Mixture Model-Based Classification
"This is a great overview of the field of model-based clustering and classification by one of its leading developers. McNicholas provides a resource that I am certain will be used by researchers in statistics and related disciplines for quite some time. The discussion of mixtures with heavy tails and asymmetric distributions will place this text as the authoritative, modern reference in the mixture modeling literature." (Douglas Steinley, University of Missouri)

Mixture Model-Based Classification is the first monograph devoted to mixture model-based approaches to clustering and classification. This is both a book for established researchers and newcomers to the field. A history of mixture models as a tool for classification is provided and Gaussian mixtures are considered extensively, including mixtures of factor analyzers and other approaches for high-dimensional data. Non-Gaussian mixtures are considered, from mixtures with components that parameterize skewness and/or concentration, right up to mixtures of multiple scaled distributions. Several other important topics are considered, including mixture approaches for clustering and classification of longitudinal data as well as discussion about how to define a cluster

Paul D. McNicholas is the Canada Research Chair in Computational Statistics at McMaster University, where he is a Professor in the Department of Mathematics and Statistics. His research focuses on the use of mixture model-based approaches for classification, with particular attention to clustering applications, and he has published extensively within the field. He is an associate editor for several journals and has served as a guest editor for a number of special issues on mixture models.

1121869360
Mixture Model-Based Classification
"This is a great overview of the field of model-based clustering and classification by one of its leading developers. McNicholas provides a resource that I am certain will be used by researchers in statistics and related disciplines for quite some time. The discussion of mixtures with heavy tails and asymmetric distributions will place this text as the authoritative, modern reference in the mixture modeling literature." (Douglas Steinley, University of Missouri)

Mixture Model-Based Classification is the first monograph devoted to mixture model-based approaches to clustering and classification. This is both a book for established researchers and newcomers to the field. A history of mixture models as a tool for classification is provided and Gaussian mixtures are considered extensively, including mixtures of factor analyzers and other approaches for high-dimensional data. Non-Gaussian mixtures are considered, from mixtures with components that parameterize skewness and/or concentration, right up to mixtures of multiple scaled distributions. Several other important topics are considered, including mixture approaches for clustering and classification of longitudinal data as well as discussion about how to define a cluster

Paul D. McNicholas is the Canada Research Chair in Computational Statistics at McMaster University, where he is a Professor in the Department of Mathematics and Statistics. His research focuses on the use of mixture model-based approaches for classification, with particular attention to clustering applications, and he has published extensively within the field. He is an associate editor for several journals and has served as a guest editor for a number of special issues on mixture models.

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Mixture Model-Based Classification

Mixture Model-Based Classification

by Paul D. McNicholas
Mixture Model-Based Classification
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Mixture Model-Based Classification

Mixture Model-Based Classification

by Paul D. McNicholas

Overview

"This is a great overview of the field of model-based clustering and classification by one of its leading developers. McNicholas provides a resource that I am certain will be used by researchers in statistics and related disciplines for quite some time. The discussion of mixtures with heavy tails and asymmetric distributions will place this text as the authoritative, modern reference in the mixture modeling literature." (Douglas Steinley, University of Missouri)

Mixture Model-Based Classification is the first monograph devoted to mixture model-based approaches to clustering and classification. This is both a book for established researchers and newcomers to the field. A history of mixture models as a tool for classification is provided and Gaussian mixtures are considered extensively, including mixtures of factor analyzers and other approaches for high-dimensional data. Non-Gaussian mixtures are considered, from mixtures with components that parameterize skewness and/or concentration, right up to mixtures of multiple scaled distributions. Several other important topics are considered, including mixture approaches for clustering and classification of longitudinal data as well as discussion about how to define a cluster

Paul D. McNicholas is the Canada Research Chair in Computational Statistics at McMaster University, where he is a Professor in the Department of Mathematics and Statistics. His research focuses on the use of mixture model-based approaches for classification, with particular attention to clustering applications, and he has published extensively within the field. He is an associate editor for several journals and has served as a guest editor for a number of special issues on mixture models.

Product Details

ISBN-13: 9780367736958
Publisher: CRC Press
Publication date: 12/18/2020
Pages: 236
Product dimensions: 6.12(w) x 9.19(h) x (d)

About the Author

Paul D. McNicholas is the Canada Research Chair in Computational Statistics at McMaster University, where he is a Professor in the Department of Mathematics and Statistics. His research focuses on the use of mixture model-based approaches for classification, with particular attention to clustering applications, and he has published extensively within the field. He is an associate editor for several journals and has served as a guest editor for a number of special issues on mixture models.

