Numerical Analysis for Statisticians (Statistics and Computing)

Every advance in computer architecture and software tempts statisticians to tackle numerically harder problems. To do so intelligently requires a good working knowledge of numerical analysis. This book is intended to equip students to craft their own software and to understand the advantages and disadvantages of different numerical methods. Issues of numerical stability, accurate approximation, computational complexity, and mathematical modeling share the limelight in a broad yet rigorous overview of those parts of numerical analysis relevant to statisticians. Although the bulk of the book covers traditional topics from linear algebra, optimization theory, numerical integration, and Fourier analysis, several chapters highlight recent statistical developments such as wavelets, the bootstrap, hidden Markov chains, and Markov chain Monte Carlo methods. These computationally intensive methods are revolutionizing statistics. Numerical Analysis for Statisticians can serve as a graduate text for either a one or a two-semester course surveying computational statistics. With a careful selection of topics and appropriate supplementation, it can even be used at the undergraduate level. It contains enough material on optimization theory alone for a one-semester graduate course. Students mastering a substantial part of the text will be well prepared for the numerical parts of advanced topics courses in statistics. Because many of the chapters nearly self-contained, professional statisticians will also find the book useful as a reference. Kenneth Lange is Professor of Biomathematics and Human Genetics at the UCLA School of Medicine. At various times during his career, he has held appointments at the University of New Hampshire, MIT, Harvard, and the University of Michigan. While at the University of Michigan, he was the Parmacia Upjohn Foundation, Professor of Biostatistics. His research interests include human genetics, population modeling, biomedical imaging, computational statistics, and applied stochastic processes. Also available by Kenneth Lange: Mathematical and Statistical Methods for Genetic Analysis, Springer-Verlag New York Inc., 1997, 265 pp., Cloth, ISBN 0-387-949097. 1. Basic Principles Of Population Genetics : 1.1. Introduction ; 1.2. Genetics Background ; 1.3. Hardy-weinberg Equilibrium ; 1.4. Linkage Equilibrium ; 1.5. Selection ; 1.6. Balance Between Mutation And Selection ; 1.7. Problems -- 2. Counting Methods And The Em Algorithm : 2.1. Introduction ; 2.2. Gene Counting ; 2.3. Description Of The Em Algorithm ; 2.4. Ascent Property Of The Em Algorithm ; 2.5. Allele Frequency Estimation By The Em Algorithm ; 2.6. Classical Segregation Analysis By The Em Algorithm ; 2.7. Problems -- 3. Newton's Method And Scoring : 3.1. Introduction ; 3.2. Newton's Method ; 3.3. Scoring ; 3.4. Application To The Design Of Linkage Experiments ; 3.5. Quasi-newton Methods ; 3.6. The Dirichlet Distribution ; 3.7. Empirical Bayes Estimation Of Allele Frequencies ; 3.8. Problems -- 4. Hypothesis Testing And Categorical Data : 4.1. Introduction ; 4.2. Hypotheses About Genotype Frequencies ; 4.3. Other Multinomial Problems In Genetics ; 4.4. The Zmax Test ; 4.5. The Wd Statistic ; 4.6. Exact Tests Of Independence ; 4.7. Problems -- 5. Genetic Identity Coefficients : 5.1. Introduction ; 5.2. Kinship And Inbreeding Coefficients ; 5.3. Condensed Identity Coefficients ; 5.4. Generalized Kinship Coefficients ; 5.5. From Kinship To Identity Coefficients ; 5.6. Calculation Of Generalized Kinship Coefficients ; 5.7. Problems 6. Applications Of Identity Coefficients : 6.1. Introduction ; 6.2. Genotype Prediction ; 6.3. Covariance For A Quantitative Trait ; 6.4. Risk Ratios And Genetic Model Discrimination ; 6.5. An Affecteds-only Method Of Linkage Analysis ; 6.6. Problems -- 7. Computation Of Mendelian Likelihoods : 7.1. Introduction ; 7.2. Mendelian Models ; 7.3. Genotype Elimination And Allele Consolidation ; 7.4. Array Transformations And Iterated Sums ; 7.5. Array Factoring ; 7.6. Examples Of Pedigree Analysis ; 7.7. Problems -- 8. The Polygenic Model : 8.1. Introduction ; 8.2. Maximum Likelihood Estimation By Scoring ; 8.3. Application To Gc Measured Genotype Data ; 8.4. Multivariate Traits ; 8.5. Left And Right-hand Finger Ridge Counts ; 8.6. The Hypergeometric Polygenic Model ; 8.7. Application To Risk Prediction ; 8.8. Problems -- 9. Markov Chain Monte Carlo Methods : 9.1. Introduction ; 9.2. Review Of Discrete-time Makov Chains ; 9.3. The Hastings-metropolis Algorithm And Simulated Annealing ; 9.4. Descent States And Descent Graphs ; 9.5. Descent Trees And The Founder Tree Graph ; 9.6. The Descent Graph Markov Chain ; 9.7. Computing Location Scores ; 9.8. Finding A Legal Descent Graph ; 9.9. Haplotyping ; 9.10 Application To Episodic Ataxia ; 9.11. Problems -- 10. Reconstruction Of Evolutionary Trees : 10.1. Introduction ; 