Details

Matrix Algebra Useful for Statistics


Matrix Algebra Useful for Statistics


Wiley Series in Probability and Statistics 2. Aufl.

von: Shayle R. Searle, Andre I. Khuri

108,99 €

Verlag: Wiley
Format: PDF
Veröffentl.: 31.03.2017
ISBN/EAN: 9781118935163
Sprache: englisch
Anzahl Seiten: 512

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Beschreibungen

<p><b>A thoroughly updated guide to matrix algebra and it uses in statistical analysis and features SAS®, MATLAB®, and R throughout</b></p> <p>This <i>Second Edition </i>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 <i>Second Edition </i>also:</p> <ul> <li>Contains new coverage on vector spaces and linear transformations and discusses computational aspects of matrices</li> <li>Covers the analysis of balanced linear models using direct products of matrices</li> <li>Analyzes multiresponse linear models where several responses can be of interest</li> <li>Includes extensive use of SAS, MATLAB, and R throughout</li> <li>Contains over 400 examples and exercises to reinforce understanding along with select solutions</li> <li>Includes plentiful new illustrations depicting the importance of geometry as well as historical interludes</li> </ul> <p><i>Matrix Algebra Useful for Statistics, Second Edition </i>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.</p> <p><b>THE LATE SHAYLE R. SEARLE, PHD, </b>was professor emeritus of biometry at Cornell University. He was the author of <i>Linear Models for Unbalanced Data </i>and <i>Linear Models </i>and co-author of <i>Generalized, Linear, and Mixed Models, Second Edition, Matrix Algebra for Applied Economics, </i>and <i>Variance Components, </i>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.</p> <p><b>ANDRÉ I. KHURI, PHD, </b>is Professor Emeritus of Statistics at the University of Florida. He is the author of <i>Advanced Calculus with Applications in Statistics, Second Edition </i>and co-author of <i>Statistical Tests for Mixed Linear Models, </i>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.</p>
<p>PREFACE xvii</p> <p>PREFACE TO THE FIRST EDITION xix</p> <p>INTRODUCTION xxi</p> <p>ABOUT THE COMPANION WEBSITE xxxi</p> <p><b>PART I DEFINITIONS, BASIC CONCEPTS, AND MATRIX OPERATIONS 1</b></p> <p><b>1 Vector Spaces, Subspaces, and Linear Transformations 3</b></p> <p>1.1 Vector Spaces 3</p> <p>1.2 Base of a Vector Space 5</p> <p>1.3 Linear Transformations 7</p> <p><b>2 Matrix Notation and Terminology 11</b></p> <p>2.1 Plotting of a Matrix 14</p> <p>2.2 Vectors and Scalars 16</p> <p>2.3 General Notation 16</p> <p><b>3 Determinants 21</b></p> <p>3.1 Expansion by Minors 21</p> <p>3.2 Formal Definition 25</p> <p>3.3 Basic Properties 27</p> <p>3.4 Elementary Row Operations 34</p> <p>3.5 Examples 37</p> <p>3.6 Diagonal Expansion 39</p> <p>3.7 The Laplace Expansion 42</p> <p>3.8 Sums and Differences of Determinants 44</p> <p>3.9 A Graphical Representation of a 3 × 3 Determinant 45</p> <p><b>4 Matrix Operations 51</b></p> <p>4.1 The Transpose of a Matrix 51</p> <p>4.2 Partitioned Matrices 52</p> <p>4.3 The Trace of a Matrix 55</p> <p>4.4 Addition 56</p> <p>4.5 Scalar Multiplication 58</p> <p>4.6 Equality and the Null Matrix 58</p> <p>4.7 Multiplication 59</p> <p>4.8 The Laws of Algebra 74</p> <p>4.9 Contrasts With Scalar