaving holes in the distribution. Smoothing, if it is done at all, must be performed after the rule set has been generated. Locally Weighted Linear Regression An alternative approach to numeric prediction is the method of locally weighted linear regression. With model trees, the tree structure divides the instance space into regions, and a linear model is found for each of them. In effect, the training data determines how the instance space is partitioned. Locally weighted regression, on the other hand, generates local models at prediction time by giving higher weight to instances in the neighborhood of the particular test instance. More specifically, it weights the training instances according to their distance to the test instance and performs a linear regression on the weighted data. Training instances close to the test instance receive a high weight; those far away, a low one. In other words, a linear model is tailor-made for the particular test instance at hand and used to predict the instance’s class value. To use locally weighted regression, you need to decide on a distance-based weighting scheme for the training instances. A common choice is to weight the 260 CHAPTER 6 Implementations: Real Machine Learning Schemes instances according to the inverse of their Euclidean distance from the test instance. Another possibility is to use the Euclidean distance in conjunction with a Gaussian kernel function. However, there is no clear evidence that the choice of weighting function is critical. More important is the selection of a “smoothing parameter” that is used to scale the distance function—the distance is multiplied by the inverse of this parameter. If it is set to a small value, only instances very close to the test instance will receive significant weight; if it is large, more distant instances will also have a significant impact on the model. One way of choosing the smoothing parameter is to set it to the distance of the kth- nearest training instance so that its value becomes smaller as the volume of training data increases. If the weighting function is linear, say 1 – distance, the weight is 0 for all instances further than the kth-nearest one. Then the weighting function has bounded support and only the (k – 1)th-nearest neighbors need to be considered for building the linear model. The best choice of k depends on the amount of noise in the data. The more noise there is, the more neighbors should be included in the linear model. Generally, an appropriate smoothing parameter is found using cross-validation. Like model trees, locally weighted linear regression is able to approximate non- linear functions. One of its main advantages is that it is ideally suited for incremental learning: All training is done at prediction time, so new instances can be added to the training data at any time. However, like other instance-based methods, it is slow at deriving a prediction for a test instance. First, the training instances must be scanned to compute their weights; then a weighted linear regression is performed on these instances. Also, like other instance-based methods, locally weighted regres- sion provides little information about the global structure of the training dataset. Note that if the smoothing parameter is based on the kth-nearest neighbor and the weighting function gives zero weight to more distant instances, the kD-trees (page 132) and ball trees described in Section 4.7 can be used to accelerate the process of finding the relevant neighbors. Locally weighted learning is not restricted to linear regression: It can be applied with any learning technique that can handle weighted instances. In particular, you can use it for classification. Most algorithms can be easily adapted to deal with weights. The trick is to realize that (integer) weights can be simulated by creating several copies of the same instance. Whenever the learning algorithm uses an instance when computing a model, just pretend that it is accompanied by the appro- priate number of identical shadow instances. This also works if the weight is not an integer. For example, in the Naïve Bayes algorithm described in Section 4.2, multi- ply the counts derived from an instance by the instance’s weight, and—Voilà!—you have a version of Naïve Bayes that can be used for locally weighted learning. It turns out that locally weighted Naïve Bayes works quite well in practice, outperforming both Naïve Bayes itself and the k-nearest-neighbor technique. It also compares favorably with more sophisticated ways of enhancing Naïve Bayes by relaxing its intrinsic independence assumption. Locally weighted learning only assumes independence within a neighborhood, not globally in the whole instance space as standard Naïve Bayes does. 6.7 Bayesian Networks 261 In principle, locally weighted learning can also be applied to decision trees and other models that are more complex than linear regression and Naïve Bayes. However, it is less beneficial here because it is primarily a way of allowing simple models to become more flexible by allowing them to approximate arbitrary targets. If the underlying learning algorithm can already do that, there is little point in applying locally weighted learning. Nevertheless, it may improve other simple models—for example, linear support vector machines and logistic regression. Discussion Regression trees were introduced in the CART system of Breiman et al. (1984). CART, for classification and regression trees, incorporated a decision tree inducer for discrete classes like that of C4.5, as well as a scheme for inducing regression trees. Many of the techniques described in this section, such as the method of han- dling nominal attributes and the surrogate device for dealing with missing values, were included in CART. However, model trees did not appear until much more recently, being first described by Quinlan (1992). Using model trees for generating rule sets (although not partial trees) has been explored by Hall et al. (1999). A comprehensive description (and implementation) of model tree induction is given by Wang and Witten (1997). Neural nets are also commonly used for predict- ing numeric quantities, although they suffer from the disadvantage that the structures they produce are opaque and cannot be used to help understand the nature of the solution. There are techniques for producing understandable insights from the struc- ture of neural networks, but the arbitrary nature of the internal representation means that there may be dramatic variations between networks of identical architecture trained on the same data. By dividing the function being induced into linear patches, model trees provide a representation that is reproducible and at least somewhat comprehensible. There are many variations of locally weighted learning. For example, statisticians have considered using locally quadratic models instead of linear ones and have applied locally weighted logistic regression to classification problems. Also, many different potential weighting and distance functions can be found in the literature. Atkeson et al. (1997) have written an excellent survey on locally weighted learning, primarily in the context of regression problems. Frank et al. (2003) evaluated the use of locally weighted learning in conjunction with Naïve Bayes. 6.7 BAYESIAN NETWORKS The Naïve Bayes classifier of Section 4.2 and the logistic regression models of Section 4.6 both produce probability estimates rather than hard classifications. For each class value, they estimate the probability that a given instance belongs to that class. Most other types of classifiers can be coerced into yielding this kind of infor- mation if necessary. For example, probabilities can be obtained from a decision tree 262 CHAPTER 6 Implementations: Real Machine Learning Schemes by computing the relative frequency of each class in a leaf and from a decision list by examining the instances that a particular rule covers. Probability estimates are often more useful than plain predictions. They allow predictions to be ranked and their expected cost to be minimized (see Section 5.7). In fact, there is a strong argument for treating classification learning as the task of learning class probability estimates from data. What is being estimated is the con- ditional probability distribution of the values of the class attribute given the values of the other attributes. Ideally, the classification model represents this conditional distribution in a concise and easily comprehensible form. Viewed in this way, Naïve Bayes classifiers, logistic regression models, decision trees, and so on, are just alternative ways of representing a conditional probability distribution. Of course, they differ in representational power. Naïve Bayes classifiers and logistic regression models can only represent simple distributions, whereas deci- sion trees can represent—or at least approximate—arbitrary distributions. However, decision trees have their drawbacks: They fragment the training set into smaller and smaller pieces, which inevitably yields less reliable probability estimates, and they suffer from the replicated subtree problem described in Section 3.4. Rule sets go some way toward addressing these shortcomings, but the design of a good rule learner is guided by heuristics with scant theoretical justification. Does this mean that we have to accept our fate and live with these shortcomings? No! There is a statistically based alternative: a theoretically well-founded way of representing probability distributions concisely and comprehensibly in a graphical manner; the structures are called Bayesian networks. They are drawn as a network of nodes, one for each attribute, connected by directed edges in such a way that there are no cycles—a directed acyclic graph. In our explanation of how to interpret Bayesian networks and how to learn them from data, we will make some simplifying assumptions. We assume that all attributes are nominal and that there are no missing values. Some advanced learning algorithms can create new attributes in addition to the ones present in the data—so-called hidden attributes with values that cannot be observed. These can support better models if they represent salient features of the underlying problem, and Bayesian networks provide a good way of using them at prediction time. However, they make both learning and prediction far more complex and time consuming, so we will not consider them here. Making Predictions Figure 6.19 shows a simple Bayesian network for the weather data. It has a node for each of the four attributes outlook, temperature, humidity, and windy and one for the class attribute play. An edge leads from the play node to each of the other nodes. However, in Bayesian networks the structure of the graph is only half the story. Figure 6.19 shows a table inside each node. The information in the tables defines a probability distribution that is used to predict the class probabilities for any given instance. 6.7 Bayesian Networks 263 FIGURE 6.19 A simple Bayesian network for the weather data. play outlook windy play play yes no outlook temperature play yes no hot 0.238 0.385 mild 0.429 0.385 cool 0.333 0.231 temperature humidity play yes no high 0.350 0.750 normal 0.650 0.250 humidity play yes no false 0.350 0.583 true 0.650 0.417 windy sunny 0.238 0.538 overcast 0.429 0.077 rainy 0.333 0.385 yes 0.633 no 0.367 Before looking at how to compute this probability distribution, consider the information in the tables. The lower four tables (for outlook, temperature, humidity, and windy) have two parts separated by a vertical line. On the left are the values of play, and on the right are the corresponding probabilities for each value of the attri- bute represented by the node. In general, the left side contains a column for every edge pointing to the node, in this case just the play attribute. That is why the table associated with play itself does not have a left side: It has no parents. Each row of probabilities corresponds to one combination of values of the parent attributes, and the entries in the row show the probability of each value of the node’s attribute given this combination. In effect, each row defines a probability distribution over the values of the node’s attribute. The entries in a row always sum to 1. 264 CHAPTER 6 Implementations: Real Machine Learning Schemes Figure 6.20 shows a more complex network for the same problem, where three nodes (windy, temperature, and humidity) have two parents. Again, there is one column on the left for each parent and as many columns on the right as the attribute has values. Consider the first row of the table associated with the temperature node. The left side gives a value for each parent attribute, play and outlook; the right gives a probability for each value of temperature. For example, the first number (0.143) is the probability of temperature taking on the value hot, given that play and outlook have values yes and sunny, respectively. How are the tables used to predict the probability of each class value for a given instance? This turns out to be very easy because we are assuming that there are no missing values. The instance specifies a value for each attribute. For each node in the network, look up the probability of the node’s attribute value based on the row determined by its parents’ attribute values. Then just multiply all these probabilities together. For example, consider an instance with values outlook = rainy, temperature = cool, humidity = high, and windy = true. To calculate the probability for play = no, observe that the network in Figure 6.20 gives probability 0.367 from node play, 0.385 from outlook, 0.429 from temperature, 0.250 from humidity, and 0.167 from windy. The product is 0.0025. The same calculation for play = yes yields 0.0077. However, these are clearly not the final answers: The final probabilities must sum to 1, whereas 0.0025 and 0.0077 don’t. They are actually the joint probabilities Pr[play = no, E] and Pr[play = yes, E], where E denotes all the evidence given by the instance’s attribute values. Joint probabilities measure the likelihood of observ- ing an instance that exhibits the attribute values in E as well as the respective class value. They only sum to 1 if they exhaust the space of all possible attribute–value combinations, including the class attribute. This is certainly not the case in our example. The solution is quite simple (we already encountered it in Section 4.2). To obtain the conditional probabilities Pr[play = no | E] and Pr[play = yes | E], normalize the joint probabilities by dividing them by their sum. This gives probability 0.245 for play = no and 0.755 for play = yes. Just one mystery remains: Why multiply all those probabilities together? It turns out that the validity of the multiplication step hinges on a single assumption— namely that, given values for each of a node’s parents, knowing the values for any other set of nondescendants does not change the probability associated with each of the node’s possible values. In other words, other sets of nondescendants do not provide any information about the likelihood of the node’s values over and above the information provided by the parents. This can be written as Pr[ | ] Pr[ |node parents plus any other nondescendants node par= eents] which must hold for all values of the nodes and attributes involved. In statistics this property is called conditional independence. Multiplication is valid provided that each node is conditionally independent of its grandparents, great-grandparents, and 6.7 Bayesian Networks 265 FIGURE 6.20 Another Bayesian network for the weather data. windy play outlook humidity temperature play yes yes yes no no no outlook sunny overcast rainy sunny overcast rainy windy play play yes no play yes yes yes no no no play yes yes yes no no no outlook sunny overcast rainy sunny overcast rainy temperature humidity hot 0.413 0.455 0.111 0.556 0.333 0.143 mild 0.429 0.273 0.556 0.333 0.333 0.429 cool 0.429 0.273 0.333 0.111 0.333 0.429 high 0.500 0.500 0.125 0.833 0.833 0.250 normal 0.500 0.500 0.875 0.167 0.167 0.750 temperature hot mild cool hot mild cool outlook sunny 0.238 0.538 overcast 0.429 0.077 rainy 0.333 0.385 yes 0.633 no 0.367 false 0.500 0.500 0.125 0.375 0.500 0.833 true 0.500 0.500 0.875 0.625 0.500 0.167 266 CHAPTER 6 Implementations: Real Machine Learning Schemes indeed any other set of nondescendants, given its parents. The multiplication step follows as a direct result of the chain rule in probability theory, which states that the joint probability of m attributes ai can be decomposed into this product: Pr[,, ,] Pr[ , ,]a a a a a an i i i m 1 2 1 1 1 … …= − = ∏ The decomposition holds for any order of the attributes. Because our Bayesian network is an acyclic graph, its nodes can be ordered to give all ancestors of a node ai indices smaller than i. Then, because of the conditional independence assumption, Pr[,, , ] Pr[ , ,] Pr[ ]a a a a a a a am i i i m i i i m 1 2 1 1 1 1 … …= =− = = ∏ ∏ ’s parents which is exactly the multiplication rule that we applied earlier. The two Bayesian networks in Figures 6.19 and 6.20 are fundamentally different. The first (Figure 6.19) makes stronger independence assumptions because for each of its nodes the set of parents is a subset of the corresponding set of parents in the second (Figure 6.20). In fact, Figure 6.19 is almost identical to the simple Naïve Bayes classifier of Section 4.2. (The probabilities are slightly different but only because each count has been initialized to 0.5 to avoid the zero-frequency problem.) The network in Figure 6.20 has more rows in the conditional probability tables and hence more parameters; it may be a more accurate representation of the underlying domain. It is tempting to assume that the directed edges in a Bayesian network represent causal effects. But be careful! In our case, a particular value of play may enhance the prospects of a particular value of outlook, but it certainly doesn’t cause it—it is more likely to be the other way around. Different Bayesian networks can be constructed for the same problem, representing exactly the same probability distri- bution. This is done by altering the way in which the joint probability distribution is factorized to exploit conditional independencies. The network that has directed edges model causal effects is often the simplest one with the fewest parameters. Thus, human experts who construct Bayesian networks for a particular domain often benefit by representing causal effects by directed edges. However, when machine learning techniques are applied to induce models from data whose causal structure is unknown, all they can do is construct a network based on the correla- tions that are observed in the data. Inferring causality from correlation is always a dangerous business. Learning Bayesian Networks The main way to construct a learning algorithm for Bayesian networks is to define two components: a function for evaluating a given network based on the data and a 6.7 Bayesian Networks 267 method for searching through the space of possible networks. The quality of a given network is measured by the probability of the data given the network. We calculate the probability that the network accords to each instance and multiply these prob- abilities together over all instances. In practice, this quickly yields numbers too small to be represented properly (called arithmetic underflow), so we use the sum of the logarithms of the probabilities rather than their product. The resulting quantity is the log-likelihood of the network given the data. Assume that the structure of the network—the set of edges—is given. It’s easy to estimate the numbers in the conditional probability tables: Just compute the rela- tive frequencies of the associated combinations of attribute values in the training data. To avoid the zero-frequency problem each count is initialized with a constant as described in Section 4.2. For example, to find the probability that humidity = normal given that play = yes and temperature = cool (the last number of the third row of the humidity node’s table in Figure 6.20), observe from Table 1.2 (page 10) that there are three instances with this combination of attribute values in the weather data and no instances with humidity = high and the same values for play and temperature. Initializing the counts for the two values of humidity to 0.5 yields the probability (3 + 0.5)/(3 + 0 + 1) = 0.875 for humidity = normal. The nodes in the network are predetermined, one for each attribute (including the class). Learning the network structure amounts to searching through the space of possible sets of edges, estimating the conditional probability tables for each set, and computing the log-likelihood of the resulting network based on the data as a measure of the network’s quality. Bayesian network learning algorithms differ mainly in the way in which they search through the space of network structures. Some algorithms are introduced below. There is one caveat. If the log-likelihood is maximized based on the training data, it will always be better to add more edges: The resulting network will simply overfit. Various methods can be employed to combat this problem. One possibility is to use cross-validation to estimate the goodness of fit. A second is to add a penalty for the complexity of the network based on the number of parameters—that is, the total number of independent estimates in all the probability tables. For each table, the number of independent probabilities is the total number of entries minus the number of entries in the last column, which can be determined from the other columns because all rows must sum to 1. Let K be the number of parameters, LL the log-likelihood, and N the number of instances in the data. Two popular measures for evaluating the quality of a network are the Akaike Information Criterion (AIC): AIC score = − +LLK and the following MDL metric based on the MDL principle: MDL score = − +LL K N2 log In both cases the log-likelihood is negated, so the aim is to minimize these scores. 268 CHAPTER 6 Implementations: Real Machine Learning Schemes A third possibility is to assign a prior distribution over network structures and find the most likely network by combining its prior probability with the probability accorded to the network by the data. This is the “Bayesian” approach to network scoring. Depending on the prior distribution used, it can take various forms. However, true Bayesians would average over all possible network structures rather than sin- gling out one particular network for prediction. Unfortunately, this generally requires a great deal of computation. A simplified approach is to average over all network structures that are substructures of a given network. It turns out that this can be implemented very efficiently by changing the method for calculating the conditional probability tables so that the resulting probability estimates implicitly contain infor- mation from all subnetworks. The details of this approach are rather complex and will not be described here. The task of searching for a good network structure can be greatly simplified if the right metric is used for scoring. Recall that the probability of a single instance based on a network is the product of all the individual probabilities from the various conditional probability tables. The overall probability of the dataset is the product of these products for all instances. Because terms in a product are interchangable, the product can be rewritten to group together all factors relating to the same table. The same holds for the log-likelihood, using sums instead of products. This means that the likelihood can be optimized separately for each node of the network. This can be done by adding, or removing, edges from other nodes to the node that is being optimized—the only constraint is that cycles must not be introduced. The same trick also works if a local scoring metric such as AIC or MDL is used instead of plain log-likelihood, because the penalty term splits into several components, one for each node, and each node can be optimized independently. Specific Algorithms Now we move on to actual algorithms for learning Bayesian networks. One simple and very fast learning algorithm, called K2, starts with a given ordering of the attri- butes (i.e., nodes). Then it processes each node in turn and greedily considers adding edges from previously processed nodes to the current one. In each step it adds the edge that maximizes the network’s score. When there is no further improvement, attention turns to the next node. As an additional mechanism for overfitting avoid- ance, the number of parents for each node can be restricted to a predefined maximum. Because only edges from previously processed nodes are considered and there is a fixed ordering, this procedure cannot introduce cycles. However, the result depends on the initial ordering, so it makes sense to run the algorithm several times with different random orderings. The Naïve Bayes classifier is a network with an edge leading from the class attribute to each of the other attributes. When building networks for classification, it sometimes helps to use this network as a starting point for the search. This can be done in K2 by forcing the class variable to be the first one in the ordering and ini- tializing the set of edges appropriately. 6.7 Bayesian Networks 269 Another potentially helpful trick is to ensure that every attribute in the data is in the Markov blanket of the node that represents the class attribute. A node’s Markov blanket includes all its parents, children, and children’s parents. It can be shown that a node is conditionally independent of all other nodes given values for the nodes in its Markov blanket. Thus, if a node is absent from the class attribute’s Markov blanket, its value is completely irrelevant to the classification. Conversely, if K2 finds a network that does not include a relevant attribute in the class node’s Markov blanket, it might help to add an edge that rectifies this shortcoming. A simple way of doing this is to add an edge from the attribute’s node to the class node or from the class node to the attribute’s node, depending on which option avoids a cycle. A more sophisticated but slower version of K2 is not to order the nodes but to greedily consider adding or deleting edges between arbitrary pairs of nodes (all the while ensuring acyclicity, of course). A further step is to consider inverting the direc- tion of existing edges as well. As with any greedy algorithm, the resulting network only represents a local maximum of the scoring function: It is always advisable to run such algorithms several times with different random initial configurations. More sophisticated optimization strategies such as simulated annealing, tabu search, or genetic algorithms can also be used. Another good learning algorithm for Bayesian network classifiers is called tree- augmented Naïve Bayes (TAN). As the name implies, it takes the Naïve Bayes classifier and adds edges to it. The class attribute is the single parent of each node of a Naïve Bayes network. TAN considers adding a second parent to each node. If the class node and all corresponding edges are excluded from consideration, and assuming that there is exactly one node to which a second parent is not added, the resulting classifier has a tree structure rooted at the parentless node—this is where the name comes from. For this restricted type of network there is an efficient algo- rithm for finding the set of edges that maximizes the network’s likelihood based on computing the network’s maximum weighted spanning tree. This algorithm is linear in the number of instances and quadratic in the number of attributes. The type of network learned by the TAN algorithm is called a one-dependence estimator. An even simpler type of network is the superparent one-dependence estimator. Here, exactly one other node, apart from the class node, is elevated to parent status and becomes the parent of every other nonclass node. It turns out that a simple ensemble of these one-dependence estimators yields very accu- rate classifiers: In each of these estimators, a different attribute becomes the extra parent node. Then, at prediction time, class probability estimates from the dif- ferent one-dependence estimators are simply averaged. This scheme is known as AODE, for averaged one-dependence estimator. Normally, only estimators with certain supports in the data are used in the ensemble, but more sophisticated selection schemes are possible. Because no structure learning is involved for each superparent one-dependence estimator, AODE is a very efficient classifier. All the scoring metrics that we have described so far are likelihood-based in the sense that they are designed to maximize the joint probability Pr[a1, a2, …, an] for each instance. However, in classification, what we really want to maximize is the 270 CHAPTER 6 Implementations: Real Machine Learning Schemes conditional probability of the class given the values of the other attributes—in other words, the conditional likelihood. Unfortunately, there is no closed-form solution for the maximum conditional-likelihood probability estimates that are needed for the tables in a Bayesian network. On the other hand, computing the conditional likelihood for a given network and dataset is straightforward—after all, this is what logistic regression does. Thus, it has been proposed to use standard maximum- likelihood probability estimates in the network, but to use the conditional likelihood to evaluate a particular network structure. Another way of using Bayesian networks for classification is to build a separate network for each class value, based on the data pertaining to that class, and combine their predictions using Bayes’ rule. The set of networks is called a Bayesian multinet. To obtain a prediction for a particular class value, take the corresponding network’s probability and multiply it by the class’s prior probability. Do this for each class and normalize the result as we did previously. In this case we would not use the condi- tional likelihood to learn the network for each class value. All the network learning algorithms we have introduced are score-based. A dif- ferent strategy, which we will not explain here, is to piece a network together by testing individual conditional independence assertions based on subsets of the attri- butes. This is known as structure learning by conditional independence tests. Data Structures for Fast Learning Learning Bayesian networks involves a lot of counting. For each network structure considered in the search, the data must be scanned afresh to obtain the counts needed to fill out the conditional probability tables. Instead, could they be stored in a data structure that eliminated the need for scanning the data over and over again? An obvious way is to precompute the counts and store the nonzero ones in a table—say, the hash table mentioned in Section 4.5. Even so, any nontrivial dataset will have a huge number of nonzero counts. Again, consider the weather data from Table 1.2. There are five attributes, two with three values and three with two values. This gives 4 × 4 × 3 × 3 × 3 = 432 possible counts. Each component of the product corresponds to an attribute, and its contribution to the product is one more than the number of its values because the attribute may be missing from the count. All these counts can be calculated by treat- ing them as item sets, as explained in Section 4.5, and setting the minimum coverage to 1. But even without storing counts that are 0, this simple scheme runs into memory problems very quickly. The FP-growth data structure described in Section 6.3 was designed for efficient representation of data in the case of item set mining. In the following, we describe a structure that has been used for Bayesian networks. It turns out that the counts can be stored effectively in a structure called an all- dimensions (AD) tree, which is analogous to the kD-trees used for the nearest- neighbor search described in Section 4.7. For simplicity, we illustrate this using a reduced version of the weather data that only has the attributes humidity, windy, and play. Figure 6.21(a) summarizes the data. The number of possible counts is 3 × 3 × 6.7 Bayesian Networks 271 FIGURE 6.21 The weather data: (a) reduced version and (b) corresponding AD-tree. (b) any value 14 instances humidity = normal 7 instances