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A unified, coherent treatment of current classifier ensemble methods, from fundamentals of pattern recognition to ensemble feature selection, now in its second edition

The art and science of combining pattern classifiers has flourished into a prolific discipline since the first edition of Combining Pattern Classifiers was published in 2004. Dr. Kuncheva has plucked from the rich landscape of recent classifier ensemble literature the topics, methods, and algorithms that will guide the reader toward a deeper understanding of the fundamentals, design, and applications of classifier ensemble methods.

Thoroughly updated, with MATLAB code and practice data sets throughout, Combining Pattern Classifiers includes:

 Coverage of Bayes decision theory and experimental comparison of classifiers

 Essential ensemble methods such as Bagging, Random forest, AdaBoost, Random subspace, Rotation forest, Random oracle, and Error Correcting Output Code, among others

 Chapters on classifier selection, diversity, and ensemble feature selection

With firm grounding in the fundamentals of pattern recognition, and featuring more than 140 illustrations, Combining Pattern Classifiers, Second Edition is a valuable reference for postgraduate students, researchers, and practitioners in computing and engineering.



Table of Contents
Preface xv

Acknowledgements xxi

1 Fundamentals of Pattern Recognition 1

1.1 Basic Concepts: Class Feature Data Set 1

1.1.1 Classes and Class Labels 1

1.1.2 Features 2

1.1.3 Data Set 3

1.1.4 Generate Your Own Data 6

1.2 Classifier Discriminant Functions Classification Regions 9

1.3 Classification Error and Classification Accuracy 11

1.3.1 Where Does the Error Come From? Bias and Variance 11

1.3.2 Estimation of the Error 13

1.3.3 Confusion Matrices and Loss Matrices 14

1.3.4 Training and Testing Protocols 15

1.3.5 Overtraining and Peeking 17

1.4 Experimental Comparison of Classifiers 19

1.4.1 Two Trained Classifiers and a Fixed Testing Set 20

1.4.2 Two Classifier Models and a Single Data Set 22

1.4.3 Two Classifier Models and Multiple Data Sets 26

1.4.4 Multiple Classifier Models and Multiple Data Sets 27

1.5 Bayes Decision Theory 30

1.5.1 Probabilistic Framework 301.5.2 Discriminant Functions and Decision Boundaries 31

1.5.3 Bayes Error 33

1.6 Clustering and Feature Selection 35

1.6.1 Clustering 35

1.6.2 Feature Selection 37

1.7 Challenges of Real-Life Data 40

Appendix 41

1.A.1 Data Generation 41

1.A.2 Comparison of Classifiers 42

1.A.2.1 MATLAB Functions for Comparing Classifiers 42

1.A.2.2 Critical Values for Wilcoxon and Sign Test 45

1.A.3 Feature Selection 47

2 Base Classifiers 49

2.1 Linear and Quadratic Classifiers 49

2.1.1 Linear Discriminant Classifier 49

2.1.2 Nearest Mean Classifier 52

2.1.3 Quadratic Discriminant Classifier 52

2.1.4 Stability of LDC and QDC 53

2.2 Decision Tree Classifiers 55

2.2.1 Basics and Terminology 55

2.2.2 Training of Decision Tree Classifiers 57

2.2.3 Selection of the Feature for a Node 58

2.2.4 Stopping Criterion 60

2.2.5 Pruning of the Decision Tree 63

2.2.6 C4.5 and ID3 64

2.2.7 Instability of Decision Trees 64

2.2.8 Random Trees 65

2.3 The Na¿ýve Bayes Classifier 66

2.4 Neural Networks 68

2.4.1 Neurons 68

2.4.2 Rosenblatt's Perceptron 70

2.4.3 Multi-Layer Perceptron 71