Algebra 76</p> <p>4.10 Direct Sum of Matrices 77</p> <p>4.11 Direct Product of Matrices 78</p> <p>4.12 The Inverse of a Matrix 80</p> <p>4.13 Rank of a Matrix—Some Preliminary Results 82</p> <p>4.14 The Number of LIN Rows and Columns in a Matrix 84</p> <p>4.15 Determination of the Rank of a Matrix 85</p> <p>4.16 Rank and Inverse Matrices 87</p> <p>4.17 Permutation Matrices 87</p> <p><b>5 Special Matrices 97</b></p> <p>5.1 Symmetric Matrices 97</p> <p>5.2 Matrices Having All Elements Equal 102</p> <p>5.3 Idempotent Matrices 104</p> <p>5.4 Orthogonal Matrices 106</p> <p>5.5 Parameterization of Orthogonal Matrices 109</p> <p>5.6 Quadratic Forms 110</p> <p>5.7 Positive Definite Matrices 113</p> <p><b>6 Eigenvalues and Eigenvectors 119</b></p> <p>6.1 Derivation of Eigenvalues 119</p> <p>6.2 Elementary Properties of Eigenvalues 122</p> <p>6.3 Calculating Eigenvectors 125</p> <p>6.4 The Similar Canonical Form 128</p> <p>6.5 Symmetric Matrices 131</p> <p>6.6 Eigenvalues of Orthogonal and Idempotent Matrices 135</p> <p>6.7 Eigenvalues of Direct Products and Direct Sums of Matrices 138</p> <p>6.8 Nonzero Eigenvalues of AB and BA 140</p> <p><b>7 Diagonalization of Matrices 145</b></p> <p>7.1 Proving the Diagonability Theorem 145</p> <p>7.2 Other Results for Symmetric Matrices 148</p> <p>7.3 The Cayley–Hamilton Theorem 152</p> <p>7.4 The Singular-Value Decomposition 153</p> <p><b>8 Generalized Inverses 159</b></p> <p>8.1 The Moore–Penrose Inverse 159</p> <p>8.2 Generalized Inverses 160</p> <p>8.3 Other Names and Symbols 164</p> <p>8.4 Symmetric Matrices 165</p> <p><b>9 Matrix Calculus 171</b></p> <p>9.1 Matrix Functions 171</p> <p>9.2 Iterative Solution of Nonlinear Equations 174</p> <p>9.3 Vectors of Differential Operators 175</p> <p>9.4 Vec and Vech Operators 179</p> <p>9.5 Other Calculus Results 181</p> <p>9.6 Matrices with Elements That Are Complex Numbers 188</p> <p>9.7 Matrix Inequalities 189</p> <p><b>PART II APPLICATIONS OF MATRICES IN STATISTICS 199</b></p> <p><b>10 Multivariate Distributions and Quadratic Forms 201</b></p> <p>10.1 Variance-Covariance Matrices 202</p> <p>10.2 Correlation Matrices 203</p> <p>10.3 Matrices of Sums of Squares and Cross-Products 204</p> <p>10.4 The Multivariate Normal Distribution 207</p> <p>10.5 Quadratic Forms and ;;2-Distributions 208</p> <p>10.6 Computing the Cumulative Distribution Function of a Quadratic Form 213</p> <p><b>11 Matrix Algebra of Full-Rank Linear Models 219</b></p> <p>11.1 Estimation of ;; by the Method of Least Squares 220</p> <p>11.2 Statistical Properties of the Least-Squares Estimator 226</p> <p>11.3 Multiple Correlation Coefficient 229</p> <p>11.4 Statistical Properties under the Normality Assumption 231</p> <p>11.5 Analysis of Variance 233</p> <p>11.6 The Gauss–Markov Theorem 234</p> <p>11.7 Testing Linear Hypotheses 237</p> <p>11.8 Fitting Subsets of the x-Variables 246</p> <p>11.9 The Use of the R(.