windy = true 6 instances play = no 5 instances windy = true 3 instances play = no 1 instance play = no 3 instances play = no 1 instance (a) Humidity Windy Play Count high true yes 1 high true no 2 high false yes 2 high false no 2 normal true yes 2 normal true no 1 normal false yes 4 normal false no 0 3 = 27, although only eight of them are shown. For example, the count for play = no is 5 (count them!). Figure 6.21(b) shows an AD-tree for this data. Each node says how many instances exhibit the attribute values that are tested along the path from the root to that node. For example, the leftmost leaf says that there is one instance with values humidity = normal, windy = true, and play = no, and the rightmost leaf says that there are five instances with play = no. It would be trivial to construct a tree that enumerates all 27 counts explicitly. However, that would gain nothing over a plain table and is obviously not what the tree in Figure 6.21(b) does because it contains only 8 counts. There is, for example, no branch that tests humidity = high. How was the tree constructed, and how can all counts be obtained from it? Assume that each attribute in the data has been assigned an index. In the reduced version of the weather data we give humidity index 1, windy index 2, and play index 3. An AD-tree is generated by expanding each node corresponding to an attribute i with the values of all attributes that have indices j > i, with two important 272 CHAPTER 6 Implementations: Real Machine Learning Schemes restrictions: The most populous expansion for each attribute is omitted (breaking ties arbitrarily) as are expansions with counts that are zero. The root node is given index 0, so for this node all attributes are expanded, subject to the same restrictions. For example, Figure 6.21(b) contains no expansion for windy = false from the root node because with eight instances it is the most populous expansion: The value false occurs more often in the data than the value true. Similarly, from the node labeled humidity = normal there is no expansion for windy = false because false is the most common value for windy among all instances with humidity = normal. In fact, in our example the second restriction—namely that expansions with zero counts are omitted—never kicks in because the first restriction precludes any path that starts with the tests humidity = normal and windy = false, which is the only way to reach the solitary 0 in Figure 6.21(a). Each node of the tree represents the occurrence of a particular combination of attribute values. It is straightforward to retrieve the count for a combination that occurs in the tree. However, the tree does not explicitly represent many nonzero counts because the most populous expansion for each attribute is omitted. For example, the combination humidity = high and play = yes occurs three times in the data but has no node in the tree. Nevertheless, it turns out that any count can be calculated from those that the tree stores explicitly. Here’s a simple example. Figure 6.21(b) contains no node for humidity = normal, windy = true, and play = yes. However, it shows three instances with humidity = normal and windy = true, and one of them has a value for play that is different from yes. It follows that there must be two instances for play = yes. Now for a trickier case: How many times does humidity = high, windy = true, and play = no occur? At first glance it seems impossible to tell because there is no branch for humidity = high. However, we can deduce the number by calculating the count for windy = true and play = no (3) and subtracting the count for humidity = normal, windy = true, and play = no (1). This gives 2, the correct value. This idea works for any subset of attributes and any combination of attribute values, but it may have to be applied recursively. For example, to obtain the count for humidity = high, windy = false, and play = no, we need the count for windy = false and play = no and the count for humidity = normal, windy = false, and play = no. We obtain the former by subtracting the count for windy = true and play = no (3) from the count for play = no (5), giving 2, and we obtain the latter by subtracting the count for humidity = normal, windy = true, and play = no (1) from the count for humidity = normal and play = no (1), giving 0. Thus, there must be 2 – 0 = 2 instances with humidity = high, windy = false, and play = no, which is correct. AD-trees only pay off if the data contains many thousands of instances. It is pretty obvious that they do not help on the weather data. The fact that they yield no benefit on small datasets means that, in practice, it makes little sense to expand the tree all the way down to the leaf nodes. Usually, a cutoff parameter k is employed, and nodes covering fewer than k instances hold a list of pointers to these instances rather than a list of pointers to other nodes. This makes the trees smaller and more efficient to use. 6.8 Clustering 273 Discussion The K2 algorithm for learning Bayesian networks was introduced by Cooper and Herskovits (1992). Bayesian scoring metrics are covered by Heckerman et al. (1995). The TAN algorithm was introduced by Friedman et al. (1997), who also describe multinets. Grossman and Domingos (2004) show how to use the conditional like lihood for scoring networks. Guo and Greiner (2004) present an extensive compari- son of scoring metrics for Bayesian network classifiers. Bouckaert (1995) describes averaging over subnetworks. Averaged one-dependence estimators are described by Webb et al. (2005). AD-trees were introduced and analyzed by Moore and Lee (1998)—the same Andrew Moore whose work on kD-trees and ball trees was men- tioned in Section 4.7. In a more recent paper, Komarek and Moore (2000) introduce AD-trees for incremental learning that are also more efficient for datasets with many attributes. We have only skimmed the surface of the subject of learning Bayesian networks. We left open questions of missing values, numeric attributes, and hidden attributes. We did not describe how to use Bayesian networks for regression tasks. Bayesian networks are a special case of a wider class of statistical models called graphical models, which include networks with undirected edges called Markov networks. Graphical models are attracting great attention in the machine learning community today. 6.8 CLUSTERING In Section 4.8 we examined the k-means clustering algorithm in which k initial points are chosen to represent initial cluster centers, all data points are assigned to the nearest one, the mean value of the points in each cluster is computed to form its new cluster center, and iteration continues until there are no changes in the clusters. This procedure only works when the number of clusters is known in advance, and this section begins by describing what you can do if it is not. Next we take a look at techniques for creating a hierarchical clustering structure by “agglomeration”—that is, starting with individual instances and suc- cessively joining them up into clusters. Then we look at a method that works incrementally; that is, process each new instance as it appears. This method was developed in the late 1980s and embodied in a pair of systems called Cobweb (for nominal attributes) and Classit (for numeric attributes). Both come up with a hierarchical grouping of instances and use a measure of cluster “quality” called category utility. Finally, we examine a statistical clustering method based on a mixture model of different probability distributions, one for each cluster. It does not partition instances into disjoint clusters as k-means does but instead assigns instances to classes probabilistically, not deterministically. We explain the basic technique and sketch the working of a comprehensive clustering scheme called AutoClass. 274 CHAPTER 6 Implementations: Real Machine Learning Schemes Choosing the Number of Clusters Suppose you are using k-means but do not know the number of clusters in advance. One solution is to try out different possibilities and see which is best. A simple strategy is to start from a given minimum, perhaps k = 1, and work up to a small fixed maximum. Note that on the training data the “best” clustering according to the total squared distance criterion will always be to choose as many clusters as there are data points! To penalize solutions with many clusters you will have to apply something like the MDL criterion of Section 5.9. Another possibility is to begin by finding a few clusters and determining whether it is worth splitting them. You could choose k = 2, perform k-means clustering until it terminates, and then consider splitting each cluster. Computation time will be reduced considerably if the initial two-way clustering is considered irrevocable and splitting is investigated for each component independently. One way to split a cluster is to make a new seed one standard deviation away from the cluster’s center in the direction of its greatest variation, and to make a second seed the same distance in the opposite direction. (Alternatively, if this is too slow, choose a distance propor- tional to the cluster’s bounding box and a random direction.) Then apply k-means to the points in the cluster with these two new seeds. Having tentatively split a cluster, is it worthwhile retaining the split or is the original cluster equally plausible by itself? It’s no good looking at the total squared distance of all points to their cluster center—this is bound to be smaller for two subclusters. A penalty should be incurred for inventing an extra cluster, and this is a job for the MDL criterion. That principle can be applied to see whether the information required to specify the two new cluster centers, along with the informa- tion required to specify each point with respect to them, exceeds the information required to specify the original center and all the points with respect to it. If so, the new clustering is unproductive and should be abandoned. If the split is retained, try splitting each new cluster further. Continue the process until no worthwhile splits remain. Additional implementation efficiency can be achieved by combining this iterative clustering process with the kD-tree or ball tree data structure advocated in Section 4.8. Then the data points are reached by working down the tree from the root. When considering splitting a cluster, there is no need to consider the whole tree; just look at those parts of it that are needed to cover the cluster. For example, when deciding whether to split the lower left cluster in Figure 4.16(a) (below the thick line), it is only necessary to consider nodes A and B of the tree in Figure 4.16(b) because node C is irrelevant to that cluster. Hierarchical Clustering Forming an initial pair of clusters and then recursively considering whether it is worth splitting each one further produces a hierarchy that can be represented as a binary tree called a dendrogram. In fact, we illustrated a dendrogram in Figure 6.8 Clustering 275 3.11(d) (there some of the branches were three-way). The same information could be represented as a Venn diagram of sets and subsets: The constraint that the struc- ture is hierarchical corresponds to the fact that, although subsets can include one another, they cannot intersect. In some cases there exists a measure of the degree of dissimilarity between the clusters in each set; then the height of each node in the dendrogram can be made proportional to the dissimilarity between its children. This provides an easily interpretable diagram of a hierarchical clustering. An alternative to the top-down method for forming a hierarchical structure of clusters is to use a bottom-up approach, which is called agglomerative clustering. This idea was proposed many years ago and has recently enjoyed a resurgence in popularity. The basic algorithm is simple. All you need is a measure of distance (or a similarity measure) between any two clusters. (If you have a similarity measure instead, it is easy to convert that into a distance.) You begin by regarding each instance as a cluster in its own right; then find the two closest clusters, merge them, and keep on doing this until only one cluster is left. The record of mergings forms a hierarchical clustering structure—a binary dendrogram. There are numerous possibilities for the distance measure. One is the minimum distance between the clusters—the distance between their two closest members. This yields what is called the single-linkage clustering algorithm. Since this measure takes into account only the two closest members of a pair of clusters, the procedure is sensitive to outliers: The addition of just a single new instance can radically alter the entire clustering structure. Also, if we define the diameter of a cluster to be the greatest distance between its members, single-linkage clustering can produce clus- ters with very large diameters. Another measure is the maximum distance between the clusters, instead of the minimum. Two clusters are considered close only if all instances in their union are relatively similar—sometimes called the complete- linkage method. This measure, which is also sensitive to outliers, seeks compact clusters with small diameters. However, some instances may end up much closer to other clusters than they are to the rest of their own cluster. There are other measures that represent a compromise between the extremes of minimum and maximum distance between cluster members. One is to represent clusters by the centroid of their members, as the k-means algorithm does, and use the distance between centroids—the centroid-linkage method. This works well when the instances are positioned in multidimensional Euclidean space and the notion of centroid is clear, but not when all we have is a pairwise similarity measure between instances, because centroids are not instances and the similarity between them may be impossible to define. Another measure, which avoids this problem, is to calculate the average distance between each pair of members of the two clusters—the average-linkage method. Although this seems like a lot of work, you would have to calculate all pairwise distances in order to find the maximum or minimum anyway, and averaging them isn’t much additional burden. Both these measures have a technical deficiency: Their results depend on the numerical scale on which distances are measured. The minimum and maximum distance measures produce a result that depends only on the ordering 276 CHAPTER 6 Implementations: Real Machine Learning Schemes between the distances involved. In contrast, the result of both centroid-based and average-distance clustering can be altered by a monotonic transformation of all distances, even though it preserves their relative ordering. Another method, called group-average clustering, uses the average distance between all members of the merged cluster. This differs from the “average” method just described because it includes in the average pairs from the same original cluster. Finally, Ward’s clustering method calculates the increase in the sum of squares of the distances of the instances from the centroid before and after fusing two clusters. The idea is to minimize the increase in this squared distance at each clustering step. All these measures will produce the same hierarchical clustering result if the clusters are compact and well separated. However, in other cases they can yield quite different structures. Example of Hierarchical Clustering Figure 6.22 shows the result of agglomerative hierarchical clustering. (These visu- alizations have been generated using the FigTree program.1) In this case the dataset contained 50 examples of different kinds of creatures, from dolphin to mongoose, from giraffe to lobster. There was 1 numeric attribute (number of legs, ranging from 0 to 6, but scaled to the range [0, 1]) and 15 Boolean attributes such as has feathers, lays eggs, and venomous, which are treated as binary attributes with values 0 and 1 in the distance calculation. Two kinds of display are shown: a standard dendrogram and a polar plot. Figures 6.22(a) and (b) show the output from an agglomerative clusterer plotted in two dif- ferent ways, and Figures 6.22(c) and (d) show the result of a different agglomerative clusterer plotted in the same two ways. The difference is that the pair in Figures 6.22(a) and (b) was produced using the complete-linkage measure and the pair in Figures 6.22(c) and (d) was produced using the single-linkage measure. You can see that the complete-linkage method tends to produce compact clusters while the single- linkage method produces clusters with large diameters at fairly low levels of the tree. In all four visualizations the height of each node in the dendrogram is propor- tional to the dissimilarity between its children, measured as the Euclidean distance between instances. A numeric scale is provided beneath Figures 6.22(a) and (c). The total dissimilarity from root to leaf is far greater for the complete-linkage method in Figures 6.22(a) and (b) than for the single-linkage method in Figures 6.22(c) and (d) since the former involves the maximum distance and the latter the minimum distance between instances in each cluster. In the first case the total dissimilarity is a little less than 3.75, which is almost the maximum possible distance between instances—the distance between two instances that differ in 14 of the 15 attributes is 14 ≈ 3.74. In the second it is a little greater than 2 (that is, 4), which is what a difference in four Boolean attributes would produce. 1See http://tree.bio.ed.ac.uk/software/figtree/ for more information. 277 (b) crab gull kiwi cham crayfish lobster frog flea gnat housefly ladybird honeybee aardvark bear boar cheetah leopard lion lynx mongoose mink antelope buffalo deer elephant giraffe calf goathuman gorilla cavyhamster fruitbat hare mole bass catfish chub herring dogfish carp haddock dolphin chicken dove lark duck flamingo crow hawk (a) aardvark bear boar cheetah leopard lion lynx mongoose mink antelope buffalo deer elephant giraffe calf goat human gorilla cavy hamster fruitbat hare mole bass catfish chub herring dogfish carp haddock dolphin chicken dove lark duck flamingo crow hawk gull kiwi clam crab crayfish lobster frog flea gnat housefly ladybird honeybee 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0 FIGURE 6.22 Hierarchical clustering displays. Continued 278 (d) crab gull kiwi cham crayfish lobster frog flea gnat housefly ladybird honeybee aardvark bear boar cheetah leopard lion lynx mongoose antelope buffalo deer elephantgiraffe calfgoat hamster mink cavy hare mole fruitbat human gorilla dolphin bass catfish chub herring dogfish haddock carp chicken dove lark crow hawk duck flamingo (c) aardvark bear boar cheetah leopard lion lynx mongoose antelope buffalo deer elephant giraffe calf goat hamster cavy hare mink mole fruitbat human gorilla dolphin bass catfish chub herring dogfish haddock carp chicken dove lark crow hawk duck flamingo full kiwi clam crab crayfish lobster flea gnat housefly ladybird honeybee frog 2.0 1.5 1.0 0.5 0.0 FIGURE 6.22, cont’d 6.8 Clustering 279 For the complete-linkage method (Figure 6.22(a)), many elements join together at a dissimilarity of 1, which corresponds to a difference in a single Boolean attri- bute. Only one pair has a smaller dissimilarity: crab and crayfish, which differ only in the number of legs (4/6 and 6/6, respectively, after scaling). Other popular dis- similarities are 2 , 3, 4, and so on, corresponding to differences in two, three, and four Boolean attributes. For the single-linkage method (Figure 6.22(c)) that uses the minimum distance between clusters, even more elements join together at a dissimilarity of 1. Which of the two display methods—the standard dendogram and the polar plot—is more useful is a matter of taste. Although more unfamiliar at first, the polar plot spreads the visualization more evenly over the space available. Incremental Clustering Whereas the k-means algorithm iterates over the whole dataset until convergence is reached and the hierarchical method examines all the clusters present so far at each stage of merging, the clustering methods we examine next work incrementally, instance by instance. At any stage the clustering forms a tree with instances at the leaves and a root node that represents the entire dataset. In the beginning the tree consists of the root alone. Instances are added one by one, and the tree is updated appropriately at each stage. Updating may be merely a case of finding the right place to put a leaf representing the new instance, or it may involve a radical restruc- turing of the part of the tree that is affected by the new instance. The key to deciding how and where to update is a quantity called category utility that measures the overall quality of a partition of instances into clusters. We defer detailed consider- ation of how this is defined until the next section and look first at how the clustering algorithm works. The procedure is best illustrated by an example. We will use the familiar weather data again, but without the play attribute. To track progress, the 14 instances are labeled a, b, c, …, n (as in Table 4.6), and for interest we include the classes yes or no in the label—although it should be emphasized that for this artificial dataset there is little reason to suppose that the two classes of instance should fall into sepa- rate categories. Figure 6.23 shows the situation at salient points throughout the clustering procedure. At the beginning, when new instances are absorbed into the structure, they each form their own subcluster under the overall top-level cluster. Each new instance is processed by tentatively placing it in each of the existing leaves and evaluating the category utility of the resulting set of the top-level node’s children to see if the leaf is a good “host” for the new instance. For each of the first five instances, there is no such host: It is better, in terms of category utility, to form a new leaf for each instance. With the sixth it finally becomes beneficial to form a cluster, joining the new instance f with the old one—the host—e. If you look back at Table 4.6 you will see that the fifth and sixth instances are indeed very similar, differing only in the windy attribute (and play, which is being ignored here). The next example, g, is 280 CHAPTER 6 Implementations: Real Machine Learning Schemes FIGURE 6.23 Clustering the weather data. a: no a: no b: no c: yes d: yes e: yes a: no b: no c: yes d: yes e: yes f: no a: no b: no b: no a: no d: yes h: no k: yes n: no c: yes m: yes l: yes e: yes i: yes f: no g: yes j: yes c: yes d: yes e: yes f: no g: yes b: no c: yes a: no d: yes h: no e: yes f: no g: yes (a) (d) (f) (e) (b) (c) placed in the same cluster (it differs from f only in outlook). This involves another call to the clustering procedure. First, g is evaluated to see which of the five children of the root makes the best host; it turns out to be the rightmost, the one that is already a cluster. Then the clustering algorithm is invoked with this as the root, and its two children are evaluated to see which would make the better host. In this case it proves best, according to the category utility measure, to add the new instance as a subcluster in its own right. If we were to continue in this vein, there would be no possibility of any radical restructuring of the tree, and the final clustering would be excessively dependent on the ordering of examples. To avoid this, there is provision for restructuring, and you can see it come into play when instance h is added in the next step shown in Figure 6.23(e). In this case two existing nodes are merged into a single cluster: Nodes a and 6.8 Clustering 281 d are merged before the new instance h is added. One way of accomplishing this would be to consider all pairs of nodes for merging and evaluate the category utility of each pair. However, that would be computationally expensive, and would involve a lot of repeated work if it were undertaken whenever a new instance was added. Instead, whenever the nodes at a particular level are scanned for a suitable host, both the best-matching node—the one that produces the greatest category utility for the split at that level—and the runner-up are noted. The best one will form the host for the new instance (unless that new instance is better off in a cluster of its own). However, before setting to work on putting the new instance in with the host, con- sideration is given to merging the host and the runner-up. In this case, a is the preferred host and d is the runner-up. When a merge of a and d is evaluated, it turns out that it would improve the category utility measure. Consequently, these two nodes are merged, yielding a version of the fifth hierarchy before h is added. Then consideration is given to the placement of h in the new, merged node; it turns out to be best to make it a subcluster in its own right, as shown. An operation converse to merging is also implemented, called splitting. When- ever the best host is identified, and merging has not proved beneficial, consideration is given to splitting the host node. Splitting has exactly the opposite effect of merging, taking a node and replacing it with its children. For example, splitting the rightmost node in Figure 6.23(d) would raise the e, f, and g leaves up a level, making them siblings of a, b, c, and d. Merging and splitting provide an incremental way of restructuring the tree to compensate for incorrect choices caused by infelicitous ordering of examples. The final hierarchy for all 14 examples is shown in Figure 6.23(f). There are three major clusters, each of which subdivides further into its own subclusters. If the play/don’t play distinction really represented an inherent feature of the data, a single cluster would be expected for each outcome. No such clean structure is observed, although a (very) generous eye might discern a slight tendency at lower levels for yes instances to group together, and likewise for no instances. Exactly the same scheme works for numeric attributes. Category utility is defined for these as well, based on an estimate of the mean and standard deviation of the value of that attribute. Details are deferred to the next subsection. However, there is just one problem that we must attend to here: When estimating the standard devia- tion of an attribute for a particular node, the result will be zero if the node contains only one instance, as it does more often than not. Unfortunately, zero variances produce infinite values in the category utility formula. A simple heuristic solution is to impose a minimum variance on each attribute. It can be argued that because no measurement is completely precise, it is reasonable to impose such a minimum: It represents the measurement error in a single sample. This parameter is called acuity. Figure 6.24(a) shows a hierarchical clustering produced by the incremental algo- rithm for part of the iris dataset (30 instances, 10 from each class). At the top level there are two clusters (i.e., subclusters of the single node representing the whole dataset). The first contains both Iris virginicas and Iris versicolors, and the second contains only Iris setosas. The Iris setosas themselves split into two subclusters, one 282 FIGURE 6.24 Hierarchical clusterings of the iris data. (a) Versicolor Versicolor Versicolor Virginica Versicolor Virginica Versicolor Versicolor Versicolor Versicolor Versicolor Versicolor Virginica Virginica Virginica Virginica Virginica Virginica Virginica Virginica Setosa Setosa Setosa Setosa Setosa Setosa Setosa Setosa Setosa Setosa 6.8 Clustering 283 (b) Versicolor Versicolor Versicolor Versicolor Versicolor Versicolor Virginica Virginica Versicolor Versicolor Versicolor Versicolor Virginica Virginica Virginica Virginica Virginica Virginica Virginica Virginica Setosa Setosa Setosa Setosa Setosa Setosa Setosa Setosa Setosa Setosa FIGURE 6.24, cont’d with four cultivars and the other with six. The other top-level cluster splits into three subclusters, each with a fairly complex structure. Both the first and second contain only Iris versicolors, with one exception, a stray Iris virginica, in each case; the third contains only Iris virginicas. This represents a fairly satisfactory clustering of the iris data: It shows that the three genera are not artificial at all but reflect genuine differences in the data. This is, however, a slightly overoptimistic conclusion because quite a bit of experimentation with the acuity parameter was necessary to obtain such a nice division. The clusterings produced by this scheme contain one leaf for every instance. This produces an overwhelmingly large hierarchy for datasets of any reasonable size, corresponding, in a sense, to overfitting the particular dataset. Consequently, a second numerical parameter called cutoff is used to suppress growth. Some instances are deemed to be sufficiently similar to others not to warrant formation of their own child node, and this parameter governs the similarity threshold. Cutoff is specified in terms of category utility: When the increase in category utility from adding a new node is sufficiently small, that node is cut off. Figure 6.24(b) shows the same iris data, clustered with cutoff in effect. Many leaf nodes contain several instances: These are children of the parent node that have been cut off. The division into the three types of iris is a little easier to see from this hierarchy because some of the detail is suppressed. Again, however, some experi- mentation with the cutoff parameter was necessary to get this result, and in fact a sharper cutoff leads to much less satisfactory clusters. 284 CHAPTER 6 Implementations: Real Machine Learning Schemes Category Utility Now we look at how the category utility, which measures the overall quality of a partition of instances into clusters, is calculated. In Section 5.9 we learned how the MDL measure could, in principle, be used to evaluate the quality of clustering. Category utility is not MDL-based but rather resembles a kind of quadratic loss function defined on conditional probabilities. The definition of category utility is rather formidable: CU C C C C a v C a v kk i ij i ijji(,,,) Pr[ ] (Pr[ | ] Pr[ ] ) 1 2 2 2 … = = − =∑∑∑ where C1, C2, …, Ck are the k clusters; the outer summation is over these clusters; the next inner one sums over the attributes; ai is the ith attribute, and it takes on values vi1, vi2, …, which are dealt with by the sum over j. Note that the probabilities themselves are obtained by summing over all instances; thus, there is a further implied level of summation. This expression makes a great deal of sense if you take the time to examine it. The point of having a cluster is that it will give some advantage in predicting the values of attributes of instances in that cluster—that is, Pr[ai = vij | C] is a better estimate of the probability that attribute ai has value vij, for an instance in cluster C, than Pr[ai = vij] because it takes account of the cluster the instance is in. If that information doesn’t help, the clusters aren’t doing much good! So what the measure calculates, inside the multiple summation, is the amount by which that information does help in terms of the differences between squares of probabilities. This is not quite the standard squared-difference metric because that sums the squares of the differences (which produces a symmetric result) and the present measure sums the difference of the squares (which, appropriately, does not produce a symmetric result). The differences between squares of probabilities are summed over all attributes, and all their possible values, in the inner double summation. Then it is summed over all clusters, weighted by their probabilities, in the outer summation. The overall division by k is a little hard to justify because the squared differences have already been summed over the categories. It essentially provides a “per cluster” figure for the category utility that discourages overfitting. Otherwise, because the probabilities are derived by summing over the appropriate instances, the very best category utility would be obtained by placing each instance in its own cluster. Then Pr[ai = vij | C] would be 1 for the value that attribute ai actually has for the single instance in category C and 0 for all other values; the numerator of the category utility formula will end up as m a vi ijji− =∑∑ Pr[ ]2 where m is the total number of attributes. This is the greatest value that the numerator can have; thus, if it were not for the additional division by k in the category utility formula, there would never be any incentive to form clusters containing more than one member. This extra factor is best viewed as a rudimentary overfitting-avoidance heuristic. Similar clusterings are obtained if the full iris dataset of 150 instances is used. However, the results depend on the ordering of examples: Figure 6.24 was obtained by alternating the three varieties of iris in the input file. If all Iris setosas are pre- sented first, followed by all Iris versicolors and then all Iris virginicas, the resulting clusters are quite unsatisfactory. 