2.5 Support Vector Machines 73

2.5.1 Why Would It Work? 73

2.5.2 Classification Margins 74

2.5.3 Optimal Linear Boundary 76

2.5.4 Parameters and Classification Boundaries of SVM 78

2.6 The k-Nearest Neighbor Classifier (k-nn) 80

2.7 Final Remarks 82

2.7.1 Simple or Complex Models? 82

2.7.2 The Triangle Diagram 83

2.7.3 Choosing a Base Classifier for Ensembles 85

Appendix 85

2.A.1 MATLAB Code for the Fish Data 85

2.A.2 MATLAB Code for Individual Classifiers 86

2.A.2.1 Decision Tree 86

2.A.2.2 Na¿ýve Bayes 89

2.A.2.3 Multi-Layer Perceptron 90

2.A.2.4 1-nn Classifier 92

3 An Overview of the Field 94

3.1 Philosophy 94

3.2 Two Examples 98

3.2.1 The Wisdom of the "Classifier Crowdö 98

3.2.2 The Power of Divide-and-Conquer 98

3.3 Structure of the Area 100

3.3.1 Terminology 100

3.3.2 A Taxonomy of Classifier Ensemble Methods 100

3.3.3 Classifier Fusion and Classifier Selection 104

3.4 Quo Vadis? 105

3.4.1 Reinventing the Wheel? 105

3.4.2 The Illusion of Progress? 106

3.4.3 A Bibliometric Snapshot 107

4 Combining Label Outputs 111

4.1 Types of Classifier Outputs 111

4.2 A Probabilistic Framework for Combining Label Outputs 112

4.3 Majority Vote 113

4.3.1 "Democracyö in Classifier Combination 113

4.3.2 Accuracy of the Majority Vote 114

4.3.3 Limits on the Majority Vote Accuracy: An Example 117

4.3.4 Patterns of Success and Failure 119

4.3.5 Optimality of the Majority Vote Combiner 124

4.4 Weighted Majority Vote 125

4.4.1 Two Examples 126

4.4.2 Optimality of the Weighted Majority Vote Combiner 127

4.5 Na¿ýve-Bayes Combiner 128

4.5.1 Optimality of the Na¿ýve Bayes Combiner 128

4.5.2 Implementation of the NB Combiner 130

4.6 Multinomial Methods 132

4.7 Comparison of Combination Methods for Label Outputs 135

Appendix 137

4.A.1 Matan's Proof for the Limits on the Majority Vote Accuracy 137

4.A.2 Selected MATLAB Code 139

5 Combining Continuous-Valued Outputs 143

5.1 Decision Profile 143

5.2 How Do We Get Probability Outputs? 144

5.2.1 Probabilities Based on Discriminant Scores 144

5.2.2 Probabilities Based on Counts: Laplace Estimator 147

5.3 Nontrainable (Fixed) Combination Rules 150

5.3.1 A Generic Formulation 150

5.3.2 Equivalence of Simple Combination Rules 152

5.3.3 Generalized Mean Combiner 153

5.3.4 A Theoretical Comparison of Simple Combiners 156

5.3.5 Where Do They Come From? 160

5.4 The Weighted Average (Linear Combiner) 166

5.4.1 Consensus Theory 166

5.4.2 Added Error for the Weighted Mean Combination 167

5.4.3 Linear Regression 168

5.5 A Classifier as a Combiner 172

5.5.1 The Supra Bayesian Approach 172

5.5.2 Decision Templates 173

5.5.3 A Linear Classifier 175

5.6 An Example of Nine Combiners for Continuous-Valued Outputs 175

5.7 To Train or Not to Train? 176

Appendix 178

5.A.1 Theoretical Classification Error for the Simple Combiners 178

5.A.1.1 Set-up and Assumptions 178

5.A.1.2 Individual Error 180

5.A.1.3 Minimum and Maximum 180

5.A.1.4 Average (Sum) 181

5.A.1.5 Median and Majority Vote 182

5.A.1.6 Oracle 183

5.A.2 Selected MATLAB Code 183

6 Ensemble Methods 186

6.1 Bagging 186

6.1.1 The Origins: Bagging Predictors 186

6.1.2 Why Does Bagging Work? 187

6.1.3 Out-of-bag Estimates 189

6.1.4 Variants of Bagging 190

6.2 Random Forests 190

6.3 AdaBoost 192

6.3.1 The AdaBoost Algorithm 192

6.3.2 The arc-x4 Algorithm 194

6.3.3 W