10.2. Evolutionary Trees ; 10.3. Maximum Parsimony ; 10.4. Review Of Continuous-time Markov Chains ; 10.5. A Nucleotide Substitution Model ; 10.6. Maximum Likelihood Reconstruction ; 10.7. Origin Of The Eukaryotes ; 10.8. Problems 11. Radiation Hybrid Mapping : 11.1. Introduction ; 11.2. Models ; 11.3. Minimum Obligate Breaks Criterion ; 11.4. Maximum Likelihood Methods ; 11.5. Application To Haploid Data ; 11.6. Polyploid Radiation Hybrids ; 11.7. Maximum Likelihood Under Polyploidy ; 11.8. Obligate Breaks Under Polyploidy ; 11.9. Bayesian Methods ; 11.10. Applications To Diploid Data ; 11.11. Problems -- 12. Models Of Recombination : 12.1. Introduction ; 12.2. Mather's Formula And Its Generalization ; 12.3. Count-location Model ; 12.4. Stationary Renewal Models ; 12.5. Poisson-skip Model ; 12.6. Chiasma Interference ; 12.7. Application To Drosophila Data ; 12.8. Problems -- 13. Poisson Approximation : 13.1. Introduction ; 13.2. Poisson Approximation To The Wd Statistic ; 13.3. Construction Of Somatic Cell Hybrid Panels ; 13.4. Biggest Marker Gap ; 13.5 Randomness Of Restriction Sites ; 13.6. Dna Sequence Matching ; 13.7. Problems. Kenneth Lange. Includes Bibliographical References And Index. This book, like many books, was born in frustration. When in the fall of 1994 I set out to teach a second course in computational statistics to d- toral students at the University of Michigan, none of the existing texts seemed exactly right. On the one hand, the many decent, even inspiring, books on elementary computational statistics stress the nuts and bolts of using packaged programs and emphasize model interpretation more than numerical analysis. On the other hand, the many theoretical texts in - merical analysis almost entirely neglect the issues of most importance to statisticians. TheclosestbooktomyidealwastheclassicaltextofKennedy and Gentle [2]. More than a decade and a half after its publication, this book still has many valuable lessons to teach statisticians. However, upon re?ecting on the rapid evolution of computational statistics, I decided that the time was ripe for an update. The book you see before you represents a biased selection of those topics in theoretical numerical analysis most relevant to statistics. By intent this book is not a compendium of tried and trusted algorithms, is not a c- sumer's guide to existing statistical software, and is not an exposition of computer graphics or exploratory data analysis. My focus on principles of numerical analysis is intended to equip students to craft their own software and to understand the advantages and disadvantages of di?erent numerical methods. Issues of numerical stability, accurate approximation, compu- tional complexity, and mathematical modeling share the limelight and take precedence over philosophical questions of statistical inference. Cover Page......Page 1 Title Page......Page 4 Edition Info......Page 5 2 Power Series Expansions......Page 10 5 Solution of Nonlinear Equations......Page 11 9 Splines......Page 12 14 Constrained Optimization......Page 13 18 The Finite Fourier Transform......Page 14 23 Finite-State Markov Chains......Page 15 Index......Page 16 Preface......Page 6 1 Recurrence Relations......Page 17 2 Power Series Expansions......Page 28 3 Continued Fraction Expansions......Page 41 4 Asymptotic Expansions......Page 53 5 Solution of Nonlinear Equations......Page 69 6 Vector and Matrix Norms......Page 84 7 Linear Regression and Matrix Inversion......Page 95 8 Eigenvalues and Eigenvectors......Page 108 9 Splines......Page 119 10 The EM Algorithm......Page 131 11 Newton’s Method and Scoring......Page 146 12 Variations on the EM Theme......Page 159 13 Convergence of Optimization Algorithms......Page 176 14 Constrained Optimization......Page 193 15 Concrete Hilbert Spaces......Page 207 16 Quadrature Methods......Page 223 17 The Fourier Transform......Page 237 18 The Finite Fourier Transform......Page 251 19 Wavelets......Page 268 20 Generating Random Deviates......Page 285 21 Independent Monte Carlo......Page 302 22 Bootstrap Calculations......Page 315 23 Finite-State Markov Chains......Page 330 24 Markov Chain Monte Carlo......Page 346 A,B......Page 360 C......Page 361 E......Page 362 F......Page 363 G,H......Page 364 K,L......Page 365 M......Page 366 N,O......Page 367 R......Page 368 T......Page 370 U,V,W......Page 371 During the past decade, geneticists have constructed detailed maps of the human genome and cloned scores of Mendelian disease genes. They now stand on the threshold of sequencing the genome in its entirety. The unprecedented insights into human disease and evolution offered by mapping and sequencing will transform medicine and agriculture. This revolution depends vitally on the contributions of applied mathematicians, statisticians, and computer scientists. Mathematical and Statistical Methods for Genetic Analysis is written to equip graduate students in the mathematical sciences to