6.8 Clustering 285 This category utility formula applies only to nominal attributes. However, it can be easily extended to numeric attributes by assuming that their distribution is normal with a given (observed) mean µ and standard deviation σ. The probability density function for an attribute a is f a a( ) exp ()= − − 1 2 2 2 2πσ µ σ The analog of summing the squares of attribute–value probabilities is Pr[ ] ( )a v f a dai ijj i i i = ⇔ =∑ ∫2 2 1 2 πσ where σi is the standard deviation of the attribute ai. Thus, for a numeric attribute we estimate the standard deviation from the data, both within the cluster (σil) and for the data over all clusters (σi), and use these in the category utility formula: CU C C C k Ck il ii( , , , ) Pr[ ]1 2 1 1 2 1 1… = − ∑∑ π σ σ Now the problem mentioned that occurs when the standard deviation estimate is zero becomes apparent: A zero standard deviation produces an infinite value of the category utility formula. Imposing a prespecified minimum variance on each attribute, the acuity, is a rough-and-ready solution to the problem. Probability-Based Clustering Some of the shortcomings of the heuristic clustering method have already become apparent: the arbitrary division by k in the category utility formula that is necessary to prevent overfitting, the need to supply an artificial minimum value for the standard deviation of clusters, and the ad hoc cutoff value to prevent every single instance from becoming a cluster in its own right. On top of this is the uncertainty inherent in incremental algorithms. To what extent is the result dependent on the order of examples? Are the local restructuring operations of merging and splitting really enough to reverse the effect of bad initial decisions caused by unlucky ordering? Does the final result represent even a local maximum of category utility? Add to this the problem that one never knows how far the final configuration is to a global maximum—and, of course, the standard trick of repeating the clustering procedure several times and choosing the best will destroy the incremental nature of the algorithm. Finally, doesn’t the hierarchical nature of the result really beg the ques- tion of which are the best clusters? There are so many clusters in Figure 6.24 that it is difficult to separate the wheat from the chaff. A more principled statistical approach to the clustering problem can overcome some of these shortcomings. From a probabilistic perspective, the goal of clustering is to find the most likely set of clusters available given the data (and, inevitably, prior expectations). Because no finite amount of evidence is enough to make a completely firm decision on the matter, instances—even training instances—should not be placed categorically in one cluster or the other: Instead, they have a certain 286 CHAPTER 6 Implementations: Real Machine Learning Schemes probability of belonging to each cluster. This helps to eliminate the brittleness that is often associated with schemes that make hard and fast judgments. The foundation for statistical clustering is a statistical model called finite mix- tures. A mixture is a set of k probability distributions, representing k clusters, that govern the attribute values for members of that cluster. In other words, each dis- tribution gives the probability that a particular instance would have a certain set of attribute values if it were known to be a member of that cluster. Each cluster has a different distribution. Any particular instance “really” belongs to one and only one of the clusters, but it is not known which one. Finally, the clusters are not equally likely: There is some probability distribution that reflects their relative populations. The simplest finite-mixture situation is when there is only one numeric attribute, which has a Gaussian or normal distribution for each cluster—but with different means and variances. The clustering problem is to take a set of instances—in this case each instance is just a number—and a prespecified number of clusters, and work out each cluster’s mean and variance and the population distribution between the clusters. The mixture model combines several normal distributions, and its probability density function looks like a mountain range with a peak for each component. Figure 6.25 shows a simple example. There are two clusters, A and B, and each has a normal distribution with means and standard deviations µA and σA for cluster A and µB and σB for cluster B. Samples are taken from these distributions, using cluster A with probability pA and cluster B with probability pB (where pA + pB = 1), resulting in a dataset like that shown. Now, imagine being given the dataset without the classes—just the numbers—and being asked to determine the five parameters that characterize the model: µA, σA, µB, σB, and pA (the parameter pB can be calculated directly from pA). That is the finite-mixture problem. FIGURE 6.25 A two-class mixture model. 30 40 50 60 70 A B 6.8 Clustering 287 If you knew which of the two distributions each instance came from, finding the five parameters would be easy—just estimate the mean and standard deviation for the cluster A samples and the cluster B samples separately, using the formulas µ = + + +x x x n n1 2 … and σ µ µ µ2 1 2 2 2 2 1= − + − + + − − ()()()x x x n n… (The use of n – 1 rather than n as the denominator in the second formula is a technicality of sampling: It makes little difference in practice if n is used instead.) Here, x1, x2, …, xn are the samples from the distribution A or B. To estimate the fifth parameter pA, just take the proportion of the instances that are in the A cluster. If you knew the five parameters, finding the probabilities that a given instance comes from each distribution would be easy. Given an instance x, the probability that it belongs to cluster A is Pr[ | Pr[ | ] ( ; , )A] A Pr[A] Pr[ ] Pr[ ] AAAx x x f x p x= × = µ σ where f (x; µA, σA) is the normal distribution function for cluster A—that is, f x e x (;,) () µ σ πσ µ σ= − −1 2 2 22 The denominator Pr[x] will disappear: We calculate the numerators for both Pr[A | x] and Pr[B | x] and normalize them by dividing by their sum. This whole procedure is just the same as the way numeric attributes are treated in the Naïve Bayes learning scheme of Section 4.2. And the caveat explained there applies here too: Strictly speaking, f (x; µA, σA) is not the probability Pr[x | A] because the probability of x being any particular real number is zero, but the normalization process makes the final result correct. Note that the final outcome is not a particular cluster but rather the probabilities with which x belongs to cluster A and cluster B. The EM Algorithm The problem is that we know neither of these things: not the distribution that each training instance came from nor the five mixture model parameters. So we adopt the procedure used for the k-means clustering algorithm and iterate. Start with initial guesses for the five parameters, use them to calculate the cluster probabilities for each instance, use these probabilities to reestimate the param- eters, and repeat. (If you prefer, you can start with guesses for the classes of the instances instead.) This is called the EM algorithm, for expectation maxi- mization. The first step—calculation of the cluster probabilities, which are the “expected” class values—is “expectation”; the second, calculation of the distribu- tion parameters, is “maximization” of the likelihood of the distributions given the data available. 288 CHAPTER 6 Implementations: Real Machine Learning Schemes A slight adjustment must be made to the parameter estimation equations to account for the fact that it is only cluster probabilities, not the clusters themselves, that are known for each instance. These probabilities just act like weights. If wi is the probability that instance i belongs to cluster A, the mean and standard deviation for cluster A are µA = + + + + + + w x w x w x w w w n n n 1 1 2 2 1 2 … … and σ µ µ µ A 2 1 1 2 2 2 2 2 1 2 = − + − + + − + + + w x w x w x w w w n n n ()()()… … where now the xi are all the instances, not just those belonging to cluster A. (This differs in a small detail from the estimate for the standard deviation given later: If all weights are equal, the denominator is n rather than n – 1. Technically speaking, this is a “maximum- likelihood” estimator for the variance whereas the previous formula is for an “unbiased” estimator. The difference is not important in practice.) Now consider how to terminate the iteration. The k-means algorithm stops when the classes of the instances don’t change from one iteration to the next—a “fixed point” has been reached. In the EM algorithm things are not quite so easy: The algorithm converges toward a fixed point but never actually gets there. We can see how close it is getting by calculating the overall likelihood that the data came from this dataset, given the values for the five parameters. This overall likelihood is obtained by multiplying the probabilities of the individual instances i: p x p xi i i A BA BPr[ | ] Pr[ | ]+( )∏ where the probabilities given the clusters A and B are determined from the normal distribution function f(x; µ, σ). This overall likelihood is a measure of the “goodness” of the clustering and increases at each iteration of the EM algorithm. Again, there is a technical difficulty with equating the probability of a particular value of x with f (x; µ, σ), and in this case the effect does not disappear because no probability normalization operation is applied. The upshot is that the likelihood expression is not a probability and does not necessarily lie between 0 and 1; nevertheless, its magnitude still reflects the quality of the clustering. In practical implementations its logarithm is calculated instead: This is done by summing the logarithms of the individual components, avoiding multiplications. But the overall conclusion still holds: You should iterate until the increase in log-likelihood becomes negligible. For example, a practical implementation might iterate until the difference between successive values of log-likelihood is less than 10–10 for 10 successive iterations. Typically, the log-likelihood will increase very sharply over the first few iterations and then converge rather quickly to a point that is virtually stationary. Although the EM algorithm is guaranteed to converge to a maximum, this is a local maximum and may not necessarily be the same as the global maximum. For a better chance of obtaining the global maximum, the whole procedure should be repeated several times, with different initial guesses for the parameter values. The overall log-likelihood figure can be used to compare the different final configurations obtained: Just choose the largest of the local maxima. 6.8 Clustering 289 Extending the Mixture Model Now that we have seen the Gaussian mixture model for two distributions, let’s consider how to extend it to more realistic situations. The basic method is just the same, but because the mathematical notation becomes formidable we will not develop it in full detail. Changing the algorithm from two-class problems to multiclass problems is completely straightforward as long as the number k of normal distributions is given in advance. The model can easily be extended from a single numeric attribute per instance to multiple attributes as long as independence between attributes is assumed. The probabilities for each attribute are multiplied together to obtain the joint probability for the instance, just as in the Naïve Bayes method. When the dataset is known in advance to contain correlated attributes, the inde- pendence assumption no longer holds. Instead, two attributes can be modeled jointly by a bivariate normal distribution, in which each has its own mean value but the two standard deviations are replaced by a “covariance matrix” with four numeric parameters. There are standard statistical techniques for estimating the class prob- abilities of instances and for estimating the means and covariance matrix given the instances and their class probabilities. Several correlated attributes can be handled using a multivariate distribution. The number of parameters increases with the square of the number of jointly varying attributes. With n independent attributes, there are 2n parameters, a mean and a standard deviation for each. With n covariant attributes, there are n + n(n + 1)/2 parameters, a mean for each, and an n × n covariance matrix that is symmetric and therefore involves n(n + 1)/2 different quantities. This escala- tion in the number of parameters has serious consequences for overfitting, as we will explain later. To cater for nominal attributes, the normal distribution must be abandoned. Instead, a nominal attribute with v possible values is characterized by v numbers representing the probability of each one. A different set of numbers is needed for every class; kv parameters in all. The situation is very similar to the Naïve Bayes method. The two steps of expectation and maximization correspond exactly to opera- tions we have studied before. Expectation—estimating the cluster to which each instance belongs given the distribution parameters—is just like determining the class of an unknown instance. Maximization—estimating the parameters from the classi- fied instances—is just like determining the attribute–value probabilities from the training instances, with the small difference that in the EM algorithm instances are assigned to classes probabilistically rather than categorically. In Section 4.2 we encountered the problem that probability estimates can turn out to be zero, and the same problem occurs here too. Fortunately, the solution is just as simple—use the Laplace estimator. Naïve Bayes assumes that attributes are independent—that is the reason why it is called “naïve.” A pair of correlated nominal attributes with v1 and v2 possible values, respectively, can be replaced by a single covariant attribute with v1v2 pos- sible values. Again, the number of parameters escalates as the number of dependent 290 CHAPTER 6 Implementations: Real Machine Learning Schemes attributes increases, and this has implications for probability estimates and overfitting. The presence of both numeric and nominal attributes in the data to be clustered presents no particular problem. Covariant numeric and nominal attributes are more difficult to handle, and we will not describe them here. Missing values can be accommodated in various different ways. In principle, they should be treated as unknown and the EM process adapted to estimate them as well as the cluster means and variances. A simple way is to replace them by means or modes in a preprocessing step. With all these enhancements, probabilistic clustering becomes quite sophisti- cated. The EM algorithm is used throughout to do the basic work. The user must specify the number of clusters to be sought, the type of each attribute (numeric or nominal), which attributes are to be modeled as covarying, and what to do about missing values. Moreover, different distributions can be used. Although the normal distribution is usually a good choice for numeric attributes, it is not suitable for attributes (such as weight) that have a predetermined minimum (0 in the case of weight) but no upper bound; in this case a “log-normal” distribution is more appro- priate. Numeric attributes that are bounded above and below can be modeled by a “log-odds” distribution. Attributes that are integer counts rather than real values are best modeled by the “Poisson” distribution. A comprehensive system might allow these distributions to be specified individually for each attribute. In each case, the distribution involves numeric parameters—probabilities of all possible values for discrete attributes and mean and standard deviation for continuous ones. In this section we have been talking about clustering. But you may be thinking that these enhancements could be applied just as well to the Naïve Bayes algorithm too—and you’d be right. A comprehensive probabilistic modeler could accom- modate both clustering and classification learning, nominal and numeric attributes with a variety of distributions, various possibilities of covariation, and different ways of dealing with missing values. The user would specify, as part of the domain knowledge, which distributions to use for which attributes. Bayesian Clustering However, there is a snag: overfitting. You might say that if we are not sure which attributes are dependent on each other, why not be on the safe side and specify that all the attributes are covariant? The answer is that the more parameters there are, the greater the chance that the resulting structure is overfitted to the training data— and covariance increases the number of parameters dramatically. The problem of overfitting occurs throughout machine learning, and probabilistic clustering is no exception. There are two ways that it can occur: through specifying too large a number of clusters and through specifying distributions with too many parameters. The extreme case of too many clusters occurs when there is one for every data point: Clearly, that will be overfitted to the training data. In fact, in the mixture model, problems will occur whenever any of the normal distributions becomes so 6.8 Clustering 291 narrow that the cluster is centered on just one data point. Consequently, implementa- tions generally insist that clusters contain at least two different data values. Whenever there are a large number of parameters, the problem of overfitting arises. If you were unsure of which attributes were covariant, you might try out different possibilities and choose the one that maximized the overall probability of the data given the clustering that was found. Unfortunately, the more parameters there are, the larger the overall data probability will tend to be—not necessarily because of better clustering but because of overfitting. The more parameters there are to play with, the easier it is to find a clustering that seems good. It would be nice if somehow you could penalize the model for introducing new parameters. One principled way of doing this is to adopt a Bayesian approach in which every parameter has a prior probability distribution. Then, whenever a new parameter is introduced, its prior probability must be incorporated into the overall likelihood figure. Because this will involve multiplying the overall likelihood by a number less than 1—the prior probability—it will automatically penalize the addi- tion of new parameters. To improve the overall likelihood, the new parameters will have to yield a benefit that outweighs the penalty. In a sense, the Laplace estimator that was introduced in Section 4.2, and whose use we advocated earlier to counter the problem of zero probability estimates for nominal values, is just such a device. Whenever observed probabilities are small, the Laplace estimator exacts a penalty because it makes probabilities that are zero, or close to zero, greater, and this will decrease the overall likelihood of the data. Making two nominal attributes covariant will exacerbate the problem of small probabilities. Instead of v1 + v2 parameters, where v1 and v2 are the number of pos- sible values, there are now v1v2, greatly increasing the chance of a large number of small estimated probabilities. In fact, the Laplace estimator is tantamount to using a particular prior distribution for the introduction of new parameters. The same technique can be used to penalize the introduction of large numbers of clusters, just by using a prespecified prior distribution that decays sharply as the number of clusters increases. AutoClass is a comprehensive Bayesian clustering scheme that uses the finite-mixture model with prior distributions on all the param- eters. It allows both numeric and nominal attributes and uses the EM algorithm to estimate the parameters of the probability distributions to best fit the data. Because there is no guarantee that the EM algorithm converges to the global optimum, the procedure is repeated for several different sets of initial values. But that is not all. AutoClass considers different numbers of clusters and can consider different amounts of covariance and different underlying probability distribution types for the numeric attributes. This involves an additional, outer level of search. For example, it initially evaluates the log-likelihood for 2, 3, 5, 7, 10, 15, and 25 clusters: After that, it fits a log-normal distribution to the resulting data and randomly selects from it more values to try. As you might imagine, the overall algorithm is extremely computation intensive. In fact, the actual implementation starts with a prespecified time bound and continues to iterate as long as time allows. Give it longer and the results may be better! 292 CHAPTER 6 Implementations: Real Machine Learning Schemes Rather than showing just the most likely clustering to the user, it may be best to present all of them, weighted by probability. Recently, fully Bayesian techniques for hierarchical clustering have been developed that produce as output a probability distribution over possible hierarchical structures representing a dataset. Figure 6.26 is a visualization, known as a DensiTree, that shows the set of all trees for a par- ticular dataset in a triangular shape. The tree is best described in terms of its “clades,” a biological term from the Greek klados meaning branch, for a group of the same species that includes all ancestors. Here, there are five clearly distin- guishable clades. The first and fourth correspond to a single leaf, while the fifth has two leaves that are so distinct they might be considered clades in their own right. The second and third clades each have five leaves, and there is large uncer- tainty in their topology. Such visualizations make it easy for people to grasp the possible hierarchical clusterings of their data, at least in terms of the big picture. Discussion The clustering methods that have been described produce different kinds of output. All are capable of taking new data in the form of a test set and classifying it accord- ing to clusters that were discovered by analyzing a training set. However, the hierarchical and incremental clustering methods are the only ones that generate an explicit knowledge structure that describes the clustering in a way that can be visualized and reasoned about. The other algorithms produce clusters that could be visualized in instance space if the dimensionality were not too high. FIGURE 6.26 DensiTree showing possible hierarchical clusterings of a given dataset. A B E L H I C D N G K M F J 6.8 Clustering 293 If a clustering method were used to label the instances of the training set with cluster numbers, that labeled set could then be used to train a rule or decision tree learner. The resulting rules or tree would form an explicit description of the classes. A probabilistic clustering scheme could be used for the same purpose, except that each instance would have multiple weighted labels and the rule or decision tree learner would have to be able to cope with weighted instances—as many can. Another application of clustering is to fill in any values of the attributes that may be missing. For example, it is possible to make a statistical estimate of the value of unknown attributes of a particular instance, based on the class distribution for the instance itself and the values of the unknown attributes for other examples. All the clustering methods we have examined make, at some level, a basic assumption of independence among the attributes. AutoClass does allow the user to specify in advance that two or more attributes are dependent and should be modeled with a joint probability distribution. (There are restrictions, however: Nominal attributes may vary jointly, as may numeric attributes, but not both together. Moreover, missing values for jointly varying attributes are not catered for.) It may be advantageous to preprocess a dataset to make the attributes more independent, using statistical techniques such as the principal components transform described in Section 7.3. Note that joint variation that is specific to particular classes will not be removed by such techniques; they only remove overall joint variation that runs across all classes. Our description of how to modify k-means to find a good value of k by repeat- edly splitting clusters and seeing whether the split is worthwhile follows the X-means algorithm of Moore and Pelleg (2000). However, instead of the MDL principle, they use a probabilistic scheme called the Bayes Information Criterion (Kass and Wasserman, 1995). Efficient agglomerative methods for hierarchical clustering were developed by Day and Edelsbrünner (1984), and the ideas are described in recent books (Duda et al., 2001; Hastie et al., 2009). The incremental clustering procedure, based on the merging and splitting operations, was introduced in systems called Cobweb for nominal attributes (Fisher, 1987) and Classit for numeric attributes (Gennari et al., 1990). Both are based on a measure of category utility that had been defined previously (Gluck and Corter, 1985). The AutoClass program is described by Cheeseman and Stutz (1995). Two implementations have been pro- duced: the original research implementation, written in LISP, and a follow-up public implementation in C that is 10 or 20 times faster but somewhat more restricted—for example, only the normal-distribution model is implemented for numeric attributes. DensiTrees were developed by Bouckaert (2010). A hierarchical clustering method called BIRCH (balanced iterative reducing and clustering using hierarchies) has been developed specifically for large multidimen- sional datasets, where it is necessary for efficient operation to minimize input–output costs (Zhang et al., 1996). It incrementally and dynamically clusters multidimen- sional metric data points, seeking the best clustering within given memory and time constraints. It typically finds a good clustering with a single scan of the data, which can then be improved by further scans. 294 CHAPTER 6 Implementations: Real Machine Learning Schemes 6.9 SEMISUPERVISED LEARNING When introducing the machine learning process in Chapter 2 (page 40), we drew a sharp distinction between supervised and unsupervised learning—classification and clustering. In this chapter we have studied a lot of techniques for both. Recently, researchers have begun to explore territory between the two, sometimes called semisupervised learning, in which the goal is classification but the input contains both unlabeled and labeled data. You can’t do classification without labeled data, of course, because only the labels tell what the classes are. But it is sometimes attrac- tive to augment a small amount of labeled data with a large pool of unlabeled data. It turns out that the unlabeled data can help you learn the classes. How can this be? First, why would you want it? Many situations present huge volumes of raw data, but assigning classes is expensive because it requires human insight. Text mining provides some classic examples. Suppose you want to classify web pages into pre- defined groups. In an academic setting you might be interested in faculty pages, graduate student pages, course information pages, research group pages, and depart- ment pages. You can easily download thousands, or millions, of relevant pages from university web sites. But labeling the training data is a laborious manual process. Or suppose your job is to use machine learning to spot names in text, differentiating between personal names, company names, and place names. You can easily down- load megabytes, or gigabytes, of text, but making this into training data by picking out the names and categorizing them can only be done manually. Cataloging news articles, sorting electronic mail, learning users’ reading interests—the applications are legion. Leaving text aside, suppose you want to learn to recognize certain famous people in television broadcast news. You can easily record hundreds or thousands of hours of newscasts, but again labeling is manual. In any of these scenarios it would be enormously attractive to be able to leverage a large pool of unlabeled data to obtain excellent performance from just a few labeled examples, particularly if you were the graduate student who had to do the labeling! Clustering for Classification How can unlabeled data be used to improve classification? Here’s a simple idea. Use Naïve Bayes to learn classes from a small labeled dataset and then extend it to a large unlabeled dataset using the EM iterative clustering algorithm from the previous section. The procedure is this. First, train a classifier using the labeled data. Second, apply it to the unlabeled data to label it with class probabilities (the “expectation” step). Third, train a new classifier using the labels for all the data (the “maximization” step). Fourth, iterate until convergence. You could think of this as iterative clustering, where starting points and cluster labels are gleaned from the labeled data. The EM procedure guarantees finding model parameters that have equal or greater likelihood at each iteration. The key question, which can only be answered empirically, is whether these higher likelihood parameter estimates will improve classification accuracy. 6.9 Semisupervised Learning 295 Intuitively, this might work well. Consider document classification. Certain phrases are indicative of the classes. Some of them occur in labeled documents, whereas others occur only in unlabeled ones. There are, however, probably some documents that contain both, and the EM procedure uses these to generalize the learned model to use phrases that do not appear in the labeled dataset. For example, both supervisor and Ph.D. topic might indicate a graduate student’s home page. Suppose only the former phrase occurs in the labeled documents. EM iteratively generalizes the model to correctly classify documents that contain just the latter. This might work with any classifier and any iterative clustering algorithm. But it is basically a bootstrapping procedure, and you must take care to ensure that the feedback loop is a positive one. Using probabilities rather than hard decisions seems beneficial because it allows the procedure to converge slowly instead of jumping to conclusions that may be wrong. Naïve Bayes, together with the basic probabilistic EM procedure, is a particularly apt choice because the two share the same fundamental assumption: independence between attributes or, more precisely, conditional independence between attributes given the class. Of course, the independence assumption is universally violated. Even our little example used the two-word phrase Ph.D. topic, whereas actual implementations would likely use individual words as attributes—and the example would have been far less compelling if we had substituted either of the single terms Ph.D. or topic. The phrase Ph.D. students is probably more indicative of faculty rather than graduate student home pages; the phrase research topic is probably less discriminating. It is the very fact that Ph.D. and topic are not conditionally independent given the class that makes the example work: It is their combination that characterizes graduate student pages. Nevertheless, coupling Naïve Bayes and EM in this manner works well in the domain of document classification. In a particular classification task it attained the performance of a traditional learner using fewer than one-third of the labeled training instances, as well as five times as many unlabeled ones. This is a good tradeoff when labeled instances are expensive but unlabeled ones are virtually free. With a small number of labeled documents, classification accuracy can be improved dramatically by incorporating many unlabeled ones. Two refinements to the procedure have been shown to improve performance. The first is motivated by experimental evidence showing that when there are many labeled documents the incorporation of unlabeled data may reduce rather than increase accuracy. Hand-labeled data is (or should be) inherently less noisy than automatically labeled data. The solution is to introduce a weighting parameter that reduces the contribution of the unlabeled data. This can be incorporated into the maximization step of EM by maximizing the weighted likelihood of the labeled and unlabeled instances. When the parameter is close to 0, unlabeled documents have little influence on the shape of EM’s hill-climbing surface; when it is close to 1, the algorithm reverts to the original version in which the surface is equally affected by both kinds of document. 