|.) Notation in Hypothesis Testing 247</p> <p><b>12 Less-Than-Full-Rank Linear Models 253</b></p> <p>12.1 General Description 253</p> <p>12.2 The Normal Equations 256</p> <p>12.3 Solving the Normal Equations 257</p> <p>12.4 Expected Values and Variances 259</p> <p>12.5 Predicted y-Values 260</p> <p>12.6 Estimating the Error Variance 261</p> <p>12.7 Partitioning the Total Sum of Squares 262</p> <p>12.8 Analysis of Variance 263</p> <p>12.9 The R(⋅|⋅) Notation 265</p> <p>12.10 Estimable Linear Functions 266</p> <p>12.11 Confidence Intervals 272</p> <p>12.12 Some Particular Models 272</p> <p>12.13 The R(⋅|⋅) Notation (Continued) 277</p> <p>12.14 Reparameterization to a Full-Rank Model 281</p> <p><b>13 Analysis of Balanced Linear Models Using Direct Products of Matrices 287</b></p> <p>13.1 General Notation for Balanced Linear Models 289</p> <p>13.2 Properties Associated with Balanced Linear Models 293</p> <p>13.3 Analysis of Balanced Linear Models 298</p> <p><b>14 Multiresponse Models 313</b></p> <p>14.1 Multiresponse Estimation of Parameters 314</p> <p>14.2 Linear Multiresponse Models 316</p> <p>14.3 Lack of Fit of a Linear Multiresponse Model 318</p> <p><b>PART III MATRIX COMPUTATIONS AND RELATED SOFTWARE 327</b></p> <p><b>15 SAS/IML 329</b></p> <p>15.1 Getting Started 329</p> <p>15.2 Defining a Matrix 329</p> <p>15.3 Creating a Matrix 330</p> <p>15.4 Matrix Operations 331</p> <p>15.5 Explanations of SAS Statements Used Earlier in the Text 354</p> <p><b>16 Use of MATLAB in Matrix Computations 363</b></p> <p>16.1 Arithmetic Operators 363</p> <p>16.2 Mathematical Functions 364</p> <p>16.3 Construction of Matrices 365</p> <p>16.4 Two- and Three-Dimensional Plots 371</p> <p><b>17 Use of R in Matrix Computations 383</b></p> <p>17.1 Two- and Three-Dimensional Plots 396</p> <p>Exercises 408</p> <p>APPENDIX 413</p> <p>INDEX 475</p> <p> </p>
<p>"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." <b>Mathematical Reviews, Sept 2017</b></p>
<p> <b>The late Shayle R. Searle, PhD,</b> was professor emeritus of biometry at Cornell University. He was the author of <i>Linear Models for Unbalanced Data</i> and <i>Linear Models</i> and co-author of <i>Generalized, Linear, and Mixed Models, Second Edition, Matrix Algebra for Applied Economics, and Variance Components,</i> 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. <p><b>André I. Khuri, PhD,</b> is Professor Emeritus of Statistics at the University of Florida. He is the author of <i>Advanced Calculus with Applications in Statistics, Second Edition</i> and co-author of <i>Statistical Tests for Mixed Linear Models,</i> 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.
<p> <b>A thoroughly updated guide to matrix algebra and its uses in statistical analysis and features SAS®, MATLAB®, and R throughout</b> <p> This <i>Second Edition</i> 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 <i>Second Edition</i> also: <ul> <li>Contains new coverage on vector spaces and linear transformations and discusses computational aspects of matrices</li> <li>Covers the analysis of balanced linear models using direct products of matrices</li> <li>Analyzes multiresponse linear models where several responses can be of interest</li> <li>Includes extensive use of SAS, MATLAB, and R throughout</li> <li>Contains over 400 examples and exercises to reinforce understanding along with select solutions</li> <li>Includes plentiful new illustrations depicting the importance of geometry as well as historical interludes </li> </ul> <p><i>Matrix Algebra Useful for Statistics, Second Edition</i> 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. <br>

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