Matrix Algebra Useful for Statistics (eBook)

Shayle R. Searle, Andre I. Khuri (Autoren)

eBook Download: EPUB

2017 | 2. Auflage
John Wiley & Sons (Verlag)
978-1-118-93515-6 (ISBN)

Infos

A thoroughly updated guide to matrix algebra and it uses in statistical analysis and features SAS®, MATLAB®, and R throughout

This Second Edition addresses matrix algebra that is useful in the statistical analysis of data as well as within statistics as a whole. The material is presented in an explanatory style rather than a formal theorem-proof format and is self-contained. Featuring numerous applied illustrations, numerical examples, and exercises, the book has been updated to include the use of SAS, MATLAB, and R for the execution of matrix computations. In addition, André I. Khuri, who has extensive research and teaching experience in the field, joins this new edition as co-author. The Second Edition also:

• Contains new coverage on vector spaces and linear transformations and discusses computational aspects of matrices
• Covers the analysis of balanced linear models using direct products of matrices
• Analyzes multiresponse linear models where several responses can be of interest
• Includes extensive use of SAS, MATLAB, and R throughout
• Contains over 400 examples and exercises to reinforce understanding along with select solutions
• Includes plentiful new illustrations depicting the importance of geometry as well as historical interludes

Matrix Algebra Useful for Statistics, Second Edition is an ideal textbook for advanced undergraduate and first-year graduate level courses in statistics and other related disciplines. The book is also appropriate as a reference for independent readers who use statistics and wish to improve their knowledge of matrix algebra.

THE LATE SHAYLE R. SEARLE, PHD, was professor emeritus of biometry at Cornell University. He was the author of Linear Models for Unbalanced Data and Linear Models and co-author of Generalized, Linear, and Mixed Models, Second Edition, Matrix Algebra for Applied Economics, and Variance Components, all published by Wiley. Dr. Searle received the Alexander von Humboldt Senior Scientist Award, and he was an honorary fellow of the Royal Society of New Zealand.

ANDRÉ I. KHURI, PHD, is Professor Emeritus of Statistics at the University of Florida. He is the author of Advanced Calculus with Applications in Statistics, Second Edition and co-author of Statistical Tests for Mixed Linear Models, all published by Wiley. Dr. Khuri is a member of numerous academic associations, among them the American Statistical Association and the Institute of Mathematical Statistics.

The late Shayle R. Searle, PhD, was professor emeritus of biometry at Cornell University. He was the author of Linear Models for Unbalanced Data and Linear Models and co-author of Generalized, Linear, and Mixed Models, Second Edition, Matrix Algebra for Applied Economics, and Variance Components, all published by Wiley. Dr. Searle received the Alexander von Humboldt Senior Scientist Award, and he was an honorary fellow of the Royal Society of New Zealand.

André I. Khuri, PhD, is Professor Emeritus of Statistics at the University of Florida. He is the author of Advanced Calculus with Applications in Statistics, Second Edition and co-author of Statistical Tests for Mixed Linear Models, all published by Wiley. Dr. Khuri is a member of numerous academic associations, among them the American Statistical Association and the Institute of Mathematical Statistics.

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A thoroughly updated guide to matrix algebra and it uses in statistical analysis and features SAS , MATLAB , and R throughout This Second Edition addresses matrix algebra that is useful in the statistical analysis of data as well as within statistics as a whole. The material is presented in an explanatory style rather than a formal theorem-proof format and is self-contained. Featuring numerous applied illustrations, numerical examples, and exercises, the book has been updated to include the use of SAS, MATLAB, and R for the execution of matrix computations. In addition, Andr I. Khuri, who has extensive research and teaching experience in the field, joins this new edition as co-author. The Second Edition also: Contains new coverage on vector spaces and linear transformations and discusses computational aspects of matrices Covers the analysis of balanced linear models using direct products of matrices Analyzes multiresponse linear models where several responses can be of interest Includes extensive use of SAS, MATLAB, and R throughout Contains over 400 examples and exercises to reinforce understanding along with select solutions Includes plentiful new illustrations depicting the importance of geometry as well as historical interludes Matrix Algebra Useful for Statistics, Second Edition is an ideal textbook for advanced undergraduate and first-year graduate level courses in statistics and other related disciplines. The book is also appropriate as a reference for independent readers who use statistics and wish to improve their knowledge of matrix algebra. THE LATE SHAYLE R. SEARLE, PHD, was professor emeritus of biometry at Cornell University. He was the author of Linear Models for Unbalanced Data and Linear Models and co-author of Generalized, Linear, and Mixed Models, Second Edition, Matrix Algebra for Applied Economics, and Variance Components, all published by Wiley. Dr. Searle received the Alexander von Humboldt Senior Scientist Award, and he was an honorary fellow of the Royal Society of New Zealand. ANDR I. KHURI, PHD, is Professor Emeritus of Statistics at the University of Florida. He is the author of Advanced Calculus with Applications in Statistics, Second Edition and co-author of Statistical Tests for Mixed Linear Models, all published by Wiley. Dr. Khuri is a member of numerous academic associations, among them the American Statistical Association and the Institute of Mathematical Statistics.