296 CHAPTER 6 Implementations: Real Machine Learning Schemes The second refinement is to allow each class to have several clusters. As explained in the previous section, the EM clustering algorithm assumes that the data is generated randomly from a mixture of different probability distributions, one per cluster. Until now, a one-to-one correspondence between mixture com- ponents and classes has been assumed. In many circumstances, including docu- ment classification, this is unrealistic because most documents address multiple topics. With several clusters per class, each labeled document is initially assigned randomly to each of its components in a probabilistic fashion. The maximization step of the EM algorithm remains as before, but the expectation step is modi- fied not only to probabilistically label each example with the classes but to probabilistically assign it to the components within the class. The number of clusters per class is a parameter that depends on the domain and can be set by cross-validation. Co-training Another situation in which unlabeled data can improve classification performance is when there are two different and independent perspectives on the classification task. The classic example again involves documents, this time web documents, where the two perspectives are the content of a web page and the links to it from other pages. These two perspectives are well known to be both useful and different: Successful web search engines capitalize on them both using secret recipes. The text that labels a link to another web page gives a revealing clue as to what that page is about—perhaps even more revealing than the page’s own content, particularly if the link is an independent one. Intuitively, a link labeled my advisor is strong evidence that the target page is a faculty member’s home page. The idea, called co-training, is this. Given a few labeled examples, first learn a different model for each perspective—in this case a content-based and a hyperlink- based model. Then use each one separately to label the unlabeled examples. For each model, select the example that it most confidently labels as positive and the one it most confidently labels as negative, and add these to the pool of labeled examples. Better yet, maintain the ratio of positive and negative examples in the labeled pool by choosing more of one kind than the other. In either case, repeat the whole procedure, training both models on the augmented pool of labeled examples, until the unlabeled pool is exhausted. There is some experimental evidence, using Naïve Bayes throughout as the learner, that this bootstrapping procedure outperforms one that employs all the fea- tures from both perspectives to learn a single model from the labeled data. It relies on having two different views of an instance that are redundant but not completely correlated. Various domains have been proposed, from spotting celebrities in tele- vised newscasts using video and audio separately to mobile robots with vision, sonar, and range sensors. The independence of the views reduces the likelihood of both hypotheses agreeing on an erroneous label. 6.9 Semisupervised Learning 297 EM and Co-training On datasets with two feature sets that are truly independent, experiments have shown that co-training gives better results than using EM as described previously. Even better performance, however, can be achieved by combining the two into a modified version of co-training called co-EM. Co-training trains two classifiers representing different perspectives, A and B, and uses both to add new examples to the training pool by choosing whichever unlabeled examples they classify most positively or negatively. The new examples are few in number and deterministically labeled. Co-EM, on the other hand, trains classifier A on the labeled data and uses it to probabilistically label all the unlabeled data. Next it trains classifier B on both the labeled data and the unlabeled data with classifier A’s tentative labels, and then it probabilistically relabels all the data for use by classifier A. The process iterates until the classifiers converge. This procedure seems to perform consistently better than co-training because it does not commit to the class labels that are generated by classifiers A and B but rather reestimates their probabilities at each iteration. The range of applicability of co-EM, like co-training, is still limited by the requirement for multiple independent perspectives. But there is some experimental evidence to suggest that even when there is no natural split of features into indepen- dent perspectives, benefits can be achieved by manufacturing such a split and using co-training—or, better yet, co-EM—on the split data. This seems to work even when the split is made randomly; performance could surely be improved by engineering the split so that the feature sets are maximally independent. Why does this work? Researchers have hypothesized that these algorithms succeed in part because the split makes them more robust to the assumptions that their underlying classifiers make. There is no particular reason to restrict the base classifier to Naïve Bayes. Support vector machines probably represent the most successful technology for text catego- rization today. However, for the EM iteration to work it is necessary that the classifier labels the data probabilistically; it must also be able to use probabilistically weighted examples for training. Support vector machines can easily be adapted to do both. We explained how to adapt learning algorithms to deal with weighted instances in Section 6.6, under Locally Weighted Linear Regression (page 258). One way of obtaining probability estimates from support vector machines is to fit a one- dimensional logistic model to the output, effectively performing logistic regression as described in Section 4.6 on the output. Excellent results have been reported for text classification using co-EM with the support vector machine (SVM) classifier. It outperforms other variants of SVM and seems quite robust to varying proportions of labeled and unlabeled data. Discussion Nigam et al. (2000) thoroughly explored the idea of clustering for classification, showing how the EM clustering algorithm can use unlabeled data to improve an initial classifier built by Naïve Bayes. The idea of co-training is older: Blum 298 CHAPTER 6 Implementations: Real Machine Learning Schemes and Mitchell (1998) pioneered it and developed a theoretical model for the use of labeled and unlabeled data from different independent perspectives. Nigam and Ghani (2000) analyzed the effectiveness and applicability of co-training, relating it to the traditional use of standard expectation maximization to fill in missing values; they also introduced the co-EM algorithm. Up to this point, co-training and co-EM have been applied mainly to small two-class problems. Ghani (2002) used error-correcting output codes to address multiclass situations with many classes. Brefeld and Scheffer (2004) extended co-EM to use a support vector machine rather than Naïve Bayes. 6.10 MULTI-INSTANCE LEARNING All the techniques described in this chapter so far are for the standard machine learning scenario where each example consists of a single instance. Before moving on to methods for transforming the input data in Chapter 7, we revisit the more complex setting of multi-instance learning, in which each example consists of a bag of instances instead. We describe approaches that are more advanced than the simple techniques discussed in Section 4.9. First, we consider how to convert multi-instance learning to single-instance learning by transforming the data. Then we discuss how to upgrade single-instance learning algorithms to the multi-instance case. Finally, we take a look at some methods that have no direct equivalent in single-instance learning. Converting to Single-Instance Learning Section 4.9 (page 142) presented some ways of applying standard single-instance learning algorithms to multi-instance data by aggregating the input or the output. Despite their simplicity, these techniques often work surprisingly well in practice. Nevertheless, there are clearly situations in which they will fail. Consider the method of aggregating the input by computing the minimum and maximum values of numeric attributes present in the bag and treating the result as a single instance. This will yield a huge loss of information because attributes are condensed to summary statistics individually and independently. Can a bag be converted to a single instance without discarding quite so much information? The answer is yes, although the number of attributes that are present in the so-called “condensed” representation may increase substantially. The basic idea is to partition the instance space into regions and create one attribute per region in the single-instance representation. In the simplest case, attributes can be Boolean: If a bag has at least one instance in the region corresponding to a particular attribute the value of the attribute is set to true; otherwise, it is set to false. However, to preserve more information the condensed representation could instead contain numeric attri- butes, the values of which are counts that indicate how many instances of the bag lie in the corresponding region. 6.10 Multi-Instance Learning 299 Regardless of the exact types of attributes that are generated, the main problem is to come up with a partitioning of the input space. A simple approach is to partition it into hypercubes of equal size. Unfortunately, this only works when the space has very few dimensions (i.e., attributes): The number of cubes required to achieve a given granularity grows exponentially with the dimension of the space. One way to make this approach more practical is to use unsupervised learning. Simply take all the instances from all the bags in the training data, discard their class labels, and form a big single-instance dataset; then process it with a clustering technique such as k-means. This will create regions corresponding to the different clusters (k regions, in the case of k-means). Then, for each bag, create one attribute per region in the condensed representation and use it as described previously. Clustering is a rather heavy-handed way to infer a set of regions from the training data because it ignores information about class membership. An alternative approach that often yields better results is to partition the instance space using decision tree learning. Each leaf of a tree corresponds to one region of instance space. But how can a decision tree be learned when the class labels apply to entire bags of instances rather than to individual instances? The approach described under Aggregating the Output in Section 4.9 can be used: Take the bag’s class label and attach it to each of its instances. This yields a single-instance dataset, ready for decision tree learning. Many of the class labels will be incorrect—the whole point of multi-instance learn- ing is that it is not clear how bag-level labels relate to instance-level ones. However, these class labels are only being used to obtain a partition of instance space. The next step is to transform the multi-instance dataset into a single-instance one that represents how instances from each bag are distributed throughout the space. Then another single-instance learning method is applied—perhaps, again, decision tree learning—that determines the importance of individual attributes in the condensed representation which correspond to regions in the original space. Using decision trees and clustering yields “hard” partition boundaries, where an instance either does or does not belong to a region. Such partitions can also be obtained using a distance function, combined with some reference points, by assign- ing instances to their closest reference point. This implicitly divides the space into regions, each corresponding to one reference point. (In fact, this is exactly what happens in k-means clustering: The cluster centers are the reference points.) But there is no fundamental reason to restrict attention to hard boundaries: We can make the region membership function “soft” by using distance—transformed into a simi- larity score—to compute attribute values in the condensed representation of a bag. All that is needed is some way of aggregating the similarity scores between each bag and reference point into a single value—for example, by taking the maximum similarity between each instance in that bag and the reference point. In the simplest case, each instance in the training data can be used as a refer- ence point. That creates a large number of attributes in the condensed representation, but it preserves much of the information from a bag of instances in its correspond- ing single-instance representation. This method has been successfully applied to multi-instance problems. 300 CHAPTER 6 Implementations: Real Machine Learning Schemes Regardless of how the approach is implemented, the basic idea is to convert a bag of instances into a single one by describing the distribution of instances from this bag in instance space. Alternatively, ordinary learning methods can be applied to multi-instance data by aggregating the output rather than the input. Section 4.9 described a simple way: Join instances of bags in the training data into a single dataset by attaching bag-level class labels to them, perhaps weighting instances to give each bag the same total weight. A single-instance classification model can then be built. At classification time, predictions for individual instances are combined—for example, by averaging predicted class probabilities. Although this approach often works well in practice, attaching bag-level class labels to instances is simplistic. Generally, the assumption in multi-instance learn- ing is that only some of the instances—perhaps just one—are responsible for the class label of the associated bag. How can the class labels be corrected to yield a more accurate representation of the true underlying situation? This is obviously a difficult problem; if it were solved, it would make little sense to investigate other approaches to multi-instance learning. One method that has been applied is iterative: Start by assigning each instance its bag’s class label and learn a single- instance classification model; then replace the instances’ class labels by the pre- dicted labels of this single-instance classification model for these instances. Repeat the whole procedure until the class labels remain unchanged from one iteration to the next. Some care is needed to obtain sensible results. For example, suppose every instance in a bag were to receive a class label that differs from the bag’s label. Such a situation should be prevented by forcing the bag’s label on at least one instance— for example, the one with the largest predicted probability for this class. This iterative approach has been investigated for the original multi-instance scenario with two class values, where a bag is positive if and only if one of its instances is positive. In that case it makes sense to assume that all instances from negative bags are truly negative and modify only the class labels of instances from positive bags. At prediction time, bags are classified as positive if one of their instances is classified as positive. Upgrading Learning Algorithms Tackling multi-instance learning by modifying the input or output so that single- instance schemes can be applied is appealing because there is a large battery of such techniques that can then be used directly, without any modification. However, it may not be the most efficient approach. An alternative is to adapt the internals of a single- instance algorithm to the multi-instance setting. This can be done in a particularly elegant fashion if the algorithm in question only considers the data through applica- tion of a distance (or similarity) function, as with nearest-neighbor classifiers or support vector machines. These can be adapted by providing a distance (or similar- ity) function for multi-instance data that computes a score between two bags of instances. In the case of kernel-based methods such as support vector machines, the similar- ity must be a proper kernel function that satisfies certain mathematical properties. One that has been used for multi-instance data is the so-called set kernel. Given a kernel function for pairs of instances that support vector machines can apply to single-instance data—for example, one of the kernel functions considered in Section 6.4—the set kernel sums it over all pairs of instances from the two bags being com- pared. This idea is generic and can be applied with any single-instance kernel function. Nearest-neighbor learning has been adapted to multi-instance data by applying variants of the Hausdorff distance, which is defined for sets of points. Given two bags and a distance function between pairs of instances—for example, the Euclidean distance—the Hausdorff distance between the bags is the largest distance from any instance in one bag to its closest instance in the other bag. It can be made more robust to outliers by using the nth-largest distance rather than the maximum. For learning algorithms that are not based on similarity scores, more work is required to upgrade them to multi-instance data. There are multi-instance algo- rithms for rule learning and for decision tree learning, but we will not describe them here. Adapting algorithms to the multi-instance case is more straightforward if the algorithm concerned is essentially a numerical optimization strategy that is applied to the parameters of some function by minimizing a loss function on the training data. Logistic regression (Section 4.6) and multilayer perceptrons (Section 6.4) fall into this category; both have been adapted to multi-instance learning by augmenting them with a function that aggregates instance-level predictions. The so-called “soft maximum” is a differentiable function that is suitable for this purpose: It aggregates instance-level predictions by taking their (soft) maximum as the bag-level prediction. Dedicated Multi-Instance Methods Some multi-instance learning schemes are not based directly on single-instance algorithms. Here is an early technique that was specifically developed for the drug activity prediction problem mentioned in Section 2.2 (page 49), in which instances are conformations of a molecule and a molecule (i.e., a bag) is considered positive if and only if it has at least one active conformation. The basic idea is to learn a single hyperrectangle that contains at least one instance from each positive bag in the training data and no instances from any negative bags. Such a rectangle encloses an area of instance space where all positive bags overlap, but it contains no negative instances—an area that is common to all active molecules but not represented in any inactive ones. The particular drug activity data originally considered was high- dimensional, with 166 attributes describing each instance. In such a case it is com- putationally difficult to find a suitable hyperrectangle. Consequently, a heuristic approach was developed that is tuned to this particular problem. Other geometric shapes can be used instead of hyperrectangles. Indeed, the same basic idea has been applied using hyperspheres (balls). Training instances are treated 6.10 Multi-Instance Learning 301 302 CHAPTER 6 Implementations: Real Machine Learning Schemes as potential ball centers. For each one, a radius is found that yields the smallest number of errors for the bags in the training data. The original multi-instance assumption is used to make predictions: A bag is classified as positive if and only if it has at least one instance inside the ball. A single ball is generally not powerful enough to yield good classification performance. However, this method is not intended as a standalone algorithm. Rather, it is advocated as a “weak” learner to be used in conjunction with boosting algorithms (see Section 8.4) to obtain a powerful ensemble classifier—an ensemble of balls. The dedicated multi-instance methods discussed so far have hard decision bound- aries: An instance either falls inside or outside a ball or hyperrectangle. Other multi- instance algorithms use soft concept descriptions couched in terms of probability theory. The so-called diverse-density method is a classic example, again designed with the original multi-instance assumption in mind. Its basic and most commonly used form learns a single reference point in instance space. The probability that an instance is positive is computed from its distance to this point: It is 1 if the instance coincides with the reference point and decreases with increasing distance from this point, usually based on a bell-shaped function. The probability that a bag is positive is obtained by combining the individual probabilities of the instances it contains, generally using the “noisy-OR” function. This is a probabilistic version of the logical OR. If all instance-level probabilities are 0, the noisy-OR value—and thus the bag-level probability—is 0; if at least one instance-level probability is 1, the value is 1; otherwise, the value falls somewhere in between. The diverse density is defined as the probability of the class labels of the bags in the training data, computed based on this probabilistic model. It is maximized when the reference point is located in an area where positive bags overlap and no negative bags are present, just as for the two geometric methods discussed previ- ously. A numerical optimization routine such as gradient ascent can be used to find the reference point that maximizes the diverse-density measure. In addition to the location of the reference point, implementations of diverse density also optimize the scale of the distance function in each dimension because generally not all attributes are equally important. This can improve predictive performance significantly. Discussion Condensing the input data by aggregating information into simple summary statistics is a well-known technique in multirelational learning, used in the RELAGGS system by Krogel and Wrobel (2002); multi-instance learning can be viewed as a special case of this more general setting (de Raedt, 2008). The idea of replacing simple summary statistics by region-based attributes, derived from partitioning the instance space, was explored by Weidmann et al. (2003) and Zhou and Zhang (2007). Using reference points to condense bags was investigated by Chen et al. (2006) and evalu- ated in a broader context by Foulds and Frank (2008). Andrews et al. (2003) pro- posed manipulating the class labels of individual instances using an iterative learning 6.11 Weka Implementations 303 process for learning support vector machine classifiers based on the original multi- instance assumption. Nearest-neighbor learning based on variants of the Hausdorff distance was inves- tigated by Wang and Zucker (2000). Gärtner et al. (2002) experimented with the set kernel to learn support vector machine classifiers for multi-instance data. Multi- instance algorithms for rule and decision tree learning, which are not covered here, have been described by Chevaleyre and Zucker (2001) and Blockeel et al. (2005). Logistic regression has been adapted for multi-instance learning by Xu and Frank (2004) and Ray and Craven (2005); multilayer perceptrons have been adapted by Ramon and de Raedt (2000). Hyperrectangles and spheres were considered as concept descriptions for multi- instance learning by Dietterich et al. (1997) and Auer and Ortner (2004), respec- tively. The diverse-density method is the subject of Maron’s (1998) Ph.D. thesis, and is also described in Maron and Lozano-Peréz (1997). The multi-instance literature makes many different assumptions regarding the type of concept to be learned, defining, for example, how the bag-level and instance- level class labels are connected, starting with the original assumption that a bag is labeled positive if and only if one of its instances is positive. A review of assump- tions in multi-instance learning can be found in Foulds and Frank (2010). 6.11 WEKA IMPLEMENTATIONS For classifiers, see Section 11.4 and Table 11.5. For clustering methods, see Section 11.6 and Table 11.7. • Decision trees: • J48 (implementation of C4.5) • SimpleCart (minimum cost-complexity pruning à la CART) • REPTree (reduced-error pruning) • Classification rules: • JRip (RIPPER rule learner) • Part (rules from partial decision trees) • Ridor (ripple-down rule learner) • Association rules (see Section 11.7 and Table 11.8): • FPGrowth (frequent-pattern trees) • GeneralizedSequentialPatterns (find large item trees in sequential data) • Linear models and extensions: • SMO and variants for learning support vector machines • LibSVM (uses third-party libsvm library) • MultilayerPerceptron • RBFNetwork (radial-basis function network) • SPegasos (SVM using stochastic gradient descent) 304 CHAPTER 6 Implementations: Real Machine Learning Schemes • Instance-based learning: • IBk (k-nearest neighbour classifier) • KStar (generalized distance functions) • NNge (rectangular generalizations) • Numeric prediction: • M5P (model trees) M5Rules (rules from model trees) • LWL (locally weighted learning) • Bayesian networks: • BayesNet • AODE, WAODE (averaged one-dependence estimator) • Clustering: • XMeans • Cobweb (includes Classit) • EM • Multi-instance learning: • MISVM (iterative method for learning SVM by relabeling instances) • MISMO (SVM with multi-instance kernel) • CitationKNN (nearest-neighbor method with Hausdorff distance) • MILR (logistic regression for multi-instance data) • MIOptimalBall (learning balls for multi-instance classification) • MIDD (the diverse-density method using the noisy-OR function) 305Data Mining: Practical Machine Learning Tools and Techniques Copyright © 2011 Elsevier Inc. All rights of reproduction in any form reserved. CHAPTER 7 Data Transformations In Chapter 6 we examined a vast array of machine learning methods: decision trees, classification and association rules, linear models, instance-based schemes, numeric prediction techniques, Bayesian networks, clustering algorithms, and semisupervised and multi-instance learning. All are sound, robust techniques that are eminently applicable to practical data mining problems. But successful data mining involves far more than selecting a learning algorithm and running it over your data. For one thing, many learning schemes have various parameters, and suitable values must be chosen for these. In most cases, results can be improved markedly by a suitable choice of parameter values, and the appropriate choice depends on the data at hand. For example, decision trees can be pruned or unpruned, and in the former case a pruning parameter may have to be chosen. In the k-nearest-neighbor method of instance-based learning, a value for k will have to be chosen. More generally, the learning scheme itself will have to be chosen from the range of schemes that are available. In all cases, the right choices depend on the data itself. It is tempting to try out several learning schemes and several parameter values on your data, and see which works best. But be careful! The best choice is not nec- essarily the one that performs best on the training data. We have repeatedly cautioned about the problem of overfitting, where a learned model is too closely tied to the particular training data from which it was built. It is incorrect to assume that per- formance on the training data faithfully represents the level of performance that can be expected on the fresh data to which the learned model will be applied in practice. Fortunately, we have already encountered the solution to this problem in Chapter 5. There are two good methods for estimating the expected true perfor- mance of a learning scheme: the use of a large dataset that is quite separate from the training data, in the case of plentiful data, and cross-validation (see Section 5.3), if data is scarce. In the latter case, a single tenfold cross-validation is typically used in practice, although to obtain a more reliable estimate the entire procedure should be repeated 10 times. Once suitable parameters have been chosen for the learning scheme, use the whole training set—all the avail- able training instances—to produce the final learned model that is to be applied to fresh data. 306 CHAPTER 7 Data Transformations Note that the performance obtained with the chosen parameter value during the tuning process is not a reliable estimate of the final model’s performance, because the final model potentially overfits the data that was used for tuning. To ascertain how well it will perform, you need yet another large dataset that is quite separate from any data used during learning and tuning. The same is true for cross-validation: You need an “inner” cross-validation for parameter tuning and an “outer” cross- validation for error estimation. With tenfold cross-validation, this involves running the learning scheme 100 times. To summarize: When assessing the performance of a learning scheme, any parameter tuning that goes on should be treated as though it were an integral part of the training process. There are other important processes that can materially improve success when applying machine learning techniques to practical data mining problems, and these are the subject of this chapter. They constitute a kind of data engineering—engineering the input data into a form suitable for the learning scheme chosen and engineering the output to make it more effective. You can look on them as a bag of tricks that you can apply to practical data mining problems to enhance the chance of success. Sometimes they work; other times they don’t—and at the present state of the art, it’s hard to say in advance whether they will or not. In an area such as this, where trial and error is the most reliable guide, it is particularly important to be resourceful and have an understanding of what the tricks are. In this chapter we examine six different ways in which the input can be massaged to make it more amenable for learning methods: attribute selection, attribute dis- cretization, data projections, sampling, data cleansing, and converting multiclass problems to two-class ones. Consider the first, attribute selection. In many practical situations there are far too many attributes for learning schemes to handle, and some of them—perhaps the overwhelming majority—are clearly irrelevant or redundant. Consequently, the data must be preprocessed to select a subset of the attributes to use in learning. Of course, many learning schemes themselves try to select attributes appropriately and ignore irrelevant or redundant ones, but in practice their perfor- mance can frequently be improved by preselection. For example, experiments show that adding useless attributes causes the performance of learning schemes such as decision trees and rules, linear regression, instance-based learners, and clustering methods to deteriorate. Discretization of numeric attributes is absolutely essential if the task involves numeric attributes but the chosen learning scheme can only handle categorical ones. Even schemes that can handle numeric attributes often produce better results, or work faster, if the attributes are prediscretized. The converse situation, in which categorical attributes must be represented numerically, also occurs (although less often), and we describe techniques for this case, too. Data projection covers a variety of techniques. One transformation, which we have encountered before when looking at relational data in Chapter 2 and support vector machines in Chapter 6, is to add new, synthetic attributes whose purpose is to present existing information in a form that is suitable for the machine learning scheme to pick up on. More general techniques that do not depend so intimately 7.1 Attribute Selection 307 on the semantics of the particular data mining problem at hand include principal components analysis and random projections. We also cover partial least-squares regression as a data projection technique for regression problems. Sampling the input is an important step in many practical data mining applica- tions, and is often the only way in which really large-scale problems can be handled. Although it is fairly simple, we include a brief section on techniques of sampling, including a way of incrementally producing a random sample of a given size when the total size of the dataset is not known in advance. Unclean data plagues data mining. We emphasized in Chapter 2 the necessity of getting to know your data: understanding the meaning of all the different attri- butes, the conventions used in coding them, the significance of missing values and duplicate data, measurement noise, typographical errors, and the presence of systematic errors—even deliberate ones. Various simple visualizations often help with this task. There are also automatic methods of cleansing data, of detecting outliers, and of spotting anomalies, which we describe, including a class of tech- niques referred to as one-class learning in which only a single class of instances is available at training time. Finally, we examine techniques for refining the output of learning schemes that estimate class probabilities by recalibrating the estimates that they make. This is primarily of importance when accurate probabilities are required, as in cost-sensitive classification, though it can also improve classification performance. 