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Finite-dimensional optimization problems occur throughout the
mathematical sciences. The majority of these problems cannot be
solved analytically. This introduction to optimization attempts to
strike a balance between presentation of mathematical theory and
development of numerical algorithms. Building on students' skills
in calculus and linear algebra, the text provides a rigorous
exposition without undue abstraction. Its stress on statistical
applications will be especially appealing to graduate students of
statistics and biostatistics. The intended audience also includes
students in applied mathematics, computational biology, computer
science, economics, and physics who want to see rigorous
mathematics combined with real applications. In this second edition
the emphasis remains on finite-dimensional optimization. New
material has been added on the MM algorithm, block descent and
ascent, and the calculus of variations. Convex calculus is now
treated in much greater depth. Advanced topics such as the Fenchel
conjugate, subdifferentials, duality, feasibility, alternating
projections, projected gradient methods, exact penalty methods, and
Bregman iteration will equip students with the essentials for
understanding modern data mining techniques in high dimensions.
Finite-dimensional optimization problems occur throughout the
mathematical sciences. The majority of these problems cannot be
solved analytically. This introduction to optimization attempts to
strike a balance between presentation of mathematical theory and
development of numerical algorithms. Building on students' skills
in calculus and linear algebra, the text provides a rigorous
exposition without undue abstraction. Its stress on statistical
applications will be especially appealing to graduate students of
statistics and biostatistics. The intended audience also includes
students in applied mathematics, computational biology, computer
science, economics, and physics who want to see rigorous
mathematics combined with real applications. In this second edition
the emphasis remains on finite-dimensional optimization. New
material has been added on the MM algorithm, block descent and
ascent, and the calculus of variations. Convex calculus is now
treated in much greater depth. Advanced topics such as the Fenchel
conjugate, subdifferentials, duality, feasibility, alternating
projections, projected gradient methods, exact penalty methods, and
Bregman iteration will equip students with the essentials for
understanding modern data mining techniques in high dimensions.
Applied Probability presents a unique blend of theory and
applications, with special emphasis on mathematical modeling,
computational techniques, and examples from the biological
sciences. It can serve as a textbook for graduate students in
applied mathematics, biostatistics, computational biology, computer
science, physics, and statistics. Readers should have a working
knowledge of multivariate calculus, linear algebra, ordinary
differential equations, and elementary probability theory.
Chapter 1 reviews elementary probability and provides a brief
survey of relevant results from measure theory. Chapter 2 is an
extended essay on calculating expectations. Chapter 3 deals with
probabilistic applications of convexity, inequalities, and
optimization theory. Chapters 4 and 5 touch on combinatorics and
combinatorial optimization. Chapters 6 through 11 present core
material on stochastic processes. If supplemented with appropriate
sections from Chapters 1 and 2, there is sufficient material for a
traditional semester-long course in stochastic processes covering
the basics of Poisson processes, Markov chains, branching
processes, martingales, and diffusion processes. The second edition
adds two new chapters on asymptotic and numerical methods and an
appendix that separates some of the more delicate mathematical
theory from the steady flow of examples in the main text.
Besides the two new chapters, the second edition includes a more
extensive list of exercises, many additions to the exposition of
combinatorics, new material on rates of convergence to equilibrium
in reversible Markov chains, a discussion of basic reproduction
numbers in population modeling, and better coverage of Brownian
motion. Because many chapters are nearly self-contained,
mathematical scientists from a variety of backgrounds will find
Applied Probability 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 equips 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 most relevant to statisticians. In this
second edition, the material on optimization has been completely
rewritten. There is now an entire chapter on the MM algorithm in
addition to more comprehensive treatments of constrained
optimization, penalty and barrier methods, and model selection via
the lasso. There is also new material on the Cholesky
decomposition, Gram-Schmidt orthogonalization, the QR
decomposition, the singular value decomposition, and reproducing
kernel Hilbert spaces. The discussions of the bootstrap,
permutation testing, independent Monte Carlo, and hidden Markov
chains are updated, and a new chapter on advanced MCMC topics
introduces students to Markov random fields, reversible jump MCMC,
and convergence analysis in Gibbs sampling. Numerical Analysis for
Statisticians can serve as a graduate text for a course surveying
computational statistics. With a careful selection of topics and
appropriate supplementation, it can be used at the undergraduate
level. It contains enough material for a graduate course on
optimization theory. Because many chapters are nearly
self-contained, professional statisticians will also find the book
useful as a reference.