Rezensionen

"Matrix Algebra Useful for Statistics, Second Edition is an ideal textbook for advanced undergraduate and first-year graduate level courses in statistics and other related disciplines. The book is also appropriate as a reference for independent readers who use statistics and wish to improve their knowledge of matrix algebra." Mathematical Reviews, Sept 2017

Textauszug

Introduction

Historical Perspectives on Matrix Algebra

It is difficult to determine the origin of matrices from the historical point of view. Given the association between matrices and simultaneous linear equations, it can be argued that the history of matrices goes back to at least the third century BC. The Babylonians used simultaneous linear equations to study problems that pertained to agriculture in the fertile region between the Tigris and Euphrates rivers in ancient Mesopotamia (present day Iraq). They inscribed their findings, using a wedge-shaped script, on soft clay tablets which were later baked in ovens resulting in what is known as cuneiform tablets (see Figures 1 and 2).

This form of writing goes back to about 3000 BC (see Knuth, 1972). For example, a tablet dating from around 300 BC was found to contain a description of a problem that could be formulated in terms of two simultaneous linear equations in two variables. The description referred to two fields whose total area, the rate of production of grain per field, and their total yield were all given. It was required to determine the area of each field (see O’Connor and Robertson, 1996). The ancient Chinese also dealt with simultaneous linear equations between 200 BC and 100 BC in studying, for example, corn production. In fact, the text, Nine Chapters on the Mathematical Art, which was written during the Han Dynasty, played an important role in the development of mathematics in China. It was a practical handbook of mathematics consisting of 246 problems that pertained to engineering, surveying, trade, and taxation issues (see O’Connor and Robertson, 2003).

The modern development of matrices and matrix algebra did not materialize until the nineteenth century with the work of several mathematicians, including Augustin-Louis Cauchy, Ferdinand Georg Frobenius, Carl Friedrich Gauss, Arthur Cayley, and James Joseph Sylvester, among others. The use of the word “matrix” was first introduced by Sylvester in 1850. This terminology became more common after the publication of Cayley’s (1858) memoir on the theory of matrices. In 1829, Cauchy gave the first valid proof that the eigenvalues of a symmetric matrix must be real. He was also instrumental in creating the theory of determinants in his 1812 memoir. Frobenius (1877) wrote an important monograph in which he provided a unifying theory of matrices that combined the work of several other mathematicians. Hawkins (1974) described Frobenius’ paper as representing “an important landmark in the history of the theory of matrices.” Hawkins (1975) discussed Cauchy's work and its historical significance to the consideration of algebraic eigenvalue problems during the 18th century.

Figure 0.1 A Cuneiform Tablet with 97 Linear Equations (YBC4695-1). Yale Babylonian Collection, Yale University Library, New Haven, CT.

Science historians and mathematicians have regarded Cayley as the founder of the theory of matrices. His 1858 memoir was considered “the foundation upon which other mathematicians were able to erect the edifice we now call the theory of matrices” (see Hawkins, 1974, p. 561). Cayley was interested in devising a contracted notation to represent a system of m linear equations in n variables of the form

where the aij’s are given as coefficients. Cayley and other contemporary algebraists proposed replacing the m equations with a single matrix equation such as

Figure 0.2 An Old Babylonian Mathematical Text with Linear Equations (YBC4695-2). Yale Babylonian Collection, Yale University Library, New Haven, CT.

Cayley regarded such a scheme as an operator acting upon the variables, x1, x2, …, xn to produce the variables y1, y2, …, ym. This is a multivariable extension of the action of the single coefficient a upon x to produce a x, except that the rules associated with such an extension are different from the single variable case. This led to the development of the algebra of matrices.

Even though Cayley left his mark on the history of matrices, it should be pointed that his role in this endeavor was perhaps overrated by historians to the point of eclipsing the contribution of other mathematicians in the eighteenth and nineteenth centuries. Hawkins (1974) indicated that the ideas Cayley expressed in his 1858 memoir were not particularly original. He cited work by Laguerre (Edmond Nicolas Laguerre), Frobenius, and other mathematicians who had developed similar ideas during the same period, but without a knowledge of Cayley’s memoir. This conclusion was endorsed by Farebrother (1997) and Grattan-Guiness (1994). It is perhaps more accurate to conclude, as Hawkins (1975, p. 570) did, that “the history of matrix theory involved the efforts of many mathematicians, that it was indeed an international undertaking.” Higham (2008) provided an interesting commentary on the work of Cayley and Sylvester. He indicated that the multi-volume collected works of Cayley and Sylvester were both freely available online at the University of Michigan Historical Mathematics Collection by using the URL, http://quod.lib.umich.edu/u/umhistmath/ (for Cayley, use http://name.umdl.umich.edu/ABS3153.0013.001, and for Sylvester, use http://name.umdl.umich.edu/AAS8085.0002.001).