7.1 ATTRIBUTE SELECTION Most machine learning algorithms are designed to learn which are the most appro- priate attributes to use for making their decisions. For example, decision tree methods choose the most promising attribute to split on at each point and should— in theory—never select irrelevant or unhelpful attributes. Having more features should surely—in theory—result in more discriminating power, never less. “What’s the difference between theory and practice?” an old question asks. The answer goes, “There is no difference between theory and practice—in theory. But in practice, there is.” Here there is too: In practice, adding irrelevant or distracting attributes to a dataset often confuses machine learning systems. Experiments with a decision tree learner (C4.5) have shown that adding to stan- dard datasets a random binary attribute generated by tossing an unbiased coin impacts classification performance, causing it to deteriorate (typically by 5 to 10% in the situations tested). This happens because at some point in the trees that are learned, the irrelevant attribute is invariably chosen to branch on, causing random errors when test data is processed. How can this be when decision tree learners are cleverly designed to choose the best attribute for splitting at each node? The reason is subtle. As you proceed further down the tree, less and less data is available to help make the selection decision. At some point, with little data, the random attribute will look good just by chance. Because the number of nodes at each level increases 308 CHAPTER 7 Data Transformations exponentially with depth, the chance of the rogue attribute looking good somewhere along the frontier multiplies up as the tree deepens. The real problem is that you inevitably reach depths at which only a small amount of data is available for attribute selection. If the dataset were bigger it wouldn’t necessarily help—you’d probably just go deeper. Divide-and-conquer tree learners and separate-and-conquer rule learners both suffer from this effect because they inexorably reduce the amount of data on which they base judgments. Instance-based learners are very susceptible to irrelevant attri- butes because they always work in local neighborhoods, taking just a few training instances into account for each decision. Indeed, it has been shown that the number of training instances needed to produce a predetermined level of performance for instance-based learning increases exponentially with the number of irrelevant attri- butes present. Naïve Bayes, by contrast, does not fragment the instance space and robustly ignores irrelevant attributes. It assumes by design that all attributes are condi- tionally independent of one another, an assumption that is just right for random “dis- tracter” attributes. But through this very same assumption, Naïve Bayes pays a heavy price in other ways because its operation is damaged by adding redundant attributes. The fact that irrelevant distracters degrade the performance of state-of-the-art decision tree and rule learners is, at first, surprising. Even more surprising is that relevant attributes can also be harmful. For example, suppose that in a two-class dataset a new attribute was added that had the same value as the class to be predicted most of the time (65%) and the opposite value the rest of the time, randomly dis- tributed among the instances. Experiments with standard datasets have shown that this can cause classification accuracy to deteriorate (by 1 to 5% in the situations tested). The problem is that the new attribute is (naturally) chosen for splitting high up in the tree. This has the effect of fragmenting the set of instances available at the nodes below so that other choices are based on sparser data. Because of the negative effect of irrelevant attributes on most machine learning schemes, it is common to precede learning with an attribute selection stage that strives to eliminate all but the most relevant attributes. The best way to select rel- evant attributes is manually, based on a deep understanding of the learning problem and what the attributes actually mean. However, automatic methods can also be useful. Reducing the dimensionality of the data by deleting unsuitable attributes improves the performance of learning algorithms. It also speeds them up, although this may be outweighed by the computation involved in attribute selection. More important, dimensionality reduction yields a more compact, more easily interpre- table representation of the target concept, focusing the user’s attention on the most relevant variables. Scheme-Independent Selection When selecting a good attribute subset, there are two fundamentally different approaches. One is to make an independent assessment based on general characteris- tics of the data; the other is to evaluate the subset using the machine learning algorithm 7.1 Attribute Selection 309 that will ultimately be employed for learning. The first is called the filter method because the attribute set is filtered to produce the most promising subset before learn- ing commences. The second is called the wrapper method because the learning algo- rithm is wrapped into the selection procedure. Making an independent assessment of an attribute subset would be easy if there were a good way of determining when an attribute was relevant to choosing the class. However, there is no universally accepted measure of relevance, although several different ones have been proposed. One simple scheme-independent method of attribute selection is to use just enough attributes to divide up the instance space in a way that separates all the training instances. For example, if just one or two attributes are used, there will generally be several instances that have the same combination of attribute values. At the other extreme, the full set of attributes will likely distinguish the instances uniquely so that no two instances have the same values for all attributes. (This will not necessarily be the case, however; datasets sometimes contain instances with the same attribute values but different classes.) It makes intuitive sense to select the smallest attribute subset that serves to distinguish all instances uniquely. This can easily be found using an exhaustive search, although at considerable computational expense. Unfortunately, this strong bias toward consistency of the attribute set on the training data is statisti- cally unwarranted and can lead to overfitting—the algorithm may go to unnecessary lengths to repair an inconsistency that was in fact merely caused by noise. Machine learning algorithms can be used for attribute selection. For instance, you might first apply a decision tree algorithm to the full dataset and then select only those attributes that are actually used in the tree. While this selection would have no effect at all if the second stage merely built another tree, it will have an effect on a different learning algorithm. For example, the nearest-neighbor algorithm is notoriously sus- ceptible to irrelevant attributes, and its performance can be improved by using a deci- sion tree builder as a filter for attribute selection first. The resulting nearest-neighbor scheme can also perform better than the decision tree algorithm used for filtering. As another example, the simple 1R scheme described in Chapter 4 has been used to select the attributes for a decision tree learner by evaluating the effect of branching on different attributes (although an error-based method such as 1R may not be the optimal choice for ranking attributes, as we will see later when covering the related problem of supervised discretization). Often the decision tree performs just as well when only the two or three top attributes are used for its construction— and it is much easier to understand. In this approach, the user determines how many attributes to use for building the decision tree. Another possibility is to use an algorithm that builds a linear model—for example, a linear support vector machine—and ranks the attributes based on the size of the coefficients. A more sophisticated variant applies the learning algorithm repeatedly. It builds a model, ranks the attributes based on the coefficients, removes the lowest-ranked one, and repeats the process until all attributes have been removed. This method of recursive feature elimination has been found to yield better results on certain datasets (e.g., when identifying important genes for cancer classification) than simply ranking attributes based on a single model. With both 310 CHAPTER 7 Data Transformations methods it is important to ensure that the attributes are measured on the same scale; otherwise, the coefficients are not comparable. Note that these techniques just produce a ranking; another method must be used to determine the appropriate number of attributes to use. Attributes can be selected using instance-based learning methods too. You could sample instances randomly from the training set and check neighboring records of the same and different classes—“near hits” and “near misses.” If a near hit has a different value for a certain attribute, that attribute appears to be irrelevant and its weight should be decreased. On the other hand, if a near miss has a different value, the attribute appears to be relevant and its weight should be increased. Of course, this is the standard kind of procedure used for attribute weighting for instance-based learning, described in Section 6.5. After repeating this operation many times, selec- tion takes place: Only attributes with positive weights are chosen. As in the standard incremental formulation of instance-based learning, different results will be obtained each time the process is repeated, because of the different ordering of examples. This can be avoided by using all training instances and taking into account all near hits and near misses of each. A more serious disadvantage is that the method will not detect an attribute that is redundant because it is correlated with another attribute. In the extreme case, two identical attributes would be treated in the same way, either both selected or both rejected. A modification has been suggested that appears to go some way toward addressing this issue by taking the current attribute weights into account when computing the nearest hits and misses. Another way of eliminating redundant attributes as well as irrelevant ones is to select a subset of attributes that individually correlate well with the class but have little intercorrelation. The correlation between two nominal attributes A and B can be measured using the symmetric uncertainty: UABHAHBHAB HAHB(,)()()(,) ()()= + − + 2 where H is the entropy function described in Section 4.3. The entropies are based on the probability associated with each attribute value; H(A, B), the joint entropy of A and B, is calculated from the joint probabilities of all combinations of values of A and B. The symmetric uncertainty always lies between 0 and 1. Correlation-based feature selection determines the goodness of a set of attributes using UACUAAj j i j ji (,)(,)∑ ∑∑ where C is the class attribute and the indices i and j range over all attributes in the set. If all m attributes in the subset correlate perfectly with the class and with one another, the numerator becomes m and the denominator m2, which is also m. Thus, the measure is 1, which turns out to be the maximum value it can attain (the minimum is 0). Clearly, this is not ideal, because we want to avoid redundant attributes. However, any subset of this set will also have value 1. When using this criterion to search for a good subset of attributes, it makes sense to break ties in favor of the smallest subset. 7.1 Attribute Selection 311 Searching the Attribute Space Most methods for attribute selection involve searching the space of attributes for the subset that is most likely to predict the class best. Figure 7.1 illustrates the attribute space for the—by now all-too-familiar—weather dataset. The number of possible attribute subsets increases exponentially with the number of attributes, making an exhaustive search impractical on all but the simplest problems. Typically, the space is searched greedily in one of two directions: top to bottom and bottom to top in the figure. At each stage, a local change is made to the current attribute subset by either adding or deleting a single attribute. The downward direc- tion, where you start with no attributes and add them one at a time, is called forward selection. The upward one, where you start with the full set and delete attributes one at a time, is backward elimination. In forward selection, each attribute that is not already in the current subset is tentatively added to it, and the resulting set of attributes is evaluated—using, for example, cross-validation, as described in the following section. This evaluation produces a numeric measure of the expected performance of the subset. The effect of adding each attribute in turn is quantified by this measure, the best one is chosen, FIGURE 7.1 Attribute space for the weather dataset. outlook temperature humidity windy outlook temperature outlook humidity outlook windy temperature humidity temperature windy humidity windy outlook temperature humidity outlook temperature windy outlook humidity windy temperature humidity windy outlook temperature humidity windy 312 CHAPTER 7 Data Transformations and the procedure continues. However, if no attribute produces an improvement when added to the current subset, the search ends. This is a standard greedy search procedure and guarantees to find a locally—but not necessarily globally—optimal set of attributes. Backward elimination operates in an entirely analogous fashion. In both cases a slight bias is often introduced toward smaller attribute sets. This can be done for forward selection by insisting that if the search is to continue, the evaluation measure must not only increase, but must increase by at least a small predetermined quantity. A similar modification works for backward elimination. More sophisticated search schemes exist. Forward selection and backward elimi- nation can be combined into a bidirectional search; again, one can begin either with all the attributes or with none of them. Best-first search is a method that does not just terminate when the performance starts to drop but keeps a list of all attribute subsets evaluated so far, sorted in order of the performance measure, so that it can revisit an earlier configuration instead. Given enough time it will explore the entire space, unless this is prevented by some kind of stopping criterion. Beam search is similar but truncates its list of attribute subsets at each stage so that it only contains a fixed number—the beam width—of most promising candidates. Genetic algorithm search procedures are loosely based on the principle of natural selection: They “evolve” good feature subsets by using random perturbations of a current list of candidate subsets and combining them based on performance. Scheme-Specific Selection The performance of an attribute subset with scheme-specific selection is measured in terms of the learning scheme’s classification performance using just those attri- butes. Given a subset of attributes, accuracy is estimated using the normal procedure of cross-validation described in Section 5.3. Of course, other evaluation methods such as performance on a holdout set (Section 5.3) or the bootstrap estimator (Section 5.4) could be equally well used. The entire attribute selection process is rather computation intensive. If each evaluation involves a tenfold cross-validation, the learning procedure must be exe- cuted 10 times. With m attributes, the heuristic forward selection or backward elimi- nation multiplies evaluation time by a factor proportional to m2 in the worst case. For more sophisticated searches, the penalty will be far greater, up to 2m for an exhaustive algorithm that examines each of the 2m possible subsets. Good results have been demonstrated on many datasets. In general terms, back- ward elimination produces larger attribute sets than forward selection but better classification accuracy in some cases. The reason is that the performance measure is only an estimate, and a single optimistic estimate will cause both of these search procedures to halt prematurely—backward elimination with too many attributes and forward selection with not enough. But forward selection is useful if the focus is on understanding the decision structures involved, because it often reduces the number of attributes with only a small effect on classification accuracy. Experience seems 7.1 Attribute Selection 313 to show that more sophisticated search techniques are not generally justified, although they can produce much better results in certain cases. One way to accelerate the search process is to stop evaluating a subset of attributes as soon as it becomes apparent that it is unlikely to lead to higher accuracy than another candidate subset. This is a job for a paired statistical significance test, performed between the classifier based on this subset and all the other candidate classifiers based on other subsets. The performance difference between two classifiers on a particular test instance can be −1, 0, or 1 depending on, respectively, whether the first classifier is worse than, the same as, or better than the second on that instance. A paired t-test (described in Section 5.5) can be applied to these figures over the entire test set, effectively treating the results for each instance as an independent estimate of the difference in performance. Then the cross-validation for a classifier can be prematurely terminated as soon as it turns out to be significantly worse than another, which, of course, may never happen. We might want to discard classifiers more aggressively by modifying the t-test to compute the probability that one classifier is better than another classifier by at least a small user- specified threshold. If this probability becomes very small, we can discard the former classifier on the basis that it is very unlikely to perform substantially better than the latter. This methodology is called race search and can be implemented with different underlying search strategies. When it is used with forward selection, we race all possible single-attribute additions simultaneously and drop those that do not perform well enough. In backward elimination, we race all single-attribute deletions. Schemata search is a more complicated method specifically designed for racing; it runs an iterative series of races that each determine whether or not a particular attribute should be included. The other attributes for this race are included or excluded randomly at each point in the evaluation. As soon as one race has a clear winner, the next iteration of races begins, using the winner as the starting point. Another search strategy is to rank the attributes first using, for example, their information gain (assuming they are discrete), and then race the ranking. In this case the race includes no attributes, the top-ranked attribute, the top two attributes, the top three, and so on. A simple method for accelerating a scheme-specific search is to preselect a given number of attributes by ranking them first using a criterion like the information gain and discarding the rest before applying scheme-specific selection. This has been found to work surprisingly well on high-dimensional datasets such as gene expres- sion and text categorization data, where only a couple of hundred of attributes are used instead of several thousands. In the case of forward selection, a slightly more sophisticated variant is to restrict the number of attributes available for expanding the current attribute subset to a fixed-sized subset chosen from the ranked list of attributes—creating a sliding window of attribute choices—rather than making all (unused) attributes available for consideration in each step of the search process. Whatever way you do it, scheme-specific attribute selection by no means yields a uniform improvement in performance. Because of the complexity of the process, which is greatly increased by the feedback effect of including a target machine learning algorithm in the attribution selection loop, it is quite hard to predict the conditions under which it will turn out to be worthwhile. As in many machine 314 CHAPTER 7 Data Transformations learning situations, trial and error using your own particular source of data is the final arbiter. There is one type of classifier for which scheme-specific attribute selection is an essential part of the learning process: the decision table. As mentioned in Section 3.1, the entire problem of learning decision tables consists of selecting the right attributes to be included. Usually this is done by measuring the table’s cross- validation performance for different subsets of attributes and choosing the best- performing subset. Fortunately, leave-one-out cross-validation is very cheap for this kind of classifier. Obtaining the cross-validation error from a decision table derived from the training data is just a matter of manipulating the class counts associated with each of the table’s entries, because the table’s structure doesn’t change when instances are added or deleted. The attribute space is generally searched by best-first search because this strategy is less likely to get stuck in a local maximum than others, such as forward selection. Let’s end our discussion with a success story. Naïve Bayes is a learning method for which a simple scheme-specific attribute selection approach has shown good results. Although this method deals well with random attributes, it has the potential to be misled when there are dependencies among attributes, and particu- larly when redundant ones are added. However, good results have been reported using the forward selection algorithm—which is better able to detect when a redundant attribute is about to be added than the backward elimination approach— in conjunction with a very simple, almost “naïve,” metric that determines the quality of an attribute subset to be simply the performance of the learned algo- rithm on the training set. As was emphasized in Chapter 5, training set perfor- mance is certainly not a reliable indicator of test set performance. Nevertheless, experiments show that this simple modification to Naïve Bayes markedly improves its performance on those standard datasets for which it does not do so well as tree- or rule-based classifiers, and does not have any negative effect on results on datasets on which Naïve Bayes already does well. Selective Naïve Bayes, as this learning method is called, is a viable machine learning technique that performs reliably and well in practice. 7.2 DISCRETIZING NUMERIC ATTRIBUTES Some classification and clustering algorithms deal with nominal attributes only and cannot handle ones measured on a numeric scale. To use them on general datasets, numeric attributes must first be “discretized” into a small number of distinct ranges. Even learning algorithms that handle numeric attributes sometimes process them in ways that are not altogether satisfactory. Statistical clustering methods often assume that numeric attributes have a normal distribution—often not a very plausible assumption in practice—and the standard extension of the Naïve Bayes classifier for numeric attributes adopts the same assumption. Although most decision tree and decision rule learners can handle numeric attributes, some implementations work 7.2 Discretizing Numeric Attributes 315 much more slowly when numeric attributes are present because they repeatedly sort the attribute values. For all these reasons the question arises, what is a good way to discretize numeric attributes into ranges before any learning takes place? We have already encountered some methods for discretizing numeric attributes. The 1R learning scheme described in Chapter 4 uses a simple but effective tech- nique: Sort the instances by the attribute’s value and assign the value into ranges at the points that the class value changes—except that a certain minimum number of instances in the majority class (six) must lie in each of the ranges, which means that any given range may include a mixture of class values. This is a “global” method of discretization that is applied to all continuous attributes before learning starts. Decision tree learners, on the other hand, deal with numeric attributes on a local basis, examining attributes at each node of the tree when it is being constructed to see whether they are worth branching on, and only at that point deciding on the best place to split continuous attributes. Although the tree-building method we examined in Chapter 6 only considers binary splits of continuous attributes, one can imagine a full discretization taking place at that point, yielding a multiway split on a numeric attribute. The pros and cons of the local versus global approach are clear. Local discretization is tailored to the actual context provided by each tree node, and will produce different discretizations of the same attribute at different places in the tree if that seems appropriate. However, its decisions are based on less data as tree depth increases, which compromises their reliability. If trees are developed all the way out to single-class leaves before being pruned back, as with the normal technique of backward pruning, it is clear that many discretization decisions will be based on data that is grossly inadequate. When using global discretization before applying a learning scheme, there are two possible ways of presenting the discretized data to the learner. The most obvious is to treat discretized attributes like nominal ones: Each discretization interval is represented by one value of the nominal attribute. However, because a discretized attribute is derived from a numeric one, its values are ordered, and treating it as nominal discards this potentially valuable ordering information. Of course, if a learn- ing scheme can handle ordered attributes directly, the solution is obvious: Each discretized attribute is declared to be of type “ordered.” If the learning scheme cannot handle ordered attributes, there is still a simple way of enabling it to exploit the ordering information: Transform each discretized attribute into a set of binary attributes before the learning scheme is applied. If the discretized attribute has k values, it is transformed into k − 1 binary attributes. If the original attribute’s value is i for a particular instance, the first i − 1 of these new attributes are set to false and the remainder are set to true. In other words, the (i − 1)th binary attribute represents whether the discretized attribute is less than i. If a decision tree learner splits on this attribute, it implicitly utilizes the ordering information it encodes. Note that this transformation is independent of the particular discretization method being applied: It is simply a way of coding an ordered attribute using a set of binary attributes. 316 CHAPTER 7 Data Transformations Unsupervised Discretization There are two basic approaches to the problem of discretization. One is to quantize each attribute in the absence of any knowledge of the classes of the instances in the training set—so-called unsupervised discretization. The other is to take the classes into account when discretizing—supervised discretization. The former is the only possibility when dealing with clustering problems where the classes are unknown or nonexistent. The obvious way of discretizing a numeric attribute is to divide its range into a predetermined number of equal intervals: a fixed, data-independent yardstick. This is frequently done at the time when data is collected. But, like any unsupervised discretization method, it runs the risk of destroying distinctions that would have turned out to be useful in the learning process by using gradations that are too coarse or, that by unfortunate choices of boundary, needlessly lump together many instances of different classes. Equal-width binning often distributes instances very unevenly: Some bins contain many instances while others contain none. This can seriously impair the ability of the attribute to help build good decision structures. It is often better to allow the intervals to be of different sizes, choosing them so that the same number of training examples fall into each one. This method, called equal-frequency binning, divides the attribute’s range into a predetermined number of bins based on the distribution of examples along that axis—sometimes called histogram equalization because if you take a histogram of the contents of the resulting bins it will be completely flat. If you view the number of bins as a resource, this method makes the best use of it. However, equal-frequency binning is still oblivious to the instances’ classes, and this can cause bad boundaries. For example, if all instances in a bin have one class, and all instances in the next higher bin have another except for the first, which has the original class, surely it makes sense to respect the class divisions and include that first instance in the previous bin, sacrificing the equal-frequency property for the sake of homogeneity. Supervised discretization—taking classes into account during the process—certainly has advantages. Nevertheless, it has been found that equal-frequency binning can yield excellent results, at least in conjunction with the Naïve Bayes learning scheme, when the number of bins is chosen in a data-dependent fashion by setting it to the square root of the number of instances. This method is called proportional k-interval discretization. Entropy-Based Discretization Because the criterion used for splitting a numeric attribute during the formation of a decision tree works well in practice, it seems a good idea to extend it to more general discretization by recursively splitting intervals until it is time to stop. In Chapter 6 we saw how to sort the instances by the attribute’s value and consider, for each possible splitting point, the information gain of the resulting split. To 7.2 Discretizing Numeric Attributes 317 discretize the attribute, once the first split is determined, the splitting process can be repeated in the upper and lower parts of the range, and so on, recursively. To see this working in practice, we revisit the example given in Section 6.1 for discretizing the temperature attribute of the weather data, of which the values are as follows: 64 65 68 69 70 71 72 75 80 81 83 85 yes no yes yes yes no no yes yes yes no yes yes no (Repeated values have been collapsed together.) The information gain for each of the 11 possible positions for the breakpoint is calculated in the usual way. For example, the information value of the test temperature < 71.5, which splits the range into four yes and two no versus five yes and three no, is info( ) info info[,],[,] ( ) ([,])( ) ([,]) .4 2 5 3 6 14 4 2 8 14 5 3 0 93= × + × = 99 bits This represents the amount of information required to specify the individual values of yes and no given the split. We seek a discretization that makes the subintervals as pure as possible; thus, we choose to split at the point where the information value is smallest. (This is the same as splitting where the information gain, defined as the difference between the information value without the split and that with the split, is largest.) As before, we place numeric thresholds halfway between the values that delimit the boundaries of a concept. The graph labeled A in Figure 7.2 shows the information values at each possible cut point at this first stage. The cleanest division—the smallest information value—is at a temperature of 84 (0.827 bits), which separates off just the very final value, a no instance, from the preceding list. The instance classes are written below the hori- zontal axis to make interpretation easier. Invoking the algorithm again on the lower range of temperatures, from 64 to 83, yields the graph labeled B. This has a minimum at 80.5 (0.800 bits), which splits off the next two values, both yes instances. Again invoking the algorithm on the lower range, now from 64 to 80, produces the graph labeled C (shown dotted to help distinguish it from the others). The minimum is at 77.5 (0.801 bits), splitting off another no instance. Graph D has a minimum at 73.5 (0.764 bits), splitting off two yes instances. Graph E (again dashed, purely to make it more easily visible), for the temperature range 64 to 72, has a minimum at 70.5 (0.796 bits), which splits off two no and one yes. Finally, graph F, for the range 64 to 70, has a minimum at 66.5 (0.4 bits). The final discretization of the temperature attribute is shown in Figure 7.3. The fact that recursion only ever occurs in the first interval of each split is an artifact of this example: In general, both the upper and lower intervals will have to be split further. Underneath each division is the label of the graph in Figure 7.2 that is responsible for it, and below that the actual value of the split point. It can be shown theoretically that a cut point that minimizes the information value will never occur between two instances of the same class. This leads to a useful 318 CHAPTER 7 Data Transformations FIGURE 7.3 The result of discretizing the temperature attribute. 