Table of Contents

List of Figures xiii

List of Tables xvii

Preface xxiii

1 Introduction 1

1.1 Classification 1

1.2 Finite Mixture Models 3

1.3 Model-Based Clustering, Classification, and Discriminant Analysis 4

1.4 Comparing Partitions 6

1.5 R Packages 8

1.6 Datasets 9

1.7 Outline of the Contents of This Monograph 9

2 Mixtures of Multivariate Gaussian Distributions 11

2.1 Historical Development 11

2.2 Parameter Estimation 14

2.2.1 Model-Based Clustering 14

2.2.2 Model-Based Classification 15

2.2.3 Model-Based Discriminant Analysis 17

2.2.4 Initialization via Deterministic Annealing 18

2.2.5 Stopping Rules 18

2.3 Gaussian Parsimonious Clustering Models 20

2.4 Model Selection 24

2.5 Merging Gaussian Components 25

2.6 Illustrations 27

2.6.1 x2 Data 27

2.6.2 Banknote Data 29

2.6.3 Female Voles Data 31

2.6.4 Italian Olive Oil Data 33

2.7 Comments 36

3 Mixtures of Factor Analyzers and Extensions 39

3.1 Factor Analysis 39

3.1.1 The Model 39

3.1.2 An EM Algorithm for the Factor Analysis Model 40

3.1.3 Woodbury Identity 42

3.1.4 Comments 43

3.2 Mixture of Factor Analyzers 43

3.3 Parsimonious Gaussian Mixture Models 44

3.3.1 A Family of Eight Models 44

3.3.2 Parameter Estimation 44

3.3.3 Comments 48

3.4 Expanded Parsimonious Gaussian Mixture Models 49

3.4.1 A Family of Twelve Models 49

3.4.2 Parameter Estimation 50

3.5 Mixture of Common Factor Analyzers 52

3.5.1 The Model 52

3.5.2 Parameter Estimation 52

3.5.3 Discussion 55

3.6 Illustrations 55

3.6.1 x2 Data 55

3.6.2 Italian Wine Data 56

3.6.3 Italian Olive Oil Data 58

3.6.4 Alon Colon Cancer Data 59

3.7 Comments 61

4 Dimension Reduction and High-Dimensional Data 63

4.1 Implicit and Explicit Approaches 63

4.2 PGMM Family in High-Dimensional Applications 64

4.3 VSCC 65

4.4 Clustvarsel and selvarclust 67

4.5 GMMDR 68

4.6 HD-GMM 69

4.7 Illustrations 71

4.7.1 Coffee Data 71

4.7.2 Leptograpsus Crabs 71

4.7.3 Banknote Data 74

4.7.4 Wisconsin Breast Cancer Data 74

4.7.5 Leukaemia Data 75

4.8 Comments 76

5 Mixtures of Distributions with Varying Tail Weight 79

5.1 Mixtures of Multivariate t-Distributions 79

5.2 Mixtures of Power Exponential Distributions 82

5.3 Illustrations 89

5.3.1 Overview 89

5.3.2 x2 Data 89

5.3.3 Body Data 89

5.3.4 Diabetes Data 90

5.3.5 Female Voles Data 92

5.3.6 Leptograpsus Crabs Data 93

5.4 Comments 94

6 Mixtures of Generalized Hyperbolic Distributions 99

6.1 Overview 99

6.2 Generalized Inverse Gaussian Distribution 99

6.2.1 A Parameterization 99

6.2.2 An Alternative Parameterization 100

6.3 Mixtures of Shifted Asymmetric Laplace Distributions 101

6.3.1 Shifted Asymmetric Laplace Distribution 101

6.3.2 Parameter Estimation 102

6.3.3 SAL Mixtures versus Gaussian Mixtures 104

6.4 Mixture of Generalized Hyperbolic Distributions 106

6.4.1 Generalized Hyperbolic Distribution 106

6.4.2 Parameter Estimation 108

6.5 Mixture of Generalized Hyperbolic Factor Analyzers 111

6.5.1 The Model 111

6.5.2 Parameter Estimation 111

6.5.3 Analogy with the Gaussian Solution 114

6.6 Illustrations 115

6.6.1 Old Faithful Data 115

6.6.2 Yeast Data 116

6.6.3 Italian Wine Data 118

6.6.4 Liver Data 118

6.7 A Note on Normal Variance-Mean Mixtures 119

6.8 Comments 120

7 Mixtures of Multiple Scaled Distributions 123

7.1 Overview 123

7.2 Mixture of Multiple Scaled t-Distributions 124

7.3 Mixture of Multiple Scaled SAL Distributions 126

7.4 Mixture of Multiple Scaled Generalized Hyperbolic Distributions 127

7.5 Mixture of Coalesced Generalized Hyperbolic Distributions 128

7.6 Cluster Convexity 129

7.7 Illustrations 132

7.7.1 Bankruptcy Data 132

7.7.2 Other Clustering Examples 133

7.7.3 Classification and Discriminant Analysis Examples 135

7.8 Comments 137

8 Methods for Longitudinal Data 139

8.1 Modified Cholesky Decomposition 139

8.2 Gaussian Mixture Modelling of Longitudinal Data 140

8.2.1 The Model 140

8.2.2 Model Fitting 141

8.2.2.1 VEA Model 141

8.2.2.2 EVI Model 144

8.2.3 Constraining Sub-Diagonals of Tg 145

8.2.3.1 VdEA Model 145

8.2.3.2 EdVI Model 147

8.2.4 Modelling the Component Means 147

8.3 Using t-Mixtures 148

8.4 Illustrations 150

8.4.1 Clustering 150

8.4.2 Classification 153

8.5 Comments 156

9 Miscellania 157

9.1 On the Definition of a Cluster 157

9.2 What Is the Best Way to Perform Clustering, Classification, and Discriminant Analysis? 159

9.3 Mixture Model Averaging 162

9.4 Robust Clustering 164

9.5 Clustering Categorical Data 166

9.6 Cluster-Weighted Models 168

9.7 Mixed-Type Data 169

9.7.1 A Mixture of Latent Variables Model 169

9.7.2 Illustration: Urinary System Disease Diagnosis 170

9.8 Alternatives to the EM Algorithm 172

9.8.1 Variational Bayes Approximations 172

9.8.2 Other Approaches 173

9.9 Challenges and Open Questions 174

Appendix 177

A.1 Linear Algebra Results 177

A.2 Matrix Calculus Results 178

A.3 Method of Lagrange Multipliers 179

References 181

Index 205

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