The history of determinants can be traced to methods used by the ancient Chinese and Japanese to solve a system of linear equations. Seki Kōwa, a distinguished Japanese mathematician of the seventeenth century, discovered the expansion of a determinant in solving simultaneous equations (see, e.g., Smith, 1958, p. 440). However, the methods used by the Chinese and the Japanese did not resemble the methods used nowadays in dealing with determinants. In the West, the theory of determinants is believed to have originated with the German mathematician, Gottfried Leibniz, in the seventeenth century, several years after the work of Seki Kōwa. However, the actual development of this theory did not begin until 1750 with the publication of the book by Gabriel Cramer. In fact, the method of solving a system of n linear equations in n unknowns by means of determinants is known as Cramer’s rule. The term “determinant” was first introduced by Gauss in 1801 in connection with quadratic forms. In 1812, Cauchy developed the theory of determinants as is known today. Cayley was the first to introduce the present-day notation of a determinant, namely, of vertical bars enclosing a square matrix, in a paper he wrote in 1841. So, just as in the case of matrices, the history of determinants was an international undertaking shaped by the efforts of many mathematicians. For more interesting facts about the history of determinants, see Miller (1930) and Price (1947).

The Introduction of Matrices Into Statistics

The entry of matrices into statistics was slow. Farebrother (1999) indicated that matrix algebra was not to emerge until the early part of the twentieth century. Even then, determinants were used in place of matrices in solving equations which were written in longhand. Searle (2000, p. 25) indicated that the year 1930 was a good starting point for the entry of matrices into statistics. That was the year of Volume 1 of the Annals of Mathematical Statistics, its very first paper, Wicksell (1930), being “Remarks on Regression.” The paper considered finding the least-squares estimates for a linear regression model with one independent variable. The normal equations for getting the model’s parameter estimates were expressed in terms of determinants only. No matrices were used. Today, such a subject would have been replete with matrices. Lengthy arguments and numerous equations were given to describe computational methods for general regression models, even in some of the books that appeared in the early 1950s. The slowness of the use of matrices in statistics was partially attributed to the difficulty in producing numerical results in situations involving, for example, regression models with several variables. In particular, the use of a matrix inverse posed a considerable computational difficulty before the advent of computers which came about in only the last 50 years. Today, such computational tasks are carried out quickly and effortlessly for a matrix of a reasonable size using a computer software. During his graduate student days at Cornell University in 1959, Searle (2000) recalled the great excitement he and other classmates in a small computer group had felt when they inverted a 10-by-10 matrix in 7 minutes. At that time this was considered a remarkable feat considering that only a year or two earlier, a friend had inverted a 40-by-40 matrix by hand using electric Marchant or Monroe calculators. That task took 6 weeks! An early beginning to more advanced techniques to inverting a matrix was the Doolittle method, as was described in Anderson and Bancroft (1952, Chapter 15). It is interesting to note that this method was introduced in the U.S. Coast and Geodetic Survey Report of 1878 (see Doolittle, 1881).

Alexander Craig Aitken made important contributions to promoting the use of matrix algebra in statistics in the 1930s. He was a brilliant mathematician from New Zealand with a phenomenal mental capability. It was reported that he could recite the irrational number π to 707 decimal places and multiply two nine-digit numbers in his head in 30 seconds. He was therefore referred to...

Produkt-Details

• Erscheint lt. Verlag: 10.4.2017
• Reihe/Serie: Wiley Series in Probability and Statistics
• Reihe/Serie: Wiley Series in Probability and Statistics
• Reihe/Serie: Wiley Series in Probability and Statistics
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Algebra
• Themenwelt: Mathematik / Informatik ► Mathematik ► Statistik
• Themenwelt: Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik
• Themenwelt: Technik
• Schlagworte: Angewandte Wahrscheinlichkeitsrechnung u. Statistik • applied illustrations • Applied Probability & Statistics • balanced linear models • linear transformations • <p>matrix algebra • MATLAB • matrices • multiresponse linear models</p> • R • R (Programm) • SAS • Statistical Analysis • Statistical Software / R • Statistics • Statistics - Text & Reference • Statistik • Statistik / Lehr- u. Nachschlagewerke • Statistiksoftware / R • vector spaces
• ISBN-10: 1-118-93515-2 / 1118935152
• ISBN-13: 978-1-118-93515-6 / 9781118935156