64 65 68 69 70 71 72 75 80 81 83 85 yes no yes yes yes no no yes yes yes no yes yes no F E D C B A 66.5 70.5 73.5 77.5 80.5 84 FIGURE 7.2 Discretizing the temperature attribute using the entropy method. 0.4 0.2 0.6 0 0.8 1 65 70 75 80 85 ABCDE F no yesyes no yes yes yes no yes yes no yes yes no optimization: It is only necessary to consider potential divisions that separate instances of different classes. Notice that if class labels were assigned to the intervals based on the majority class in the interval, there would be no guarantee that adjacent intervals would receive different labels. You might be tempted to consider merging intervals with the same majority class (e.g., the first two intervals of Figure 7.3), but as we will see later this is not a good thing to do in general. The only problem left to consider is the stopping criterion. In the temperature example most of the intervals that were identified were “pure” in that all their instances had the same class, and there is clearly no point in trying to split such an 7.2 Discretizing Numeric Attributes 319 interval. (Exceptions were the final interval, which we tacitly decided not to split, and the interval from 70.5 to 73.5.) In general, however, things are not so straightforward. A good way to stop the entropy-based splitting discretization procedure turns out to be the MDL principle that we encountered in Chapter 5 (page 183). In accordance with that principle, we want to minimize the size of the “theory” plus the size of the information necessary to specify all the data given that theory. In this case, if we do split, the “theory” is the splitting point, and we are comparing the situation in which we split with that in which we do not. In both cases we assume that the instances are known but their class labels are not. If we do not split, the classes can be trans- mitted by encoding each instance’s label. If we do, we first encode the split point (in log2[N − 1] bits, where N is the number of instances), then the classes of the instances below that point, and then the classes of those above it. You can imagine that if the split is a good one—say, all the classes below it are yes and all those above are no—then there is much to be gained by splitting. If there is an equal number of yes and no instances, each instance costs 1 bit without splitting but hardly more than 0 bits with splitting—it is not quite 0 because the class values associated with the split itself must be encoded, but this penalty is amortized across all the instances. In this case, if there are many examples, the penalty of having to encode the split point will be far outweighed by the information that is saved by splitting. We emphasized in Section 5.9 that when applying the MDL principle, the devil is in the details. In the relatively straightforward case of discretization, the situation is tractable although not simple. The amounts of information can be obtained exactly under certain reasonable assumptions. We will not go into the details, but the upshot is that the split dictated by a particular cut point is worthwhile if the information gain for that split exceeds a certain value that depends on the number of instances N, the number of classes k, the entropy of the instances E, the entropy of the instances in each subinterval E1 and E2, and the number of classes represented in each subinterval k1 and k2: gain N N kE k E k E N k > − + − − + +log ( ) log ( )2 2 1 1 2 21 3 2 The first component is the information needed to specify the splitting point; the second is a correction due to the need to transmit the classes that correspond to the upper and lower subintervals. When applied to the temperature example, this criterion prevents any splitting at all. The first split removes just the final example, and, as you can imagine, very little actual information is gained by this when transmitting the classes—in fact, the MDL criterion will never create an interval containing just one example. Failure to discretize temperature effectively disbars it from playing any role in the final decision structure because the same discretized value will be given to all instances. In this situation, this is perfectly appropriate: Temperature does not occur in good decision trees or rules for the weather data. In effect, failure to discretize is tantamount to attribute selection. 320 CHAPTER 7 Data Transformations Other Discretization Methods The entropy-based method with the MDL stopping criterion is one of the best general techniques for supervised discretization. However, many other methods have been investigated. For example, instead of proceeding top-down by recursively splitting intervals until some stopping criterion is satisfied, you could work bottom-up, first placing each instance into its own interval and then considering whether to merge adjacent intervals. You could apply a statistical criterion to see which would be the best two intervals to merge, and merge them if the statistic exceeds a certain preset confidence level, repeating the operation until no potential merge passes the test. The χ2 test is a suitable one and has been used for this purpose. Instead of specifying a preset significance threshold, more complex techniques are available to determine an appropriate level automatically. A rather different approach is to count the number of errors that a discretization makes when predicting each training instance’s class, assuming that each interval receives the majority class. For example, the 1R method described earlier is error- based—it focuses on errors rather than the entropy. However, the best possible discretization in terms of error count is obtained by using the largest possible number of intervals, and this degenerate case should be avoided by restricting the number of intervals in advance. Let’s consider the best way to discretize an attribute into k intervals in a way that minimizes the number of errors. The brute-force method of finding this is exponential in k and therefore infeasible. However, there are much more efficient schemes that are based on the idea of dynamic programming. Dynamic program- ming applies not just to the error count measure but to any given additive impurity function, and it can find the partitioning of N instances into k intervals in a way that minimizes the impurity in time proportional to kN2. This gives a way of finding the best entropy-based discretization, yielding a potential improvement in the quality of the discretization (in practice a negligible one) over the recursive entropy-based method described previously. The news for error-based discretization is even better because there is an algorithm that can be used to minimize the error count in time linear in N. Entropy-Based versus Error-Based Discretization Why not use error-based discretization, since the optimal discretization can be found very quickly? The answer is that there is a serious drawback to error-based discreti- zation: It cannot produce adjacent intervals with the same label (such as the first two of Figure 7.3). The reason is that merging two such intervals will not affect the error count, but it will free up an interval that can be used elsewhere to reduce the error count. Why would anyone want to generate adjacent intervals with the same label? The reason is best illustrated with an example. Figure 7.4 shows the instance space for a simple two-class problem with two numeric attributes ranging from 0 to 1. Instances 7.2 Discretizing Numeric Attributes 321 belong to one class (the dots) if their first attribute (a1) is less than 0.3, or if the first attribute is less than 0.7 and their second attribute (a2) is less than 0.5. Otherwise, they belong to the other class (triangles). The data in Figure 7.4 has been artificially generated according to this rule. Now suppose we are trying to discretize both attributes with a view to learning the classes from the discretized attributes. The very best discretization splits a1 into three intervals (0–0.3, 0.3–0.7, and 0.7–1) and a2 into two intervals (0–0.5 and 0.5–1). Given these nominal attributes, it will be easy to learn how to tell the classes apart with a simple decision tree or rule algorithm. Discretizing a2 is no problem. For a1, however, the first and last intervals will have opposite labels (dot and tri- angle, respectively). The second will have whichever label happens to occur most in the region from 0.3 to 0.7 (it is in fact dot for the data in Figure 7.4). Either way, this label must inevitably be the same as one of the adjacent labels—of course, this is true whatever the class probability happens to be in the middle region. Thus, this discretization will not be achieved by any method that minimizes the error counts because such a method cannot produce adjacent intervals with the same label. The point is that what changes as the value of a1 crosses the boundary at 0.3 is not the majority class but the class distribution. The majority class remains dot. The distribution, however, changes markedly, from 100% before the boundary to just over 50% after it. And the distribution changes again as the boundary at 0.7 is crossed, from 50 to 0%. Entropy-based discretization methods are sensitive to FIGURE 7.4 Class distribution for a two-class, two-attribute problem. 0 0.2 0.4 0.6 0.8 1 a1 0 0.2 0.4 0.6 0.8 1 a 2 322 CHAPTER 7 Data Transformations changes in the distribution even though the majority class does not change. Error- based methods are not sensitive. Converting Discrete Attributes to Numeric Attributes There is a converse problem to discretization. Some learning algorithms—notably the nearest-neighbor instance-based method and numeric prediction techniques involving regression—naturally handle only attributes that are numeric. How can they be extended to nominal attributes? In instance-based learning, as described in Section 4.7 (page 132), discrete attributes can be treated as numeric by defining as 0 the “distance” between two nominal values that are the same and as 1 the distance between two values that are different, regardless of the actual values involved. Rather than modifying the distance function, this can be achieved by an attribute transformation: Replace a k-valued nominal attribute by k synthetic binary attributes, one for each value indicating whether the attribute has that value or not. If the attributes have equal weight, this achieves the same effect on the distance function. The distance is insensitive to the attribute values because only “same” or “different” information is encoded, not the shades of difference that may be associated with the various possible values of the attribute. More subtle distinctions can be made if the attri- butes have weights reflecting their relative importance. If the values of the attribute can be ordered, more possibilities arise. For a numeric prediction problem, the average class value corresponding to each value of a nominal attribute can be calculated from the training instances and used to determine an ordering—this technique was introduced for model trees in Section 6.6 (page 253). (It is hard to come up with an analogous way of ordering attribute values for a clas- sification problem.) An ordered nominal attribute can be replaced by an integer in the obvious way, but this implies not just an ordering but also a metric on the attri- bute’s values. The implication of a metric can be avoided by creating k − 1 synthetic binary attributes for a k-valued nominal attribute, in the manner described on page 315. This encoding still implies an ordering among different values of the attribute— adjacent values differ in just one of the synthetic attributes whereas distant ones differ in several—but does not imply an equal distance between the attribute values. 7.3 PROJECTIONS Resourceful data miners have a toolbox full of techniques, such as discretization, for transforming data. As we emphasized in Section 2.4, data mining is hardly ever a matter of simply taking a dataset and applying a learning algorithm to it. Every problem is different. You need to think about the data and what it means, and (cre- atively) examine it from diverse points of view to arrive at a suitable perspective. Transforming it in different ways can help you get started. In mathematics, a projec- tion is a kind of function or mapping that transforms data in some way. 7.3 Projections 323 You don’t have to make your own toolbox by implementing the projections yourself. Comprehensive environments for data mining, such as the one described in Part III of this book, contain a wide range of suitable tools for you to use. You do not necessarily need a detailed understanding of how they are implemented. What you do need to understand is what the tools do and how they can be applied. In Part III we list, and briefly describe, all the transformations in the Weka data mining workbench. Data often calls for general mathematical transformations of a set of attributes. It might be useful to define new attributes by applying specified mathematical functions to existing ones. Two date attributes might be subtracted to give a third attribute representing age—an example of a semantic transformation driven by the meaning of the original attributes. Other transformations might be suggested by known properties of the learning algorithm. If a linear relationship involving two attributes, A and B, is suspected, and the algorithm is only capable of axis-parallel splits (as most decision tree and rule learners are), the ratio A:B might be defined as a new attribute. The transformations are not necessarily mathematical ones, but may involve real-world knowledge such as days of the week, civic holidays, or chemical atomic numbers. They could be expressed as operations in a spreadsheet or as functions that are implemented by arbitrary computer programs. Or you can reduce several nominal attributes to one by concatenating their values, producing a single k1 × k2-valued attribute from attributes with k1 and k2 values, respectively. Discretization converts a numeric attribute to nominal, and we saw earlier how to convert in the other direction too. As another kind of transformation, you might apply a clustering procedure to the dataset and then define a new attribute with a value for any given instance that is the cluster that contains it using an arbitrary labeling for the clusters. Alternatively, with probabilistic clustering, you could augment each instance with its membership probabilities for each cluster, including as many new attributes as there are clusters. Sometimes it is useful to add noise to data, perhaps to test the robustness of a learning algorithm; to take a nominal attribute and change a given percentage of its values; to obfuscate data by renaming the relation, attribute names, and nominal and string attribute values (because it is often necessary to anonymize sensitive datasets); to randomize the order of instances or produce a random sample of the dataset by resampling it; to reduce a dataset by removing a given percentage of instances, or all instances that have certain values for nominal attributes, or numeric values above or below a certain threshold; or to remove outliers by applying a classification method to the dataset and deleting misclassified instances. Different types of input call for their own transformations. If you can input sparse data files (see Section 2.4), you may need to be able to convert datasets to nonsparse form and vice versa. Textual input and time series input call for their own specialized conversions, described in the following sections. But first we look at two general techniques for transforming data with numeric attributes into a lower-dimensional form that may be more useful for mining. 324 CHAPTER 7 Data Transformations Principal Components Analysis In a dataset with m numeric attributes, you can visualize the data as a cloud of points in m-dimensional space—the stars in the sky, a swarm of flies frozen in time, a two- dimensional scatter plot on paper. The attributes represent the coordinates of the space. But the axes you use, the coordinate system itself, is arbitrary. You can place horizontal and vertical axes on the paper and represent the points of the scatter plot using those coordinates, or you could draw an arbitrary straight line to represent the x-axis and one perpendicular to it to represent the y-axis. To record the positions of the flies you could use a conventional coordinate system with a north–south axis, an east–west axis, and an up–down axis. But other coordinate systems would do equally well. Creatures like flies don’t know about north, south, east, and west, although, being subject to gravity, they may perceive up–down as something special. And as for the stars in the sky, who’s to say what the “right” coordinate system is? Over the centuries our ancestors moved from a geocentric perspective to a heliocen- tric one to a purely relativistic one, each shift of perspective being accompanied by turbulent religious–scientific upheavals and painful reexamination of humankind’s role in God’s universe. Back to the dataset. Just as in these examples, there is nothing to stop you from transforming all the data points into a different coordinate system. But unlike these examples, in data mining there often is a preferred coordinate system, defined not by some external convention but by the very data itself. Whatever coordinates you use, the cloud of points has a certain variance in each direction, indicating the degree of spread around the mean value in that direction. It is a curious fact that if you add up the variances along each axis and then transform the points into a different coor- dinate system and do the same there, you get the same total variance in both cases. This is always true provided the coordinate systems are orthogonal—that is, each axis is at right angles to the others. The idea of principal components analysis is to use a special coordinate system that depends on the cloud of points as follows: Place the first axis in the direction of greatest variance of the points to maximize the variance along that axis. The second axis is perpendicular to it. In two dimensions there is no choice—its direction is determined by the first axis—but in three dimensions it can lie anywhere in the plane perpendicular to the first axis, and in higher dimensions there is even more choice, though it is always constrained to be perpendicular to the first axis. Subject to this constraint, choose the second axis in the way that maximizes the variance along it. And so on, choosing each axis to maximize its share of the remaining variance. How do you do this? It’s not hard, given an appropriate computer program, and it’s not hard to understand, given the appropriate mathematical tools. Technically—for those who understand the italicized terms—you calculate the covariance matrix of the mean-centered coordinates of the points and diagonalize it to find the eigenvectors. These are the axes of the transformed space, sorted in order of eigenvalue—because each eigenvalue gives the variance along its axis. 7.3 Projections 325 FIGURE 7.5 Principal components transform of a dataset: (a) variance of each component and (b) variance plot. 0% 10% 20% 30% 40% 50% 60% 70% 1 2 3 4 5 6 7 8 9 10 Percentage of Variance Component Number (b) (a) Axis Variance Cumulative 61.2% 79.2% 83.9% 87.9% 91.1% 94.0% 96.0% 97.7% 99.1% 1 61.2% 2 18.0% 3 4.7% 4 4.0% 5 3.2% 6 2.9% 7 2.0% 8 1.7% 9 1.4% 10 0.9% 100.0% Figure 7.5 shows the result of transforming a particular dataset with 10 numeric attributes, corresponding to points in 10-dimensional space. Imagine the original dataset as a cloud of points in 10 dimensions—we can’t draw it! Choose the first axis along the direction of greatest variance, the second perpendicular to it along the direction of next greatest variance, and so on. The table in the figure gives the vari- ance along each new coordinate axis in the order in which the axes were chosen. Because the sum of the variances is constant regardless of the coordinate system, they are expressed as percentages of that total. We call axes components and say that each one “accounts for” its share of the variance. Figure 7.5(b) plots the variance that each component accounts for against the component’s number. You can use all the components as new attributes for data mining, or you might want to choose just the first few, the principal components, and discard the rest. In this case, three prin- cipal components account for 84% of the variance in the dataset; seven account for more than 95%. On numeric datasets it is common to use principal components analysis prior to data mining as a form of data cleanup and attribute selection. For example, you might want to replace the numeric attributes with the principal component axes or with a subset of them that accounts for a given proportion—say, 95%—of the vari- ance. Note that the scale of the attributes affects the outcome of principal compo- nents analysis, and it is common practice to standardize all attributes to zero mean and unit variance first. 326 CHAPTER 7 Data Transformations Another possibility is to apply principal components analysis recursively in a decision tree learner. At each stage an ordinary decision tree learner chooses to split in a direction that is parallel to one of the axes. However, suppose a principal com- ponents transform is performed first, and the learner chooses an axis in the trans- formed space. This equates to a split along an oblique line in the original space. If the transform is performed afresh before each split, the result will be a multivariate decision tree with splits that are in directions that are not parallel with the axes or with one another. Random Projections Principal components analysis transforms the data linearly into a lower-dimensional space—but it’s expensive. The time taken to find the transformation (which is a matrix comprising the eigenvectors of the covariance matrix) is cubic in the number of dimensions. This makes it infeasible for datasets with a large number of attri- butes. A far simpler alternative is to use a random projection of the data into a subspace with a predetermined number of dimensions. It’s very easy to find a random projection matrix. But will it be any good? In fact, theory shows that random projections preserve distance relationships quite well on average. This means that they could be used in conjunction with kD- trees or ball trees to do approximate nearest-neighbor search in spaces with a huge number of dimensions. First transform the data to reduce the number of attributes; then build a tree for the transformed space. In the case of nearest-neighbor classifica- tion you could make the result more stable, and less dependent on the choice of random projection, by building an ensemble classifier that uses multiple random matrices. Not surprisingly, random projections perform worse than projections carefully chosen by principal components analysis when used to preprocess data for a range of standard classifiers. However, experimental results have shown that the difference is not too great—and it tends to decrease as the number of dimensions increases. And, of course, random projections are far cheaper computationally. Partial Least-Squares Regression As mentioned earlier, principal components analysis is often performed as a pre processing step before applying a learning algorithm. When the learning algorithm is linear regression, the resulting model is known as principal components regres- sion. Since principal components are themselves linear combinations of the original attributes, the output of principal components regression can be reexpressed in terms of the original attributes. In fact, if all the components are used—not just the “prin- cipal” ones—the result is the same as that obtained by applying least-squares regres- sion to the original input data. Using fewer than the full set of components results in a reduced regression. 7.3 Projections 327 Partial least-squares differs from principal components analysis in that it takes the class attribute into account, as well as the predictor attributes, when constructing a coordinate system. The idea is to calculate derived directions that, as well as having high variance, are strongly correlated with the class. This can be advantageous when seeking as small a set of transformed attributes as possible to use for supervised learning. There is a simple iterative method for computing the partial least-squares direc- tions that involves only dot product operations. Starting with input attributes that have been standardized to have zero mean and unit variance, the attribute coefficients for the first partial least-squares direction are found by taking the dot product between each attribute vector and the class vector in turn. To find the second direc- tion the same approach is used, but the original attribute values are replaced by the difference between the attribute’s value and the prediction from a simple univariate regression that uses the first direction as the single predictor of that attribute. These differences are called residuals. The process continues in the same fashion for each remaining direction, with residuals for the attributes from the previous iteration forming the input for finding the current partial least-squares direction. Here is a simple worked example that should help make the procedure clear. For the first five instances from the CPU performance data in Table 1.5 (page 15), Table 7.1(a) shows the values of CHMIN and CHMAX (after standardization to zero mean and unit variance) and PRP (not standardized). The task is to find an expression for the target attribute PRP in terms of the other two. The attribute coefficients for the first partial least-squares direction are found by taking the dot product between the class and each attribute in turn. The dot product between the PRP and CHMIN columns is −0.4472, and that between PRP and CHMAX is 22.981. Thus, the first partial least-squares direction is PLSCHMINCHMAX 1 0 4472 22 981= − +. . Table 7.1(b) shows the values for PLS 1 obtained from this formula. The next step is to prepare the input data for finding the second partial least- squares direction. To this end, PLS 1 is regressed onto CHMIN and CHMAX in Table 7.1 First Five Instances from the CPU Performance Data (a) (b) (c) chmin chmax prp pls 1 chmin chmax prp 1 1.7889 1.7678 198 39.825 0.0436 0.0008 198 2 −0.4472 −0.3536 269 −7.925 −0.0999 −0.0019 269 3 −0.4472 −0.3536 220 −7.925 −0.0999 −0.0019 220 4 −0.4472 −0.3536 172 −7.925 −0.0999 −0.0019 172 5 −0.4472 −0.7071 132 −16.05 0.2562 0.005 132 (a) original values, (b) first partial least-squares direction, and (c) residuals from the first direction. 328 CHAPTER 7 Data Transformations turn, resulting in linear equations that predict each of these attributes individually from PLS 1. The coefficients are found by taking the dot product between PLS 1 and the attribute in question, and dividing the result by the dot product between PLS 1 and itself. The resulting univariate regression equations are CHMIN PLS= 0 0438 1. CHMAX PLS = 0 0444 1. Table 7.1(c) shows the CPU data in preparation for finding the second partial least-squares direction. The original values of CHMIN and CHMAX have been replaced by residuals—that is, the difference between the original value and the output of the corresponding univariate regression equation given before (the target value PRP remains the same). The entire procedure is repeated using this data as input to yield the second partial least-squares direction, which is PLSCHMINCHMAX 2 23 6002 0 4593= − + −. . After this last partial least-squares direction has been found, the attribute residuals are all zero. This reflects the fact that, as with principal components analysis, the full set of directions account for all of the variance in the original data. When the partial least-squares directions are used as input to linear regression, the resulting model is known as a partial least-squares regression model. As with principal components regression, if all the directions are used, the solution is the same as that obtained by applying linear regression to the original data. Text to Attribute Vectors In Section 2.4 we introduced string attributes that contain pieces of text, and there remarked that the value of a string attribute is often an entire document. String attributes are basically nominal, with an unspecified number of values. If they are treated simply as nominal attributes, models can be built that depend on whether the values of two string attributes are equal or not. But that does not capture any internal structure of the string or bring out any interesting aspects of the text it represents. You could imagine decomposing the text in a string attribute into paragraphs, sentences, or phrases. Generally, however, the word is the most useful unit. The text in a string attribute is usually a sequence of words, and it is often best represented in terms of the words it contains. For example, you might transform the string attribute into a set of numeric attributes, one for each word, that represents how often each word appears. The set of words—that is, the set of new attributes—is determined from the dataset and is typically quite large. If there are several string attributes with properties that should be treated separately, the new attribute names must be distin- guished, perhaps by a user-determined prefix. Conversion into words—tokenization—is not as simple an operation as it sounds. Tokens may be formed from contiguous alphabetic sequences with nonalphabetic 7.3 Projections 329 characters discarded. If numbers are present, numeric sequences may be retained too. Numbers may involve + or − signs, may contain decimal points, and may have exponential notation—in other words, they must be parsed according to a defined number syntax. An alphanumeric sequence may be regarded as a single token. Perhaps the space character is the token delimiter; perhaps whitespace (including the tab and new-line characters) and punctuation are too. Periods can be difficult: Sometimes they should be considered part of the word (e.g., with initials, titles, abbreviations, and numbers), but sometimes they should not (e.g., if they are sentence delimiters). Hyphens and apostrophes are similarly problematic. All words may be converted to lowercase before being added to the dictionary. Words on a fixed, predetermined list of function words, or stopwords—such as the, and, and but—could be ignored. Note that stopword lists are language dependent. In fact, so are capitalization conventions (German capitalizes all nouns), number syntax (Europeans use the comma for a decimal point), punctuation conventions (Spanish has an initial question mark), and, of course, character sets. Text is complicated! Low-frequency words such as hapax legomena1 are often discarded. Sometimes it is found beneficial to keep the most frequent k words after stopwords have been removed—or perhaps the top k words for each class. Along with all these tokenization options, there is the question of what the value of each word attribute should be. The value may be the word count—the number of times the word appears in the string—or it may simply indicate the word’s presence or absence. Word frequencies could be normalized to give each document’s attribute vector the same Euclidean length. Alternatively, the frequen- cies fij for word i in document j can be transformed in various standard ways. One standard logarithmic term-frequency measure is log (1 + fij). A measure that is widely used in information retrieval is TF × IDF, or “term frequency times inverse document frequency.” Here, the term frequency is modulated by a factor that depends on how com- monly the word is used in other documents. The TF × IDF metric is typically defined as fij log number of documents number of documents that include wword i The idea is that a document is basically characterized by the words that appear often in it, which accounts for the first factor, except that words used in every document or almost every document are useless as discriminators, which accounts for the second. TF × IDF is used to refer not just to this particular formula but to a general class of measures of the same type. For example, the frequency factor fij may be replaced by a logarithmic term such as log (1 + fij). 