Written to equip students in the mathematical siences to understand
and model the epidemiological and experimental data encountered in
genetics research. This second edition expands the original edition
by over 100 pages and includes new material. Sprinkled throughout
the chapters are many new problems.
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.
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. 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 Pharmacia & Upjohn Foundation Professor of Biostatistics. His research interests include human genetics, population modeling, biomedical imaging, computational statistics, and applied stochastic processes. Springer-Verlag published his book Numerical Analysis for Statisticians in 1999.
Applied Probability presents a unique blend of theory and
applications, with special emphasis on mathematical modeling,
computational techniques, and examples from the biological
sciences. It can serve as a textbook for graduate students in
applied mathematics, biostatistics, computational biology, computer
science, physics, and statistics. Readers should have a working
knowledge of multivariate calculus, linear algebra, ordinary
differential equations, and elementary probability theory.
Chapter 1 reviews elementary probability and provides a brief
survey of relevant results from measure theory. Chapter 2 is an
extended essay on calculating expectations. Chapter 3 deals with
probabilistic applications of convexity, inequalities, and
optimization theory. Chapters 4 and 5 touch on combinatorics and
combinatorial optimization. Chapters 6 through 11 present core
material on stochastic processes. If supplemented with appropriate
sections from Chapters 1 and 2, there is sufficient material for a
traditional semester-long course in stochastic processes covering
the basics of Poisson processes, Markov chains, branching
processes, martingales, and diffusion processes. The second edition
adds two new chapters on asymptotic and numerical methods and an
appendix that separates some of the more delicate mathematical
theory from the steady flow of examples in the main text.
Besides the two new chapters, the second edition includes a more
extensive list of exercises, many additions to the exposition of
combinatorics, new material on rates of convergence to equilibrium
in reversible Markov chains, a discussion of basic reproduction
numbers in population modeling, and better coverage of Brownian
motion. Because many chapters are nearly self-contained,
mathematical scientists from a variety of backgrounds will find
Applied Probability useful as a reference
Offers an overview of the MM principle, a device for deriving
optimization algorithms satisfying the ascent or descent property.
These algorithms can: Separate the variables of a problem. Avoid
large matrix inversions. Linearize a problem. Restore symmetry.
Deal with equality and inequality constraints gracefully. Turn a
non-differentiable problem into a smooth problem. The author:
Presents the first extended treatment of MM algorithms, which are
ideal for high-dimensional optimization problems in data mining,
imaging, and genomics. Derives numerous algorithms from a broad
diversity of application areas, with a particular emphasis on
statistics, biology, and data mining. Summarizes a large amount of
literature that has not reached book form before.
Algorithms are a dominant force in modern culture, and every
indication is that they will become more pervasive, not less. The
best algorithms are undergirded by beautiful mathematics. This text
cuts across discipline boundaries to highlight some of the most
famous and successful algorithms. Readers are exposed to the
principles behind these examples and guided in assembling complex
algorithms from simpler building blocks. Algorithms from THE BOOK:
Incorporates Julia code for easy experimentation. Is written in
clear, concise prose consistent with mathematical rigour. Includes
a large number of classroom-tested exercises at the end of each
chapter. Covers background material, often omitted from
undergraduate courses, in the appendices. This textbook is aimed at
first-year graduate and advanced undergraduate students. It will
also serve as a convenient reference for professionals throughout
the mathematical sciences, physical sciences, engineering, and the
quantitative sectors of the biological and social sciences.
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