Mathematical Tools for Applied Multivariate Analysis

Preface

Acknowledgments

Chapter 1 The Nature of Multivariate Data Analysis


1.1 Introduction


1.2 Multivariate Methods in Research


1.3 A Classification of Techniques for Analyzing Associative Data


1.4 Organizing the Techniques


1.5 Illustrative Applications


1.6 Some Numerical Examples


1.7 Format of Succeeding Chapters


1.8 Summary

Review Questions

Chapter 2 Vector and Matrix Operations for Multivariate Analysis


2.1 Introduction


2.2 Vector Representation


2.3 Basic Definitions and Operations on Vectors


2.4 Matrix Representation


2.5 Basic Definitions and Operations on Matrices


2.6 Some Special Matrices


2.7 Determinants of Matrices


2.8 Applying Matrix Operations to Statistical Data


2.9 Summary

Review Questions

Chapter 3 Vector and Matrix Concepts from a Geometric Viewpoint


3.1 Introduction


3.2 Euclidean Space and Rectangular Cartesian Coordinates


3.3 Geometric Representation of Vectors


3.4 Linear Dependence of Vectors


3.5 Orthogonal Transformations


3.6 Geometric Aspects of Cross-Product Matrices and Determinants


3.7 Summary

Review Questions

Chapter 4 Linear Transformations from a Geometric Viewpoint


4.1 Introduction


4.2 Simultaneous Equations and Matrix Transformations


4.3 Matrix Inversion


4.4 Geometric Relationships Involving Matrix Transformations


4.5 Composite Transformations


4.6 Invertible Transformations and Matrix Rank


4.7 Methods for Rank Determination and Matrix Inversion


4.8 Summary

Review Questions

Chapter 5 Decomposition of Matrix Transformations: Eigenstructures and Quadratic Forms


5.1 Introduction


5.2 An Overview of Matrix Eigenstructures


5.3 Transformations of Covariance Matrices


5.4 Eigenstructure of a Symmetric Matrix


5.5 Properties of Matrix Eigenstructures


5.6 Eigenstructures and Matrix Rank


5.7 The Basic Structure of a Matrix


5.8 Quadratic Forms


5.9 Eigenstructures of Nonsymmetric Matrices in Multivariate Analysis


5.10 Summary

Review Questions

Chapter 6 Applying the Tools to Multivariate Data


6.1 Introduction


6.2 The Multiple Regression Problem


6.3 Other Forms of the General Linear Model


6.4 The Factor Analysis Problem


6.5 The Multiple Discriminant Analysis Problem


6.6 A Parting Look at Multivariate Technique Classification


6.7 Summary

Review Questions

Appendix A Symbolic Differentiation and Optimization of Multivariable Functions

A.1 Introduction

A.2 Differentiation of Functions of One Argument

A.3 Differentiation of Functions of Two Arguments

A.4 Symbolic Differentiation

A.5 Application of the Calculus to Multivariate Analysis

A.6 Summary

Review Questions

Appendix B Linear Equations and Generalized Inverses

B.1 Introduction

B.2 Simultaneous Linear Equations

B.3 Introductory Aspects of Generalized Inverses

B.4 The g Inverse

B.5 Summary

Review Questions

Answers to Numerical Problems

References

Index