1A hapax legomena is a word that only occurs once in a given corpus of text. 330 CHAPTER 7 Data Transformations Time Series In time series data, each instance represents a different time step and the attributes give values associated with that time, such as in weather forecasting or stock market prediction. You sometimes need to be able to replace an attribute’s value in the current instance by the corresponding value in some other instance in the past or the future. Even more common is to replace an attribute’s value by the difference between the current value and the value in some previous instance. For example, the difference—often called the Delta—between the current value and the preceding one is often more informative than the value itself. The first instance, for which the time-shifted value is unknown, may be removed or replaced with a missing value. The Delta value is essentially the first derivative scaled by some constant that depends on the size of the time step. Successive Delta transformations take higher derivatives. In some time series, instances do not represent regular samples; instead, the time of each instance is given by a timestamp attribute. The difference between time- stamps is the step size for that instance, and if successive differences are taken for other attributes they should be divided by the step size to normalize the derivative. In other cases, each attribute may represent a different time, rather than each instance, so that the time series is from one attribute to the next rather than one instance to the next. Then, if differences are needed, they must be taken between one attribute’s value and the next attribute’s value for each instance. 7.4 SAMPLING In many applications involving a large volume of data it is necessary to come up with a random sample of much smaller size for processing. A random sample is one in which each instance in the original dataset has an equal chance of being included. Given a batch of N instances, a sample of any desired size is easily created: Just generate uniform random integers between 1 and N and retrieve the corresponding instances until the appropriate number has been collected. This is sampling with replacement, because the same instance might be selected more than once. (In fact, we used sampling with replacement for the bootstrap algorithm in Section 5.4—page 155.) For sampling without replacement, simply note, when selecting each instance, whether it has already been chosen and, if so, discard the second copy. If the sample size is much smaller than the full dataset, there is little difference between sampling with and without replacement. Reservoir Sampling Sampling is such a simple procedure that it merits little discussion or explanation. But there is a situation in which producing a random sample of a given size becomes 7.5 Cleansing 331 a little more challenging. What if the training instances arrive one by one but the total number of them—the value of N—is not known in advance? Or suppose we need to be able to run a learning algorithm on a sample of a given size from a con- tinuous stream of instances at any time, without repeatedly performing an entire sampling operation? Or perhaps the number of training instances is so vast that it is impractical to store them all before taking a sample? All these situations call for a way of generating a random sample of an input stream without storing up all the instances and waiting for the last one to arrive before beginning the sampling procedure. Is it possible to generate a random sample of a given size and still guarantee that each instance has an equal chance of being selected? The answer is yes. Furthermore, there is a simple algorithm to do so. The idea is to use a “reservoir” of size r, the size of the sample that is to be generated. To begin, place successive instances from the input stream in the reservoir until it is full. If the stream were to stop there, we would have the trivial case of a random sample of size r from an input stream of the same size. But most likely more instances will come in. The next one should be included in the sample with probabil- ity r/(r + 1)—in fact, if the input stream were to stop there, (N = r + 1), any instance should be in the sample with this probability. Consequently, with probability r/(r + 1) we replace a random instance in the reservoir with this new instance. And we carry on in the same vein, replacing a random reservoir element with the next instance with probability r/(r + 2) and so on. In general, the ith instance in the input stream is placed into the reservoir at a random location with probability r/i. It is easy to show by induction that once this instance has been processed the probability of any particular instance being in the reservoir is just the same, namely r/i. Thus, at any point in the procedure, the reservoir contains a random sample of size r from the input stream. You can stop at any time, secure in the knowledge that the reservoir contains the desired random sample. This method samples without replacement. Sampling with replacement is a little harder, although for large datasets and small reservoirs there is little difference between the two. But if you really want a sample of size r with replacement, you could set up r independent reservoirs, each with size 1. Run the algorithm concur- rently for all of these, and at any time their union is a random sample with replacement. 7.5 CLEANSING A problem that plagues practical data mining is poor quality of the data. Errors in large databases are extremely common. Attribute values, and class values too, are frequently unreliable and corrupted. Although one way of addressing this problem is to painstakingly check through the data, data mining techniques themselves can sometimes help to solve the problem. 332 CHAPTER 7 Data Transformations Improving Decision Trees It is a surprising fact that decision trees induced from training data can often be simplified, without loss of accuracy, by discarding misclassified instances from the training set, relearning, and then repeating until there are no misclassified instances. Experiments on standard datasets have shown that this hardly affects the classifica- tion accuracy of C4.5, a standard decision tree–induction scheme. In some cases it improves slightly; in others it deteriorates slightly. The difference is rarely statisti- cally significant—and even when it is, the advantage can go either way. What the technique does affect is decision tree size. The resulting trees are invariably much smaller than the original ones, even though they perform about the same. What is the reason for this? When a decision tree–induction method prunes away a subtree, it applies a statistical test that decides whether that subtree is “justified” by the data. The decision to prune accepts a small sacrifice in classification accuracy on the training set in the belief that this will improve test-set performance. Some training instances that were classified correctly by the unpruned tree will now be misclassified by the pruned one. In effect, the decision has been taken to ignore these training instances. But that decision has only been applied locally, in the pruned subtree. Its effect has not been allowed to percolate further up the tree, perhaps resulting in different choices being made of attributes to branch on. Removing the misclassified instances from the training set and relearning the decision tree is just taking the pruning deci- sions to their logical conclusion. If the pruning strategy is a good one, this should not harm performance. And it may improve it by allowing better attribute choices to be made. It would no doubt be even better to consult a human expert. Misclassified training instances could be presented for verification, and those that were found to be wrong could be deleted—or, better still, corrected. Notice that we are assuming that the instances are not misclassified in any sys- tematic way. If instances are systematically corrupted in both training and test sets— for example, one class value might be substituted for another—it is only to be expected that training on the erroneous training set would yield better performance on the (also erroneous) test set. Interestingly enough, it has been shown that when artificial noise is added to attributes (rather than added to classes), test-set performance is improved if the same noise is added in the same way to the training set. In other words, when attribute noise is the problem, it is not a good idea to train on a “clean” set if performance is to be assessed on a “dirty” one. A learning scheme can learn to compensate for attribute noise, in some measure, if given a chance. In essence, it can learn which attributes are unreliable and, if they are all unreliable, how best to use them together to yield a more reliable result. To remove noise from attributes for the training set denies the opportunity to learn how best to combat that noise. But with class noise (rather than attribute noise), it is best to train on noise-free instances if possible, if accurate classification is the goal. 7.5 Cleansing 333 Robust Regression The problems caused by noisy data have been known in linear regression for years. Statisticians often check data for outliers and remove them manually. In the case of linear regression, outliers can be identified visually, although it is never completely clear whether an outlier is an error or just a surprising, but correct, value. Outliers dramatically affect the usual least-squares regression because the squared distance measure accentuates the influence of points far away from the regression line. Statistical methods that address the problem of outliers are called robust. One way of making regression more robust is to use an absolute-value distance measure instead of the usual squared one. This weakens the effect of outliers. Another pos- sibility is to try to identify outliers automatically and remove them from consider- ation. For example, one could form a regression line and then remove from consideration those 10% of points that lie furthest from the line. A third possibility is to minimize the median (rather than the mean) of the squares of the divergences from the regression line. It turns out that this estimator is very robust and actually copes with outliers in the x-direction as well as outliers in the y-direction, which is the normal direction one thinks for outliers. A dataset that is often used to illustrate robust regression is a graph of interna- tional telephone calls made from Belgium during the years 1950 through 1973, shown in Figure 7.6. This data is taken from the Belgian Statistical Survey published by the Ministry of Economy. The plot seems to show an upward trend over the years, but there is an anomalous group of points from 1964 to 1969. It turns out that during this period, results were mistakenly recorded as the total number of minutes of the calls. The years 1963 and 1970 are also partially affected. This error causes a large fraction of outliers in the y-direction. FIGURE 7.6 Number of international phone calls from Belgium, 1950–1973. –5 0 5 10 15 20 25 1950 1955 1960 1965 1970 1975 Least squares Least median of squares Year Phone Calls (tens of millions) 334 CHAPTER 7 Data Transformations Not surprisingly, the usual least-squares regression line is seriously affected by this anomalous data. However, the least median of squares line remains remarkably unperturbed. This line has a simple and natural interpretation. Geometrically, it cor- responds to finding the narrowest strip covering half of the observations, where the thickness of the strip is measured in the vertical direction—this strip is marked gray in Figure 7.6. The least median of squares line lies at the exact center of this band. Note that this notion is often easier to explain and visualize than the normal least- squares definition of regression. Unfortunately, there is a serious disadvantage to median-based regression techniques: They incur high computational cost, which often makes them infeasible for practical problems. Detecting Anomalies A serious problem with any form of automatic detection of apparently incorrect data is that the baby may be thrown out with the bathwater. Short of consulting a human expert, there is no way of telling whether a particular instance really is an error or whether it just does not fit the type of model that is being applied. In statistical regression, visualizations help. It will usually be visually apparent, even to the non- expert, if the wrong kind of curve is being fitted—a straight line is being fitted to data that lies on a parabola, for example. The outliers in Figure 7.6 certainly stand out to the eye. But most classification problems cannot be so easily visualized: The notion of “model type” is more subtle than a regression line. And although it is known that good results are obtained on most standard datasets by discarding instances that do not fit a decision tree model, this is not necessarily of great comfort when dealing with a particular new dataset. The suspicion will remain that perhaps the new dataset is simply unsuited to decision tree modeling. One solution that has been tried is to use several different learning schemes (e.g., a decision tree, a nearest-neighbor learner, and a linear discriminant function) to filter the data. A conservative approach is to ask that all three schemes fail to classify an instance correctly before it is deemed erroneous and removed from the data. In some cases, filtering the data in this way and using the filtered data as input to a final learning scheme gives better performance than simply using the three learning schemes and letting them vote on the outcome. Training all three schemes on the filtered data and letting them vote can yield even better results. However, there is a danger to voting techniques: Some learning algorithms are better suited to certain types of data than others, and the most appropriate scheme may simply get out- voted! We will examine a more subtle method of combining the output from different classifiers, called stacking, in Section 8.7 (page 369). The lesson, as usual, is to get to know your data and look at it in many different ways. One possible danger with filtering approaches is that they might conceivably just be sacrificing instances of a particular class (or group of classes) to improve accuracy on the remaining classes. Although there are no general ways to guard against this, it has not been found to be a common problem in practice. 7.5 Cleansing 335 Finally, it is worth noting once again that automatic filtering is a poor substitute for getting the data right in the first place. And if this is too time-consuming and expensive to be practical, human inspection could be limited to those instances that are identified by the filter as suspect. One-Class Learning In most classification problems, training data is available for all classes that can occur at prediction time, and the learning algorithm uses the data for the different classes to determine decision boundaries that discriminate between them. However, some problems exhibit only a single class of instances at training time, while at prediction time new instances with unknown class labels can belong either to this target class or to a new class that was not available during training. Then, two different predictions are possible: target, meaning that an instance belongs to the class experienced during training, and unknown, where the instance does not appear to belong to that class. This type of learning problem is known as one-class classification. In many cases, one-class problems can be reformulated into two-class ones because there is data from other classes that can be used for training. However, there are genuine one-class applications where it is impossible or inappropriate to make use of negative data during training. For example, consider password hardening, a biometric system that strengthens a computer login process by not only requiring the correct password to be typed, but also requiring that it be typed with the correct rhythm. This is a one-class problem; a single user must be verified and during train- ing time only data from that user is available—we cannot ask anyone else to provide data without supplying them with the password! Even in applications where instances from several classes are available at training time, it may be best to focus solely on the target class under consideration—if, for example, new classes may occur at prediction time that differ from all those avail- able during training. Continuing with the typing-rhythm scenario, suppose we are to recognize typists in a situation where the text is not fixed—the current typist is to be verified as who he or she claims to be from his or her rhythmic patterns on a block of free text. This task is fundamentally different from distinguishing one user from a group of other users because we must be prepared to refuse attackers that the system has never seen before. Outlier Detection One-class classification is often called outlier (or novelty) detection because the learning algorithm is being used to differentiate between data that appears normal and abnormal with respect to the distribution of the training data. Earlier in this section we talked about making regression more robust against outliers by replacing the usual squared distance measure with the absolute-value one, and about trying to detect anomalies by using several different learning schemes. 336 CHAPTER 7 Data Transformations A generic statistical approach to one-class classification is to identify outliers as instances that lie beyond a distance d from a given percentage p of the training data. Alternatively, a probability density can be estimated for the target class by fitting a statistical distribution, such as a Gaussian, to the training data; any test instances with a low probability value can be marked as outliers. The challenge is to identify an appropriate distribution for the data at hand. If this cannot be done, one can adopt a nonparametric approach such as kernel density estimation (men- tioned at the end of Section 4.2, page 99). An advantage of the density estimation approach is that the threshold can be adjusted at prediction time to obtain a suit- able rate of outliers. Multiclass classifiers can be tailored to the one-class situation by fitting a boundary around the target data and deeming instances that fall outside it to be outliers. The boundary can be generated by adapting the inner workings of exist- ing multiclass classifiers such as support vector machines. These methods rely heavily on a parameter that determines how much of the target data is likely to be classified as outliers. If it is chosen too conservatively, data in the target class will erroneously be rejected. If it is chosen too liberally, the model will overfit and reject too much legitimate data. The rejection rate usually cannot be adjusted during testing, because an appropriate parameter value needs to be chosen at training time. Generating Artificial Data Rather than modify the internal workings of a multiclass classifier to form a one- class decision boundary directly, another possibility is to generate artificial data for the outlier class and apply any off-the-shelf classifier. Not only does this allow any classifier to be used, but if the classifier produces class probability estimates the rejection rate can be tuned by altering the threshold. The most straightforward approach is to generate uniformly distributed data and learn a classifier that can discriminate this from the target. However, different decision boundaries will be obtained for different amounts of artificial data: If too much is generated it will overwhelm the target class and the learning algorithm will always predict the artificial class. This problem can be avoided if the objec- tive of learning is viewed as accurate class probability estimation rather than minimizing the classification error. For example, bagged decision trees (described in Section 8.2 (page 352), which have been shown to yield good class probability estimators, can be used. Once a class probability estimation model has been obtained in this fashion, different thresholds on the probability estimates for the target class correspond to different decision boundaries surrounding the target class. This means that, as in the density estimation approach to one-class classification, the rate of outliers can be adjusted at prediction time to yield an outcome appropriate for the application at hand. There is one significant problem. As the number of attributes increases, it quickly becomes infeasible to generate enough artificial data to obtain adequate coverage of 7.5 Cleansing 337 the instance space, and the probability that a particular artificial instance occurs inside or close to the target class diminishes to a point that makes any kind of dis- crimination impossible. The solution is to generate artificial data that is as close as possible to the target class. In this case, because it is no longer uniformly distributed, the distri- bution of this artificial data—call this the “reference” distribution—must be taken into account when computing the membership scores for the resulting one-class model. In other words, the class probability estimates of the two-class classifier must be combined with the reference distribution to obtain membership scores for the target class. To elaborate a little further, let T denote the target class for which we have training data and seek a one-class model, and A the artificial class, for which we generate data using a known reference distribution. What we would like to obtain is Pr[X | T], the density function of the target class, for any instance X—of course, we know Pr[X | A], the density function of the reference distribution. Assume for the moment that we know the true class probability function Pr[T | X]. In practice, we need to estimate this function using a class probability estimator learned from the training data. A simple application of Bayes’ rule can be used to express Pr[X | T] in terms of Pr[T], Pr[T | X], and Pr[X | A]: Pr[ | ] ( Pr[ ])Pr[ | ] Pr[ ]( Pr[ | ]) Pr[ | ]XTTTX TTXXA= − − 1 1 To use this equation in practice, choose Pr[X | A], generate a user-specified amount of artificial data from it, label it A, and combine it with instances in the training set for the target class, labeled T. The proportion of target instances is an estimate of Pr[T], and a standard learning algorithm can be applied to this two-class dataset to obtain a class probability estimator Pr[T | X]. Given that the value for Pr[X | A] can be computed for any particular instance X, everything is at hand to compute an estimate of the target density function Pr[X | T] for any instance X. To perform classification we choose an appropriate threshold, adjusted to tune the rejection rate to any desired value. One question remains, namely, how to choose the reference density Pr[X | A]. We need to be able to generate artificial data from it and to compute its value for any instance X. Another requirement is that the data it generates should be close to the target class. In fact, ideally the reference density is identical to the target density, in which case Pr[T | X] becomes a constant function that any learning algorithm should be able to induce—the resulting two-class learning problem becomes trivial. This is unrealistic because it would require us to know the density of the target class. However, this observation gives a clue as to how to proceed: Apply any density estimation technique to the target data and use the resulting function to model the artificial class. The better the match between Pr[X | A] and Pr[X | T], the easier the resulting two-class class probability estimation task becomes. In practice, given the availability of powerful methods for class probability estimation and the relative lack of such techniques for density estimation, it makes sense to apply a simple density estimation technique to the target data first to obtain Pr[X | A] and then employ a state-of-the-art class probability estimation method to the two-class problem that is obtained by combining the artificial data with the data from the target class. 338 CHAPTER 7 Data Transformations 7.6 TRANSFORMING MULTIPLE CLASSES TO BINARY ONES Recall from Chapter 6 that some learning algorithms—for example, standard sup port vector machines—only work with two-class problems. In most cases, soph isticated multiclass variants have been developed, but they may be very slow or difficult to implement. As an alternative, it is common practice to transform multi class problems into multiple two-class ones: The dataset is decomposed into several two-class problems, the algorithm is run on each one, and the outputs of the result- ing classifiers are combined. Several popular techniques can implement this idea. We begin with a very simple one that was touched on when we were discussing how to use linear regression for classification; we then move on to pairwise clas- sification and more advanced techniques—error-correcting output codes and ensem- bles of nested dichotomies—that can often be profitably applied even when the underlying learning algorithm is able to deal with multiclass problems directly. Simple Methods At the beginning of the Linear Classification section in Chapter 4 (page 125) we learned how to transform a multiclass dataset for multiresponse linear regression to perform a two-class regression for each class. The idea essentially produces several two-class datasets by discriminating each class against the union of all the other classes. This technique is commonly called one-vs.-rest (or somewhat mislead- ingly, one-vs.-all). For each class, a dataset is generated containing a copy of each instance in the original data, but with a modified class value. If the instance has the class associated with the corresponding dataset, it is tagged yes; otherwise, no. Then classifiers are built for each of these binary datasets—classifiers that output a confidence figure with their predictions; for example, the estimated probability that the class is yes. During classification, a test instance is fed into each binary classifier, and the final class is the one associated with the classifier that predicts yes most confidently. Of course, this method is sensitive to the accuracy of the confidence figures produced by the classifiers: If some classifiers have an exaggerated opinion of their own predictions, the overall result will suffer. That is why it can be impor- tant to carefully tune parameter settings in the underlying learning algorithm. For example, in standard support vector machines for classification, it is gener- ally necessary to tune the parameter C, which provides an upper bound to the influence of each support vector and controls the closeness of fit to the training data, and the value of the kernel parameter—for example, the degree of the exponent in a polynomial kernel. This can be done based on internal cross- validation. It has been found empirically that the one-vs.-rest method can be very competitive, at least in the case of kernel-based classifiers, when appropriate parameter tuning is done. Note that it may also be useful to apply techniques for calibrating confidence scores, discussed in the next section, to the individual two-class models. 7.6 Transforming Multiple Classes To Binary Ones 339 Another simple and general method for multiclass problems is pairwise classi- fication. Here, a classifier is built for every pair of classes, using only the instances from these two classes. The output on an unknown test example is based on which class receives the most votes. This scheme generally yields accurate results in terms of classification error. It can also be used to produce probability estimates by apply- ing a method called pairwise coupling, which calibrates the individual probability estimates from the different classifiers. If there are k classes, pairwise classification builds a total of k(k − 1)/2 classifiers. Although this sounds unnecessarily computation intensive, it is not. In fact, if the classes are evenly populated, a pairwise classifier is at least as quick to train as any other multiclass method. The reason is that each of the pairwise learning problems only involves instances pertaining to the two classes under consideration. If n instances are divided evenly among k classes, this amounts to 2n/k instances per problem. Suppose the learning algorithm for a two-class problem with n instances takes time proportional to n seconds to execute. Then the runtime for pairwise clas- sification is proportional to k(k − 1)/2 × 2n/k seconds, which is (k − 1)n. In other words, the method scales linearly with the number of classes. If the learning algo- rithm takes more time—say proportional to n2—the advantage of the pairwise approach becomes even more pronounced. Error-Correcting Output Codes The simple methods discussed above are often very effective. Pairwise classification in particular can be a very useful technique. It has been found that it can in some cases improve accuracy even when the underlying learning algorithm, such as a decision tree learner, can deal with multiclass problems directly. This may be due to the fact that pairwise classification actually generates an ensemble of many clas- sifiers. Ensemble learning is a well-known strategy for obtaining accurate classifiers, and we will discuss several ensemble learning methods in Chapter 8. It turns out that there are methods other than pairwise classification that can be used to generate an ensemble classifier by decomposing a multiclass problem into several two-class subtasks. The one we discuss next is based on error-correcting output codes. Two-class decompositions of multiclass problems can be viewed in terms of the so-called “output codes” they correspond to. Let us revisit the simple one-vs.