understand and model the epidemiological and experimental data encountered in genetics research. Mathematical, statistical, and computational principles relevant to this task are developed hand in hand with applications to gene mapping, risk prediction, and the testing of epidemiological hypotheses. The book includes many topics currently accessible only in journal articles, including pedigree analysis algorithms, Markov chain Monte Carlo methods, reconstruction of evolutionary trees, radiation hybrid mapping, and models of recombination. Exercise sets are included. Kenneth Lange is Professor of Biostatistics and Mathematics and the Pharmacia & Upjohn Foundations Research Professor at the University of Michigan. He has held visiting appointments at MIT and Harvard. His research interests include human genetics, population modeling, biomedical imaging, computational statistics, and applied stochastic processes. During the past decade, geneticists have cloned scores of Mendelian disease genes and constructed a rough draft of the entire human genome. The unprecedented insights into human disease and evolution offered by mapping, cloning, and sequencing will transform medicine and agriculture. This revolution depends vitally on the contributions of applied mathematicians, statisticians, and computer scientists. Mathematical and Statistical Methods for Genetic Analysis is written to equip students in the mathematical sciences to understand and model the epidemiological and experimental data encountered in genetics research. Mathematical, statistical, and computational principles relevant to this task are developed hand in hand with applications to population genetics, gene mapping, risk prediction, testing of epidemiological hypotheses, molecular evolution, and DNA sequence analysis. Many specialized topics are covered that are currently accessible only in journal articles. This second edition expands the original edition by over 100 pages and includes new material on DNA sequence analysis, diffusion processes, binding domain identification, Bayesian estimation of haplotype frequencies, case-control association studies, the gamete competition model, QTL mapping and factor analysis, the Lander-Green-Kruglyak algorithm of pedigree analysis, and codon and rate variation models in molecular phylogeny. Sprinkled throughout the chapters are many new problems. Every advance in computer architecture and software tempts statisticians to tackle numerically harder problems. To do so intelligently requires a good working knowledge of numerical analysis. This book is intended to equip students to craft their own software and to understand the advantages and disadvantages of different numerical methods. Numerical Analysis for Statisticians can serve as a graduate text for either a one- or a two-semester course surveying computational statistics. With a careful selection of topics and appropriate supplementation, it can even be used at the undergraduate level. Because many of the chapters are nearly self-contained, professional statisticians will also find the book useful as a reference. Every advance in computer architecture and software tempts statisticians to tackle numerically harder problems. To do so intelligently requires a good working knowledge of numerical analysis. This book is intended to equip students to craft their own software and to understand the advantages and disadvantages of different numerical methods and can serve as a graduate text for either a one- or a two-semester course surveying computational statistics. With a careful selection of topics and appropriate supplementation, it can even be used at the undergraduate level. Because many of the chapters are nearly self-contained, professional statisticians will also find the book useful as a reference During the past decade, geneticists have constructed detailed maps of the human genome and cloned scores of Mendelian disease genes. These insights into human disease and evolution which have transformed medicine and agriculture depend on the contributions of applied mathematicians, statisticians, and computer scientists. This work is written to help readers understand and model the epidemiological and experimental data encountered in genetics research. Numerical analysis is the study of computation and its accuracy, stability and often its implementation on a computer. This book focuses on the principles of numerical analysis and is intended to equip those readers who use statistics to craft their own software and to understand the advantages and disadvantages of different numerical methods. This Book Presents Topics In Numerical Analysis For Statisticians. It Would Be Suitable As A Text For A Graduate Course In Statistical Computing. The Focus Is On Principles Of Numerical Analysis Intended To Equip Students To Craft Their Own Software And To Understand The Advantages And Disadvantages Of Different Numerical Methods.