Multivariate Analysis

Informationen zum Autor 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. Klappentext Comprehensive Reference Work on Multivariate Analysis and its Applications The first edition of this book, by Mardia, Kent and Bibby, has been used globally for over 40 years. This second edition brings many topics up to date, with a special emphasis on recent developments. A wide range of material in multivariate analysis is covered, including the classical themes of multivariate normal theory, multivariate regression, inference, multidimensional scaling, factor analysis, cluster analysis and principal component analysis. The book also now covers modern developments such as graphical models, robust estimation, statistical learning, and high-dimensional methods. The book expertly blends theory and application, providing numerous worked examples and exercises at the end of each chapter. The reader is assumed to have a basic knowledge of mathematical statistics at an undergraduate level together with an elementary understanding of linear algebra. There are appendices which provide a background in matrix algebra, a summary of univariate statistics, a collection of statistical tables and a discussion of computational aspects. The work includes coverage of: Basic properties of random vectors, copulas, normal distribution theory, and estimation Hypothesis testing, multivariate regression, and analysis of variance Principal component analysis, factor analysis, and canonical correlation analysis Discriminant analysis, cluster analysis, and multidimensional scaling New advances and techniques, including supervised and unsupervised statistical learning, graphical models and regularization methods for high-dimensional data Although primarily designed as a textbook for final year undergraduates and postgraduate students in mathematics and statistics, the book will also be of interest to research workers and applied scientists. Zusammenfassung Multivariate AnalysisComprehensive Reference Work on Multivariate Analysis and its ApplicationsThe first edition of this book, by Mardia, Kent and Bibby, has been used globally for over 40 years. This second edition brings many topics up to date, with a special emphasis on recent developments.A wide range of material in multivariate analysis is covered, including the classical themes of multivariate normal theory, multivariate regression, inference, multidimensional scaling, factor analysis, cluster analysis and principal component analysis. The book also now covers modern developments such as graphical models, robust estimation, statistical learning, and high-dimensional methods. The book expertly blends theory and application, providing numerous worked examples and exercises at the end of each chapter. The reader is assumed to have a basic knowledge of mathematical statistics at an undergraduate level together with an elementary understanding of linear algebra. There are appendices which provide a background in matrix algebra, a summary of univariate statistics, a collection of statistical tables and a discussion of computational aspects. The work includes coverage of:* Basic properties of random vectors, copulas, normal distribution theory, and estimation* Hypothesis testing, multivariate regression, and analysis of variance* Principal component analysis, factor analysis, and canonical correlation analysis* Discriminant analysis, cluster analysis, and multidimensional scaling* New advances and techniques, including supervised and unsupervised statistical learning, graphical models and regularization methods for high-dimensional dataAlthough primarily...

List of contents

Epigraph
 
Preface xiii
 
Preface to the Second Edition xiii
 
Preface to the First Edition xv
 
Acknowledgements from First Edition xviii
 
Notation, abbreviations and key ideas xix
 

1 Introduction 1
 
1.1 Objects and Variables 1
 
1.2 Some Multivariate Problems and Techniques 2
 
1.2.1 Generalizations of univariate techniques 2
 
1.2.2 Dependence and regression 2
 
1.2.3 Linear combinations 2
 
1.2.4 Assignment and dissection 5
 
1.2.5 Building configurations 6
 
1.3 The Data Matrix 6
 
1.4 Summary Statistics 7
 
1.4.1 The mean vector and covariance matrix 8
 
1.4.2 Measures of multivariate scatter 11
 
1.5 Linear Combinations 11
 
1.5.1 The scaling transformation 12
 
1.5.2 Mahalanobis transformation 12
 
1.5.3 Principal component transformation 12
 
1.6 Geometrical Ideas 13
 
1.7 Graphical Representation 14
 
1.7.1 Univariate scatters 14
 
1.7.2 Bivariate scatters 16
 
1.7.3 Harmonic curves 16
 
1.7.4 Parallel coordinates plot 19
 
1.8 Measures of Multivariate Skewness and Kurtosis 19
 
2 Basic Properties of Random Vectors 27
 
2.1 Cumulative Distribution Functions and Probability Density Functions 27
 
2.2 Population Moments 29
 
2.2.1 Expectation and correlation 29
 
2.2.2 Population mean vector and covariance matrix 29
 
2.2.3 Mahalanobis space 31
 
2.2.4 Higher moments 31
 
2.2.5 Conditional moments 33
 
2.3 Characteristic Functions 33
 
2.4 Transformations 35
 
2.5 The Multivariate Normal Distribution 36
 
2.5.1 Definition 36
 
2.5.2 Geometry 38
 
2.5.3 Properties 38
 
2.5.4 Singular multivariate normal distribution 43
 
2.5.5 The matrix normal distribution 44
 
2.6 Random Samples 45
 
2.7 Limit Theorems 47
 
3 Non-normal Distributions 53
 
3.1 Introduction 53
 
3.2 Some Multivariate Generalizations of Univariate Distributions 53
 
3.2.1 Direct generalizations 53
 
3.2.2 Common components 54
 
3.2.3 Stochastic generalizations 55
 
3.3 Families of Distributions 56
 
3.3.1 The exponential family 56
 
3.3.2 The spherical family 57
 
3.3.3 Elliptical distributions 60
 
3.3.4 Stable distributions 62
 
3.4 Insights into skewness and kurtosis 62
 
3.5 Copulas 64
 
3.5.1 The Gaussian Copula 66
 
3.5.2 The Clayton-Mardia copula 67
 
3.5.3 Archimedean Copulas 68
 
3.5.4 Fr´echet-H¨offding Bounds 69
 
4 Normal Distribution Theory 77
 
4.1 Characterization and Properties 77
 
4.1.1 The central role of multivariate normal theory 77
 
4.1.2 A definition by characterization 78
 
4.2 Linear Forms 79
 
4.3 Transformations of Normal Data Matrices 81
 
4.4 The Wishart Distribution 83
 
4.4.1 Introduction 83
 
4.4.2 Properties of Wishart matrices 83
 
4.4.3 PartitionedWishart matrices 86
 
4.5 The Hotelling T 2 Distribution 89
 
4.6 Mahalanobis Distance 92
 
4.6.1 The two-sample Hotelling T 2 statistic 92
 
4.6.2 A decomposition of Mahalanobis distance 93
 
4.7 Statistics Based on the Wishart Distribution 95
 
4.8 Other Distributions Related to the Multivariate Normal 99
 
5 Estimation 111
 
5.1 Likelihood and Sufficiency 111
 
5.1.1 The likelihood function 111
 
5.1.2 Efficient scores and Fisher's information 112
 
5.1.3 The

Product details

• Authors: John T. Kent, Kent John T., Kanti V. Mardia, Kanti V. (University of Leeds Mardia, Mardia Kanti V., Charles C. Taylor
• Publisher: Wiley, John and Sons Ltd
• Languages: English
• Product format: Hardback
• Released: 11.07.2024
• EAN: 9781118738023
• ISBN: 978-1-118-73802-3
• No. of pages: 592
• Series: Wiley Series in Probability and Statistics
• Subjects: Natural sciences, medicine, IT, technology > Mathematics > Probability theory, stochastic theory, mathematical statistics; Statistik, Datenanalyse, Statistics, data analysis, Multivariate Analyse, Angewandte Wahrscheinlichkeitsrechnung u. Statistik, Applied Probability & Statistics, Multivariate Analysis