1st Edition

Basics of Matrix Algebra for Statistics with R




ISBN 9781498712361
Published July 6, 2015 by Chapman and Hall/CRC
248 Pages

USD $67.95

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Book Description

A Thorough Guide to Elementary Matrix Algebra and Implementation in R

Basics of Matrix Algebra for Statistics with R provides a guide to elementary matrix algebra sufficient for undertaking specialized courses, such as multivariate data analysis and linear models. It also covers advanced topics, such as generalized inverses of singular and rectangular matrices and manipulation of partitioned matrices, for those who want to delve deeper into the subject.

The book introduces the definition of a matrix and the basic rules of addition, subtraction, multiplication, and inversion. Later topics include determinants, calculation of eigenvectors and eigenvalues, and differentiation of linear and quadratic forms with respect to vectors. The text explores how these concepts arise in statistical techniques, including principal component analysis, canonical correlation analysis, and linear modeling.

In addition to the algebraic manipulation of matrices, the book presents numerical examples that illustrate how to perform calculations by hand and using R. Many theoretical and numerical exercises of varying levels of difficulty aid readers in assessing their knowledge of the material. Outline solutions at the back of the book enable readers to verify the techniques required and obtain numerical answers.

Avoiding vector spaces and other advanced mathematics, this book shows how to manipulate matrices and perform numerical calculations in R. It prepares readers for higher-level and specialized studies in statistics.

Table of Contents

Introduction
Objectives
Further Reading
Guide to Notation
An Outline Guide to R
Inputting Data to R
Summary of Matrix Operators in R
Examples of R Commands

Vectors and Matrices
Vectors
Matrices
Matrix Arithmetic
Transpose and Trace of Sums and Products
Special Matrices
Partitioned Matrices
Algebraic Manipulation of matrices
Useful Tricks
Linear and Quadratic Forms
Creating Matrices in R
Matrix Arithmetic in R
Initial Statistical Applications

Rank of Matrices
Introduction and Definitions
Rank Factorization
Rank Inequalities
Rank in Statistics

Determinants
Introduction and Definitions
Implementation in R
Properties of Determinants
Orthogonal Matrices
Determinants of Partitioned Matrices
A Key Property of Determinants

Inverses
Introduction and Definitions
Properties
Implementation in R
Inverses of Patterned Matrices
Inverses of Partitioned Matrices
General Formulae
Initial Applications Continued

Eigenanalysis of Real Symmetric Matrices
Introduction and Definitions
Eigenvectors
Implementation in R
Properties of Eigenanalyses
A Key Statistical Application: PCA
Matrix Exponential
Decompositions
Eigenanalysis of Matrices with Special Structures
Summary of Key Results

Vector and Matrix Calculus
Introduction
Differentiation of a Scalar with Respect to a Vector
Differentiation of a Scalar with Respect to a Matrix
Differentiation of a Vector with Respect to a Vector
Differentiation of a Matrix with Respect to a Scalar
Use of Eigenanalysis in Constrained Optimization

Further Topics
Introduction
Further Matrix Decompositions
Generalized Inverses
Hadamard Products
Kronecker Products and the Vec Operator

Key Applications to Statistics
Introduction
The Multivariate Normal Distribution
Principal Component Analysis
Linear Discriminant Analysis
Canonical Correlation Analysis
Classical Scaling
Linear Models

Outline Solutions to Exercises

Bibliography

Index

Exercises appear at the end of each chapter.

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Author(s)

Biography

Dr. Nick Fieller is a retired senior lecturer in the School of Mathematics and Statistics and an honorary research fellow in archaeology at the University of Sheffield. His research interests include multivariate data analysis and statistical modeling in the pharmaceutical industry, archaeology, and forensic sciences.

Reviews

"…belongs to the category of mathematics books that integrate a programming language with substantive content. On the substantive side, the author has meticulously selected matrix algebra topics that are fundamental to learning, using, and understanding statistics. In this manner, the reader is saved time by focusing on matrix mathematics which is of most relevance to statistics. In addition, an instructor also benefits from the concise introduction to matrix algebra related to statistics. Therefore, this book can easily be adopted as a matrix algebra supplemental book in a syllabus on statistics. The exercises are short but rigorous, with detailed solutions provided at the end of the book...as a traditional text to teach practical matrix algebra to students taking multivariate and more advanced statistics courses, this book can be of good use."
—Abdolvahab Khademi, University of Massachusetts, Journal of Statistical Software, July 2016

"Key features of the book include highlighting useful tricks when manipulating matrices, derivation of key results with step-by-step cross-referenced explanations and demonstrations of implementing the techniques in R using numerical examples…it is a good beginner’s guide to understanding and manipulating matrices in R. It is suitable for early year undergraduate students and anyone who wishes to be introduced to matrix algebra in R in preparation for high-level or specialised studies in statistics. The book’s collection of summaries and key results also make it a good handbook for any statistician to refer to."
—Shuangzhe Liu, Stastistical Papers, July 2016 

"… a concise and straightforward presentation of matrix algebra techniques that are commonly used in statistics. Furthermore, the book discusses how to implement numerical instances of these techniques using R. … If you have a need or desire to carry out matrix computations in R, then it is likely that here you will find the needed commands. There are several nice features … it is very easy to find the R command for carrying out a specific matrix calculation. … useful as a reference. In addition, the author provides helpful tips and tricks for working with R. Another positive feature of this book is the applications to statistics. … the inclusion of exercises facilitates the use of this book as a course text."
MAA Reviews, January 2016