Green Mathematical Tools for Applied Multivariate Analysis

1. Auflage 2014
ISBN: 978-1-4832-1404-7
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 402 Seiten, Web PDF

Mathematical Tools for Applied Multivariate Analysis provides information pertinent to the aspects of transformational geometry, matrix algebra, and the calculus that are most relevant for the study of multivariate analysis. This book discusses the mathematical foundations of applied multivariate analysis. Organized into six chapters, this book begins with an overview of the three problems in multiple regression, principal components analysis, and multiple discriminant analysis. This text then presents a standard treatment of the mechanics of matrix algebra, including definitions and operations on matrices, vectors, and determinants. Other chapters consider the topics of eigenstructures and linear transformations that are important to the understanding of multivariate techniques. This book discusses as well the eigenstructures and quadratic forms. The final chapter deals with the geometric aspects of linear transformations. This book is a valuable resource for students.

Autoren/Hrsg.

Green, Paul E.

Inhaltsverzeichnis

1;Front Cover;1
2;Mathematical Tools for Applied Multivariate Analysis;4
3;Copyright Page;6
4;Table of Contents;8
5;Dedication;7
6;Preface;12
7;Acknowledgments;14
8;CHAPTER 1. The Nature of Multivariate Data Analysis;16
8.1;1.1 INTRODUCTION;16
8.2;1.2 MULTIVARIATE METHODS IN RESEARCH;17
8.3;1.3 A CLASSIFICATION OF TECHNIQUES FOR ANALYZING ASSOCIATIVE DATA;19
8.4;1.4 ORGANIZING THE TECHNIQUES;23
8.5;1.5 ILLUSTRATIVE APPLICATIONS;26
8.6;1.6 SOME NUMERICAL EXAMPLES;29
8.7;1.7 FORMAT OF SUCCEEDING CHAPTERS;38
8.8;1.8 SUMMARY;39
8.9;REVIEW QUESTIONS;40
9;CHAPTER 2. Vector and Matrix Operations for Multivariate Analysis;41
9.1;2.1 INTRODUCTION;41
9.2;2.2 VECTOR REPRESENTATION;42
9.3;2.3 BASIC DEFINITIONS AND OPERATIONS ON VECTORS;42
9.4;2.4 MATRIX REPRESENTATION;55
9.5;2.5 BASIC DEFINITIONS AND OPERATIONS ON MATRICES;55
9.6;2.6 SOME SPECIAL MATRICES;67
9.7;2.7 DETERMINANTS OF MATRICES;73
9.8;2.8 APPLYING MATRIX OPERATIONS TO STATISTICAL DATA;84
9.9;2.9 SUMMARY;88
9.10;REVIEW QUESTIONS;89
10;CHAPTER 3. Vector and Matrix Concepts from a Geometric Viewpoint;92
10.1;3.1 INTRODUCTION;92
10.2;3.2 EUCLIDEAN SPACE AND RECTANGULAR CARTESIAN COORDINATES;93
10.3;3.3 GEOMETRIC REPRESENTATION OF VECTORS;100
10.4;3.4 LINEAR DEPENDENCE OF VECTORS;116
10.5;3.5 ORTHOGONAL TRANSFORMATIONS;126
10.6;3.6 GEOMETRIC ASPECTS OF CROSS-PRODUCT MATRICES AND DETERMINANTS;133
10.7;3.7 SUMMARY;139
10.8;REVIEW QUESTIONS;140
11;CHAPTER 4. Linear Transformations from a Geometric Viewpoint;142
11.1;4.1 INTRODUCTION;142
11.2;4.2 SIMULTANEOUS EQUATIONS AND MATRIX TRANSFORMATIONS;143
11.3;4.3 MATRIX INVERSION;151
11.4;4.4 GEOMETRIC RELATIONSHIPS INVOLVING MATRIX TRANSFORMATIONS;162
11.5;4.5 COMPOSITE TRANSFORMATIONS;171
11.6;4.6 INVERTIBLE TRANSFORMATIONS AND MATRIX RANK;178
11.7;4.7 METHODS FOR RANK DETERMINATION AND MATRIX INVERSION;191
11.8;4.8 SUMMARY;204
11.9;REVIEW QUESTIONS;205
12;CHAPTER 5. Decomposition of Matrix Transformations: Eigenstructures and Quadratic Forms;209
12.1;5.1 INTRODUCTION;209
12.2;5.2 AN OVERVIEW OF MATRIX EIGENSTRUCTURES;210
12.3;5.3 TRANSFORMATIONS OF COVARIANCE MATRICES;222
12.4;5.4 EIGENSTRUCTURE OF A SYMMETRIC MATRIX;225
12.5;5.5 PROPERTIES OF MATRIX EIGENSTRUCTURES;234
12.6;5.6 EIGENSTRUCTURES AND MATRIX RANK;240
12.7;5.7 THE BASIC STRUCTURE OF A MATRIX;245
12.8;5.8 QUADRATIC FORMS;255
12.9;5.9 EIGENSTRUCTURES OF NONSYMMETRIC MATRICES IN MULTIVARIATE ANALYSIS;262
12.10;5.10 SUMMARY;270
12.11;REVIEW QUESTIONS;271
13;CHAPTER 6. Applying the Tools to Multivariate Data;274
13.1;6.1 INTRODUCTION;274
13.2;6.2 THE MULTIPLE REGRESSION PROBLEM;275
13.3;6.3 OTHER FORMS OF THE GENERAL LINEAR MODEL;285
13.4;6.4 THE FACTOR ANALYSIS PROBLEM;287
13.5;6.5 THE MULTIPLE DISCRIMINANT ANALYSIS PROBLEM;293
13.6;6.6 A PARTING LOOK AT MULTIVARIATE TECHNIQUE CLASSIFICATION;301
13.7;6.7 SUMMARY;308
13.8;REVIEW QUESTIONS;309
14;APPENDIX A: Symbolic Differentiation and Optimization of Multivariable Functions;310
14.1;A.1 INTRODUCTION;310
14.2;A.2 DIFFERENTIATION OF FUNCTIONS OF ONE ARGUMENT;311
14.3;A.3 DIFFERENTIATION OF FUNCTIONS OF TWO ARGUMENTS;319
14.4;A.4 SYMBOLIC DIFFERENTIATION;327
14.5;A.5 APPLICATION OF THE CALCULUS TO MULTIVARIATE ANALYSIS;333
14.6;A.6 SUMMARY;335
14.7;REVIEW QUESTIONS;336
15;APPENDIX B: Linear Equations and Generalized inverses;338
15.1;B.1 INTRODUCTION;338
15.2;B.2 SIMULTANEOUS LINEAR EQUATIONS;339
15.3;B.3 INTRODUCTORY ASPECTS OF GENERALIZED INVERSES;349
15.4;B.4 THE g INVERSE;358
15.5;B.5 SUMMARY;365
15.6;REVIEW QUESTIONS;365
16;Answers to Numerical Problems;367
17;References;379
18;Index;384