-rest method to see what such codes look like. Consider a multiclass problem with four classes a, b, c, and d. The transformation can be visualized as illustrated in Table 7.2(a), where yes and no are mapped to 1 and 0, respectively. Each of the original class values is converted into a 4-bit code word, 1 bit per class, and the four clas- sifiers predict the bits independently. Interpreting the classification process in terms of these code words, errors occur when the wrong binary bit receives the highest confidence. However, we do not have to use the particular code words shown. Indeed, there is no reason why each class must be represented by 4 bits. Look instead at the code of Table 7.2(b), where classes are represented by 7 bits. When applied to a dataset, 340 CHAPTER 7 Data Transformations Table 7.2 Transforming a Multiclass Problem into a Two-Class One (a) (b) Class Class Vector Class Class Vector a 1 0 0 0 a 1 1 1 1 1 1 1 b 0 1 0 0 b 0 0 0 0 1 1 1 c 0 0 1 0 c 0 0 1 1 0 0 1 d 0 0 0 1 d 0 1 0 1 0 1 0 (a) standard method and (b) error-correcting code. seven classifiers must be built instead of four. To see what that might buy, consider the classification of a particular instance. Suppose it belongs to class a, and that the predictions of the individual classifiers are 1 0 1 1 1 1 1, respectively. Obviously, comparing this code word with those in Table 7.2(b), the second classifier has made a mistake: It predicted 0 instead of 1, no instead of yes. Comparing the predicted bits with the code word associated with each class, the instance is clearly closer to a than to any other class. This can be quantified by the number of bits that must be changed to convert the predicted code word into those of Table 7.2(b): The Hamming distance, or the discrepancy between the bit strings, is 1, 3, 3, and 5 for the classes a, b, c, and d, respectively. We can safely conclude that the second classifier made a mistake and correctly identify a as the instance’s true class. The same kind of error correction is not possible with the code words shown in Table 7.2(a) because any predicted string of 4 bits, other than these four 4-bit words, has the same distance to at least two of them. Thus, the output codes are not “error correcting.” What determines whether a code is error correcting or not? Consider the Hamming distance between the code words representing different classes. The number of errors that can be possibly corrected depends on the minimum distance between any pair of code words, say d. The code can guarantee to correct up to (d − 1)/2 1-bit errors because if this number of bits of the correct code word are flipped, it will still be the closest and will therefore be identified correctly. In Table 7.2(a) the Hamming distance for each pair of code words is 2. Thus, the minimum distance d is also 2, and we can correct no more than 0 errors! However, in the code of Table 7.2(b) the minimum distance is 4 (in fact, the distance is 4 for all pairs). That means it is guaranteed to correct 1-bit errors. We have identified one property of a good error-correcting code: The code words must be well separated in terms of their Hamming distance. Because they comprise the rows of the code table, this property is called row separation. There is a second requirement that a good error-correcting code should fulfill: column separation. The Hamming distance between every pair of columns must be large, as must the dis- tance between each column and the complement of every other column. The seven 7.6 Transforming Multiple Classes To Binary Ones 341 columns in Table 7.2(b) are separated from one another (and their complements) by at least 1 bit. Column separation is necessary because if two columns are identical (or if one is the complement of another), the corresponding classifiers will make the same errors. Error correction is weakened if the errors are correlated—in other words, if many bit positions are simultaneously incorrect. The greater the distance between columns, the more errors are likely to be corrected. With fewer than four classes it is impossible to construct an effective error- correcting code because good row separation and good column separation cannot be achieved simultaneously. For example, with three classes there are only eight possible columns (23), four of which are complements of the other four. Moreover, columns with all 0s or all 1s provide no discrimination. This leaves just three pos- sible columns, and the resulting code is not error correcting at all. (In fact, it is the standard “one-vs.-rest” encoding.) If there are few classes, an exhaustive error-correcting code, such as the one in Table 7.2(b), can be built. In an exhaustive code for the k classes, the columns comprise every possible k-bit string, except for complements and the trivial all-0 or all-1 strings. Each of the code words contains 2k−1 − 1 bits. The code is constructed as follows: The code word for the first class consists of all 1s; that for the second class has 2k−2 0s followed by 2k−2 − 1 1s; the third has 2k−3 0s followed by 2k−3 1s followed by 2k−3 0s followed by 2k−3 − 1 1s; and so on. The ith code word consists of alternating runs of 2k−i 0s and 1s, the last run being one short. With more classes, exhaustive codes are infeasible because the number of columns increases exponentially and too many classifiers have to be built. In that case, more sophisticated methods are employed, which can build a code with good error-correcting properties from a smaller number of columns. Error-correcting output codes do not work for local learning algorithms such as instance-based learners, which predict the class of an instance by looking at nearby training instances. In the case of a nearest-neighbor classifier, all output bits would be predicted using the same training instance. The problem can be circumvented by using different attribute subsets to predict each output bit, decorrelating the predictions. Ensembles of Nested Dichotomies Error-correcting output codes often produce accurate classifiers for multiclass prob- lems. However, the basic algorithm produces classifications, whereas often we would like class probability estimates as well—for example, to perform cost-sensitive classification using the minimum expected cost approach discussed in Section 5.7 (page 167). Fortunately, there is a method for decomposing multiclass problems into two-class ones that provides a natural way of computing class probability estimates, so long as the underlying two-class models are able to produce probabilities for the corresponding two-class subtasks. 342 CHAPTER 7 Data Transformations Table 7.3 Nested Dichotomy in the Form of a Code Matrix Class Class Vector a 0 0 X b 1 1 X c 0 X 0 d 1 X 0 The idea is to recursively split the full set of classes from the original multiclass problem into smaller and smaller subsets, while splitting the full dataset of instances into subsets corresponding to these subsets of classes. This yields a binary tree of classes. Consider the hypothetical four-class problem discussed earlier. At the root node is the full set of classes {a, b, c, d}, which is split into disjoint subsets, say {a, c} and {b, d}, along with the instances pertaining to these two subsets of classes. The two subsets form the two successor nodes in the binary tree. These subsets are then split further into one-element sets, yielding successors {a} and {c} for the node {a, c} and successors {b} and {d} for the node {b, d}. Once we reach one-element subsets, the splitting process stops. The resulting binary tree of classes is called a nested dichotomy because each internal node and its two successors define a dichotomy—for example, discriminat- ing between classes {a, c} and {b, d} at the root node—and the dichotomies are nested in a hierarchy. We can view a nested dichotomy as a particular type of sparse output code. Table 7.3 shows the output code matrix for the example just discussed. There is one dichotomy for each internal node of the tree structure. Thus, given that the example involves three internal nodes, there are three columns in the code matrix. In contrast to the class vectors considered before, the matrix contains elements marked X that indicate that instances of the corresponding classes are simply omitted from the associated two-class learning problems. What is the advantage of this kind of output code? It turns out that, because the decomposition is hierarchical and yields disjoint subsets, there is a simple method for computing class probability estimates for each element in the original set of multiple classes, assuming two-class estimates for each dichotomy in the hierarchy. The reason is the chain rule from probability theory, which we already encountered when discussing Bayesian networks in Section 6.7 (page 265). Suppose we want to compute the probability for class a given a particular instance x—that is, the conditional probability Pr[a | x]. This class corresponds to one of the four leaf nodes in the hierarchy of classes in the previous example. First, we learn two-class models that yield class probability estimates for the three two- class datasets at the internal nodes of the hierarchy. Then, from the two-class model at the root node, an estimate of the conditional probability Pr[{a, b} | x]—namely, that x belongs to either a or b—can be obtained. Moreover, we can obtain an estimate of Pr[{a} | x, {a, b}]—the probability that x belongs to a given that we already know 7.7 Calibrating Class Probabilities 343 that it belongs to either a or b—from the model that discriminates between the one- element sets {a} and {b}. Now, based on the chain rule, Pr[{a} | x] = Pr[{a} | {a, b}, x ] × Pr[{a, b} | x]. Thus, to compute the probability for any individual class of the original multiclass problem—any leaf node in the tree of classes—we simply multiply together the probability estimates collected from the internal nodes encoun- tered when proceeding from the root node to this leaf node: the probability estimates for all subsets of classes that contain the target class. Assuming that the individual two-class models at the internal nodes produce accurate probability estimates, there is reason to believe that the multiclass probabil- ity estimates obtained using the chain rule will generally be accurate. However, it is clear that estimation errors will accumulate, causing problems for very deep hier- archies. A more basic issue is that in the previous example we arbitrarily decided on a particular hierarchical decomposition of the classes. Perhaps there is some background knowledge regarding the domain concerned, in which case one particu- lar hierarchy may be preferable because certain classes are known to be related, but this is generally not the case. What can be done? If there is no reason a priori to prefer any particular decom- position, perhaps all of them should be considered, yielding an ensemble of nested dichotomies. Unfortunately, for any nontrivial number of classes there are too many potential dichotomies, making an exhaustive approach infeasible. But we could consider a subset, taking a random sample of possible tree structures, building two- class models for each internal node of each tree structure (with caching of models, given that the same two-class problem may occur in multiple trees), and then averag- ing the probability estimates for each individual class to obtain the final estimates. Empirical experiments show that this approach yields accurate multiclass clas- sifiers and is able to improve predictive performance even in the case of classifiers, such as decision trees, that can deal with multiclass problems directly. In contrast to standard error-correcting output codes, the technique often works well even when the base learner is unable to model complex decision boundaries. The reason is that, generally speaking, learning is easier with fewer classes so results become more successful the closer we get to the leaf nodes in the tree. This also explains why the pairwise classification technique described earlier works particularly well for simple models such as ones corresponding to hyperplanes: It creates the simplest possible dichotomies! Nested dichotomies appear to strike a useful balance between the simplicity of the learning problems that occur in pairwise classification—after all, the lowest-level dichotomies involve pairs of individual classes—and the power of the redundancy embodied in standard error-correcting output codes. 7.7 CALIBRATING CLASS PROBABILITIES Class probability estimation is obviously more difficult than classification. Given a way of generating class probabilities, classification error is minimized as long as the correct class is predicted with maximum probability. However, a method for accurate 344 CHAPTER 7 Data Transformations classification does not imply a method of generating accurate probability estimates: The estimates that yield the correct classification may be quite poor when assessed according to the quadratic (page 160) or informational (page 161) loss discussed in Section 5.6. Yet—as we have stressed several times—it is often more important to obtain accurate conditional class probabilities for a given instance than to simply place the instance into one of the classes. Cost-sensitive prediction based on the minimum expected cost approach is one example where accurate class probability estimates are very useful. Consider the case of probability estimation for a dataset with two classes. If the predicted probabilities are on the correct side of the 0.5 threshold commonly used for classification, no classification errors will be made. However, this does not mean that the probability estimates themselves are accurate. They may be systematically too optimistic—too close to either 0 or 1—or too pessimistic—not close enough to the extremes. This type of bias will increase the measured quadratic or informational loss, and will cause problems when attempting to minimize the expected cost of classifications based on a given cost matrix. Figure 7.7 demonstrates the effect of overoptimistic probability estimation for a two-class problem. The x-axis shows the predicted probability of the multinomial Naïve Bayes model from Section 4.2 (page 97) for one of two classes in a text clas- sification problem with about 1000 attributes representing word frequencies. The y-axis shows the observed relative frequency of the target class. The predicted prob- abilities and relative frequencies were collected by running a tenfold cross-validation. To estimate relative frequencies, the predicted probabilities were first discretized into 20 ranges using equal-frequency discretization. Observations corresponding to one interval were then pooled—predicted probabilities on the one hand and corre- sponding 0/1 values on the other—and the pooled values are shown as the 20 points in the plot. FIGURE 7.7 Overoptimistic probability estimation for a two-class problem. 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Observed Relative Frequencies Predicted Probabilities 7.7 Calibrating Class Probabilities 345 This kind of plot, known as a reliability diagram, shows how reliable the esti- mated probabilities are. For a well-calibrated class probability estimator, the observed curve will coincide with the diagonal. This is clearly not the case here. The Naïve Bayes model is too optimistic, generating probabilities that are too close to 0 and 1. This is not the only problem: The curve is quite far from the line that corresponds to the 0.5 threshold that is used for classification. This means that classification performance will be affected by the poor probability estimates that the model generates. The fact that we seek a curve that lies close to the diagonal makes the remedy clear: Systematic misestimation should be corrected by using post hoc calibration of the probability estimates to map the empirically observed curve into a diagonal. A coarse way of doing this is to use the data from the reliability diagram directly for calibration, and map the predicted probabilities to the observed relative frequen- cies in the corresponding discretization intervals. Data for this can be obtained using internal cross-validation or a holdout set so that the actual test data remains untouched. Discretization-based calibration is very fast. However, determining appropriate discretization intervals is not easy. With too few, the mapping is too coarse; with too many, each interval contains insufficient data for a reliable estimate of relative frequencies. However, other ways of calibrating can be devised. The key is to realize that calibrating probability estimates for two-class problems is a function estimation problem with one input—the estimated class probability—and one output—the calibrated probability. In principle, complex functions could be used to estimate the mapping—perhaps arbitrary polynomials. However, it makes sense to assume that the observed relationship is at least monotonically increasing, in which case increasing functions should be used. Assuming that the calibration function is piecewise constant and monotonically increasing, there is an efficient algorithm that minimizes the squared error between the observed class “probabilities” (which are either 0 or 1 when no binning is applied) and the resulting calibrated class probabilities. Estimating a piecewise constant monotonically increasing function is an instance of isotonic regression, for which there is a fast algorithm based on the pair-adjacent violators (PAV) approach. The data consists of estimated probabilities and 0/1 values; assume it has been sorted according to the estimated probabilities. The basic PAV algorithm iteratively merges pairs of neighboring data points that violate the monotonicity constraint by comput- ing their weighted mean—initially this will be the mean of 0/1 values—and using it to replace the original data points. This is repeated until all conflicts have been resolved. It can be shown that the order in which data points are merged does not affect the outcome of the process. The result is a function that increases monotoni- cally in a stepwise fashion. This naïve algorithm is quadratic in the number of data points, but there is a clever variant that operates in linear time. Another popular calibration method, which also presupposes a monotonic rela- tionship, is to assume a linear relation between the log-odds of the estimated class probabilities and the target class probabilities. The logistic function is appropriate 346 CHAPTER 7 Data Transformations here, and logistic regression can be used to estimate the calibration function, with the caveat that it is important to use log-odds of the estimated class probabilities rather than the raw values as the input for logistic regression. Given that logistic regression, with only two parameters, uses a simpler model than the PAV approach, it can be more appropriate when little data is available for calibration. However, with a large volume of data, PAV-based calibration is generally preferable. Logistic regression has the advantage that it can be easily applied to calibrate probabilities for multiclass problems because multiclass versions of logistic regression exist. In the case of isotonic regression it is common to use the one-vs.- rest method for problems with more than two classes, but pairwise coupling or ensembles of nested dichotomies—discussed in Section 7.6—offer an alternative. Note that situations exist in which the relationship between the estimated and true probabilities is not monotonic. However, rather than switching to a more complex calibration method—or using discretization-based calibration, which does not assume monotonicity—this should perhaps be taken as an indication that the underlying class probability estimation method is not powerful enough for the problem at hand. 7.8 FURTHER READING Attribute selection, under the term feature selection, has been investigated in the field of pattern recognition for decades. Backward elimination, for example, was introduced in the early 1960s (Marill and Green, 1963). Kittler (1978) surveys the feature-selection algorithms that have been developed for pattern recognition. Best- first search and genetic algorithms are standard artificial intelligence techniques (Winston, 1992; Goldberg, 1989). The experiments that show the performance of decision tree learners deteriorat- ing when new attributes are added are reported by John (1997), who gives a nice explanation of attribute selection. The idea of finding the smallest attribute set that carves up the instances uniquely is from Almuallin and Dietterich (1991, 1992) and was further developed by Liu and Setiono (1996). Kibler and Aha (1987) and Cardie (1993) both investigated the use of decision tree algorithms to identify features for nearest-neighbor learning; Holmes and Nevill-Manning (1995) used OneR to order features for selection. Kira and Rendell (1992) used instance-based methods to select features, leading to a scheme called Relief for Recursive Elimination of Features. Gilad-Bachrach et al. (2004) show how this scheme can be modified to work better with redundant attributes. The correlation-based feature-selection method was developed by Hall (2000). The use of wrapper methods for feature selection is from John et al. (1994) and Kohavi and John (1997), and genetic algorithms have been applied within a wrapper framework by Vafaie and DeJong (1992) and Cherkauer and Shavlik (1996). The selective Naïve Bayes learning scheme is from Langley and Sage (1994). Guyon et al. (2002) present and evaluate the recursive feature-elimination 7.8 Further Reading 347 scheme in conjunction with support vector machines. The method of raced search was developed by Moore and Lee (1994). Gütlein et al. (2009) investigate how to speed up scheme-specific selection for datasets with many attributes using simple ranking-based methods. Dougherty et al. (1995) give a brief account of supervised and unsupervised discretization, along with experimental results comparing the entropy-based method with equal-width binning and the OneR method. Frank and Witten (1999) describe the effect of using the ordering information in discretized attributes. Proportional k-interval discretization for Naïve Bayes was proposed by Yang and Webb (2001). The entropy-based method for discretization, including the use of the MDL stopping criterion, was developed by Fayyad and Irani (1993). The bottom-up statistical method using the χ2 test is from Kerber (1992), and its extension to an automatically determined significance level is described by Liu and Setiono (1997). Fulton et al. (1995) investigate the use of dynamic programming for discretization and derive the quadratic time bound for a general impurity function (e.g., entropy) and the linear one for error-based discretization. The example used for showing the weakness of error-based discretization is adapted from Kohavi and Sahami (1996), who were the first to clearly identify this phenomenon. Principal components analysis is a standard technique that can be found in most statistics textbooks. Fradkin and Madigan (2003) analyze the performance of random projections. The algorithm for partial least-squares regression is from Hastie et al. (2009). The TF × IDF metric is described by Witten et al. (1999b). The experiments on using C4.5 to filter its own training data were reported by John (1995). The more conservative approach of a consensus filter involving several different learning algorithms has been investigated by Brodley and Friedl (1996). Rousseeuw and Leroy (1987) describe the detection of outliers in statistical regres- sion, including the least median of squares method; they also present the telephone data of Figure 7.6. It was Quinlan (1986) who noticed that removing noise from the training instance’s attributes can decrease a classifier’s performance on similarly noisy test instances, particularly at higher noise levels. Barnett and Lewis (1994) address the general topic of outliers in statistical data, while Pearson (2005) describes the statistical approach of fitting a distribution to the target data. Schölkopf et al. (2000) describe the use of support vector machines for novelty detection, while Abe et al. (2006), among others, use artificial data as a second class. Combining density estimation and class probability estimation using artificial data is suggested as a generic approach to unsupervised learning by Hastie et al. (2009), and Hempstalk et al. (2008) describe it in the context of one-class classification. Hempstalk and Frank (2008) discuss the fair comparison of one-class and multiclass classification when several classes are available at training time and we want to discriminate against an entirely new class at prediction time. Vitter (1985) explored the idea of reservoir sampling; he called the method we described algorithm R. Its computational complexity is O(N), where N is the number of instances in the stream, because a random number must be generated for every instance in order to determine whether, and where, to place it in the reservoir. Vitter 348 CHAPTER 7 Data Transformations describes several other algorithms that improve on R by reducing the number of random numbers that must be generated in order to produce the sample. Rifkin and Klautau (2004) show that the one-vs.-rest method for multiclass classification can work well if appropriate parameter tuning is applied. Friedman (1996) describes the technique of pairwise classification, Fürnkranz (2002) further analyzes it, and Hastie and Tibshirani (1998) extend it to estimate probabilities using pairwise coupling. Fürnkranz (2003) evaluates pairwise classification as a technique for ensemble learning. The idea of using error-correcting output codes for classification gained wide acceptance after a paper by Dietterich and Bakiri (1995); Ricci and Aha (1998) showed how to apply such codes to nearest- neighbor classifiers. Frank and Kramer (2004) introduce ensembles of nested dichotomies for multiclass problems. Dong et al. (2005) considered using bal- anced nested dichotomies rather than unrestricted random hierarchies to reduce training time. The importance of methods for calibrating class probability estimates is now well-established. Zadrozny and Elkan (2002) applied the PAV approach and logistic regression to calibration, and also investigated how to deal with multiclass problems. Niculescu-Mizil and Caruana (2005) compared a variant of logistic regression and the PAV-based method in conjunction with a large set of underlying class probability estimators, and found that the latter is preferable for sufficiently large calibration sets. They also found that multilayer perceptrons and bagged decision trees produce well-calibrated probabilities and do not require an extra calibration step. Stout (2008) describes a linear-time algorithm for isotonic regression based on minimizing the squared error. 7.9 WEKA IMPLEMENTATIONS Attribute selection (see Section 11.8 and Tables 11.9 and 11.10): • CfsSubsetEval (correlation-based attribute subset evaluator) • ConsistencySubsetEval (measures class consistency for a given set of attributes) • ClassifierSubsetEval (uses a classifier for evaluating subsets of attributes) • SVMAttributeEval (ranks attributes according to the magnitude of the coefficients learned by a support vector machine) • ReliefF (instance-based approach for ranking attributes) • WrapperSubsetEval (uses a classifier plus cross-validation) • GreedyStepwise (forward selection and backward elimination search) • LinearForwardSelection (forward selection with a sliding window of attribute choices at each step of the search) • BestFirst (search method that uses greedy hill-climbing with backtracking) • RaceSearch (uses the race search methodology) • Ranker (ranks individual attributes according to their evaluation) 7.9 Weka Implementations 349 Learning decision tables—DecisionTable (see Section 11.4 and Table 11.5) Discretization (see Section 11.3): Discretize in Table 11.1 (provides a variety of options for unsupervised discretization) PKIDiscretize in Table 11.1 (proportional k-interval discretization) Discretize in Table 11.3 (provides a variety of options for supervised discretization) Other data transformation operations (see Section 11.3): • PrincipalComponents and RandomProjection in Table 11.1 (principal components analysis and random projections) • Operations in Table 11.1 include arithmetic operations; time-series operations; obfuscation; generating cluster membership values; adding noise; various conversions between numeric, binary, and nominal attributes; and various data-cleansing operations. • Operations in Table 11.2 include resampling and reservoir sampling. • Operations in Table 11.3 include partial least-squares transformation. • MultiClassClassifier (see Table 11.6; includes many ways of handling multiclass problems with two-class classifiers, including error-correcting output codes) • END (see Table 11.6; ensembles of nested dichotomies) This page intentionally left blank 351Data Mining: Practical Machine Learning Tools and Techniques Copyright © 2011 Elsevier Inc. All rights of reproduction in any form reserved. CHAPTER 8 Ensemble Learning Having studied how to massage the input and calibrate the output, we now turn to techniques for combining different models learned from the data. There are some surprises in store. For example, it is often advantageous to take the train- ing data and derive several different training sets from it, learn a model from each, and combine them to produce an ensemble of learned models. Indeed, techniques for doing this can be very powerful. It is, for example, possible to transform a relatively weak learning scheme into an extremely strong one (in a precise sense that we will explain). Loss of interpretability is a drawback when applying ensemble learning, but there are ways to derive intelligible structured descriptions based on what these methods learn. Finally, if several learning schemes are available, it may be advantageous not to choose the best-performing one for your dataset (using cross-validation) but to use them all and combine the results. Many of these results are quite counterintuitive, at least at first blush. How can it be a good idea to use many different models together? How can you possibly do better than choose the model that performs best? Surely, all this runs counter to Occam’s razor, which advocates simplicity? How can you possibly obtain first- class performance by combining indifferent models, as one of these techniques appears to do? But consider committees of humans, which often come up with wiser decisions than individual experts. Recall Epicurus’ view that, faced with alternative explanations, one should retain them all. Imagine a group of specialists each of whom excels in a limited domain even though none is competent across the board. In struggling to understand how these methods work, researchers have exposed all sorts of connections and links that have led to even greater improvements. 8.1 COMBINING MULTIPLE MODELS When wise people make critical decisions, they usually take into account the opin- ions of several experts rather than relying on their own judgment or that of a solitary trusted advisor. For example, before choosing an important new policy direction, a 352 CHAPTER 8 Ensemble Learning benign dictator consults widely: He or she would be ill advised to follow just one expert’s opinion blindly. In a democratic setting, discussion of different viewpoints may produce a consensus; if not, a vote may be called for. In either case, different expert opinions are being combined. In data mining, a model generated by machine learning can be regarded as an expert. Expert is probably too strong a word!—depending on the amount and quality of the training data, and whether the learning algorithm is appropriate to the problem at hand, the expert may in truth be regrettably ignorant—but we use the term nev- ertheless. An obvious approach to making decisions more reliable is to combine the output of several different models. Several machine learning techniques do this by learning an ensemble of models and using them in combination: Prominent among these are schemes called bagging, boosting, and stacking. They can all, more often than