Multivariate Analysis

Chapter 1: Introduction

• • 1.1 Objects and Variables
• • 1.2 Some Multivariate Problems and Techniques
• • 1.3 The Data Matrix
• • 1.4 Summary Statistics
• • 1.5 Linear Combinations
• • 1.6 Geometrical Ideas
• • 1.7 Graphical Representation
• • 1.8 Measures of Multivariate Skewness and Kurtosis
• • Exercises and Complements

Chapter 2: Basic Properties of Random Vectors

• • Introduction
• • 2.1 Cumulative Distribution Functions and Probability Density Functions
• • 2.2 Population Moments
• • 2.3 Characteristic Functions
• • 2.4 Transformations
• • 2.5 The Multivariate Normal Distribution
• • 2.6 Random Samples
• • 2.7 Limit Theorems
• • Exercises and Complements

Chapter 3: Nonnormal Distributions

• • 3.1 Introduction
• • 3.2 Some Multivariate Generalizations of Univariate Distributions
• • 3.3 Families of Distributions
• • 3.4 Insights into Skewness and Kurtosis
• • 3.5 Copulas
• • Exercises and Complements

Chapter 4: Normal Distribution Theory

• • 4.1 Introduction and Characterization
• • 4.2 Linear Forms
• • 4.3 Transformations of Normal Data Matrices
• • 4.4 The Wishart Distribution
• • 4.5 The Hotelling T2 Distribution
• • 4.6 Mahalanobis Distance
• • 4.7 Statistics Based on the Wishart Distribution
• • 4.8 Other Distributions Related to the Multivariate Normal
• • Exercises and Complements

Chapter 5: Estimation

• • Introduction
• • 5.1 Likelihood and Sufficiency
• • 5.2 Maximum-likelihood Estimation
• • 5.3 Robust Estimation of Location and Dispersion for Multivariate Distributions
• • 5.4 Bayesian Inference
• • Exercises and Complements

Chapter 6: Hypothesis Testing

• • 6.1 Introduction
• • 6.2 The Techniques Introduced
• • 6.3 The Techniques Further Illustrated
• • 6.4 Simultaneous Confidence Intervals
• • 6.5 The Behrens–Fisher Problem
• • 6.6 Multivariate Hypothesis Testing: Some General Points
• • 6.7 Nonnormal Data
• • 6.8 Mardia’s Nonparametric Test for the Bivariate Two-sample Problem
• • Exercises and Complements

Chapter 7: Multivariate Regression Analysis

• • 7.1 Introduction
• • 7.2 Maximum-likelihood Estimation
• • 7.3 The General Linear Hypothesis
• • 7.4 Design Matrices of Degenerate Rank
• • 7.5 Multiple Correlation
• • 7.6 Least-squares Estimation
• • 7.7 Discarding of Variables
• • Exercises and Complements

Chapter 8: Graphical Models

• • 8.1 Introduction
• • 8.2 Graphs and Conditional Independence
• • 8.3 Gaussian Graphical Models
• • 8.4 Log-linear Graphical Models
• • 8.5 Directed and Mixed Graphs
• • Exercises and Complements

Chapter 9: Principal Component Analysis

• • 9.1 Introduction
• • 9.2 Definition and Properties of Principal Components
• • 9.3 Sampling Properties of Principal Components
• • 9.4 Testing Hypotheses About Principal Components
• • 9.5 Correspondence Analysis
• • 9.6 Allometry – Measurement of Size and Shape
• • 9.7 Discarding of Variables
• • 9.8 Principal Component Regression
• • 9.9 Projection Pursuit and Independent Component Analysis
• • 9.10 PCA in High Dimensions
• • Exercises and Complements

Chapter 10: Factor Analysis

• • 10.1 Introduction
• • 10.2 The Factor Model
• • 10.3 Principal Factor Analysis
• • 10.4 Maximum-likelihood Factor Analysis
• • 10.5 Goodness-of-fit Test
• • 10.6 Rotation of Factors
• • 10.7 Factor Scores
• • 10.8 Relationships Between Factor Analysis and Principal Component Analysis
• • 10.9 Analysis of Covariance Structures
• • Exercises and Complements

Chapter 11: Canonical Correlation Analysis

• • 11.1 Introduction
• • 11.2 Mathematical Development
• • 11.3 Qualitative Data and Dummy Variables
• • 11.4 Qualitative and Quantitative Data
• • Exercises and Complements

Chapter 12: Discriminant Analysis and Statistical Learning

• • 12.1 Introduction
• • 12.2 Bayes’ Discriminant Rule
• • 12.3 The Error Rate
• • 12.4 Discrimination Using the Normal Distribution
• • 12.5 Discarding of Variables
• • 12.6 Fisher’s Linear Discriminant Function
• • 12.7 Nonparametric Distance-based Methods
• • 12.8 Classification Trees
• • 12.9 Logistic Discrimination
• • 12.10 Neural Networks
• • Exercises and Complements

Chapter 13: Multivariate Analysis of Variance

• • 13.1 Introduction
• • 13.2 Formulation of Multivariate One-way Classification
• • 13.3 The Likelihood Ratio Principle
• • 13.4 Testing Fixed Contrasts
• • 13.5 Canonical Variables and A Test of Dimensionality
• • 13.6 The Union Intersection Approach
• • 13.7 Two-way Classification
• • Exercises and Complements

Chapter 14: Cluster Analysis and Unsupervised Learning

• • 14.1 Introduction
• • 14.2 Probabilistic Membership Models
• • 14.3 Parametric Mixture Models
• • 14.4 Partitioning Methods
• • 14.5 Hierarchical Methods
• • 14.6 Distances and Similarities
• • 14.7 Grouped Data
• • 14.8 Mode Seeking
• • 14.9 Measures of Agreement
• • Exercises and Complements

Chapter 15: Multidimensional Scaling

• • 15.1 Introduction
• • 15.2 Classical Solution
• • 15.3 Duality Between Principal Coordinate Analysis and Principal Component Analysis
• • 15.4 Optimal Properties of the Classical Solution and Goodness of Fit
• • 15.5 Seriation
• • 15.6 Nonmetric Methods
• • 15.7 Goodness of Fit Measure: Procrustes Rotation
• • 15.8 Multisample Problem and Canonical Variates
• • Exercises and Complements

Chapter 16: High-dimensional Data

• • 16.1 Introduction
• • 16.2 Shrinkage Methods in Regression
• • 16.3 Principal Component Regression
• • 16.4 Partial Least Squares Regression
• • 16.5 Functional Data
• • Exercises and Complements

Chapter A: Matrix Algebra

• • A.1 Introduction
• • A.2 Matrix Operations
• • A.3 Further Particular Matrices and Types of Matrices
• • A.4 Vector Spaces, Rank, and Linear Equations
• • A.5 Linear Transformations
• • A.6 Eigenvalues and Eigenvectors
• • A.7 Quadratic Forms and Definiteness
• • A.8 Generalized Inverse
• • A.9 Matrix Differentiation and Maximization Problems
• • A.10 Geometrical Ideas

Chapter B: Univariate Statistics

• • B.1 Introduction
• • B.2 Normal Distribution
• • B.3 Chi-squared Distribution
• • B.4 F and Beta Variables
• • B.5 t Distribution
• • B.6 Poisson Distribution

Chapter C: R commands and Data

• • C.1 Basic R Commands Related to Matrices
• • C.2 R Libraries and Commands Used in Exercises and Figures
• • C.3 Data Availability

Chapter D: Tables