Multivariate Analysis

Kanti V. Mardia OBE is a Senior Research Professor in the Department of Statistics at the University of Leeds, Leverhulme Emeritus Fellow, and Visiting Professor in the Department of Statistics, University of Oxford. John T. Kent and Charles C. Taylor are both Professors in the Department of Statistics, University of Leeds.

• Epigraph xviiPreface to the Second Edition xixPreface to the First Edition xxiAcknowledgments from First Edition xxvNotation, Abbreviations, and Key Ideas xxvii1 Introduction 11.1 Objects and Variables 11.2 Some Multivariate Problems and Techniques 11.3 The Data Matrix 71.4 Summary Statistics 81.5 Linear Combinations 121.6 Geometrical Ideas 141.7 Graphical Representation 151.8 Measures of Multivariate Skewness and Kurtosis 20Exercises and Complements 222 Basic Properties of Random Vectors 25Introduction 252.1 Cumulative Distribution Functions and Probability Density Functions 252.2 Population Moments 272.3 Characteristic Functions 312.4 Transformations 322.5 The Multivariate Normal Distribution 342.6 Random Samples 412.7 Limit Theorems 42Exercises and Complements 443 Nonnormal Distributions 493.1 Introduction 493.2 Some Multivariate Generalizations of Univariate Distributions 493.3 Families of Distributions 523.4 Insights into Skewness and Kurtosis 573.5 Copulas 60Exercises and Complements 654 Normal Distribution Theory 714.1 Introduction and Characterization 714.2 Linear Forms 734.3 Transformations of Normal Data Matrices 754.4 The Wishart Distribution 774.5 The Hotelling T2 Distribution 834.6 Mahalanobis Distance 854.7 Statistics Based on the Wishart Distribution 884.8 Other Distributions Related to the Multivariate Normal 92Exercises and Complements 935 Estimation 101Introduction 1015.1 Likelihood and Sufficiency 1015.2 Maximum-likelihood Estimation 1065.3 Robust Estimation of Location and Dispersion for Multivariate Distributions 1125.4 Bayesian Inference 117Exercises and Complements 1196 Hypothesis Testing 1256.1 Introduction 1256.2 The Techniques Introduced 1276.3 The Techniques Further Illustrated 1346.4 Simultaneous Confidence Intervals 1426.5 The Behrens–Fisher Problem 1446.6 Multivariate Hypothesis Testing: Some General Points 1456.7 Nonnormal Data 1466.8 Mardia’s Nonparametric Test for the Bivariate Two-sample Problem 149Exercises and Complements 1517 Multivariate Regression Analysis 1597.1 Introduction 1597.2 Maximum-likelihood Estimation 1607.3 The General Linear Hypothesis 1627.4 Design Matrices of Degenerate Rank 1657.5 Multiple Correlation 1677.6 Least-squares Estimation 1717.7 Discarding of Variables 174Exercises and Complements 1788 Graphical Models 1838.1 Introduction 1838.2 Graphs and Conditional Independence 1848.3 Gaussian Graphical Models 1888.4 Log-linear Graphical Models 1958.5 Directed and Mixed Graphs 202Exercises and Complements 2049 Principal Component Analysis 2079.1 Introduction 2079.2 Definition and Properties of Principal Components 2079.3 Sampling Properties of Principal Components 2219.4 Testing Hypotheses About Principal Components 2279.5 Correspondence Analysis 2309.6 Allometry – Measurement of Size and Shape 2379.7 Discarding of Variables 2409.8 Principal Component Regression 2419.9 Projection Pursuit and Independent Component Analysis 2449.10 PCA in High Dimensions 247Exercises and Complements 24910 Factor Analysis 25910.1 Introduction 25910.2 The Factor Model 26010.3 Principal Factor Analysis 26410.4 Maximum-likelihood Factor Analysis 26610.5 Goodness-of-fit Test 26910.6 Rotation of Factors 27010.7 Factor Scores 27510.8 Relationships Between Factor Analysis and Principal Component Analysis 27610.9 Analysis of Covariance Structures 277Exercises and Complements 27711 Canonical Correlation Analysis 28111.1 Introduction 28111.2 Mathematical Development 28211.3 Qualitative Data and Dummy Variables 28811.4 Qualitative and Quantitative Data 290Exercises and Complements 29312 Discriminant Analysis and Statistical Learning 29712.1 Introduction 29712.2 Bayes’ Discriminant Rule 29912.3 The Error Rate 30012.4 Discrimination Using the Normal Distribution 30412.5 Discarding of Variables 31212.6 Fisher’s Linear Discriminant Function 31412.7 Nonparametric Distance-based Methods 31912.8 Classification Trees 32312.9 Logistic Discrimination 33212.10 Neural Networks 336Exercises and Complements 34213 Multivariate Analysis of Variance 35513.1 Introduction 35513.2 Formulation of Multivariate One-way Classification 35513.3 The Likelihood Ratio Principle 35613.4 Testing Fixed Contrasts 35813.5 Canonical Variables and A Test of Dimensionality 35913.6 The Union Intersection Approach 36913.7 Two-way Classification 370Exercises and Complements 37514 Cluster Analysis and Unsupervised Learning 37914.1 Introduction 37914.2 Probabilistic Membership Models 38014.3 Parametric Mixture Models 38414.4 Partitioning Methods 38614.5 Hierarchical Methods 39114.6 Distances and Similarities 39714.7 Grouped Data 40414.8 Mode Seeking 40614.9 Measures of Agreement 408Exercises and Complements 41215 Multidimensional Scaling 41915.1 Introduction 41915.2 Classical Solution 42115.3 Duality Between Principal Coordinate Analysis and Principal Component Analysis 42815.4 Optimal Properties of the Classical Solution and Goodness of Fit 42915.5 Seriation 43615.6 Nonmetric Methods 43815.7 Goodness of Fit Measure: Procrustes Rotation 44015.8 Multisample Problem and Canonical Variates 443Exercises and Complements 44416 High-dimensional Data 44916.1 Introduction 44916.2 Shrinkage Methods in Regression 45116.3 Principal Component Regression 45516.4 Partial Least Squares Regression 45716.5 Functional Data 465Exercises and Complements 473A Matrix Algebra 475A.1 Introduction 475A.2 Matrix Operations 478A.3 Further Particular Matrices and Types of Matrices 483A.4 Vector Spaces, Rank, and Linear Equations 485A.5 Linear Transformations 488A.6 Eigenvalues and Eigenvectors 488A.7 Quadratic Forms and Definiteness 495A.8 Generalized Inverse 497A.9 Matrix Differentiation and Maximization Problems 499A.10 Geometrical Ideas 501B Univariate Statistics 505B.1 Introduction 505B.2 Normal Distribution 505B.3 Chi-squared Distribution 506B.4 F and Beta Variables 506B.5 t Distribution 507B.6 Poisson Distribution 507C R commands and Data 509C.1 Basic R Commands Related to Matrices 509C.2 R Libraries and Commands Used in Exercises and Figures 510C.3 Data Availability 511D Tables 513References and Author Index 523Index 543