Series: Springer Series in Statistics
2009, Approx. 800 p., Hardcover
ISBN: 978-0-387-98134-5
Due: August 2009
This book is a thorough introduction to the most important topics in data mining and machine learning. It begins with a detailed review of classical function estimation and proceeds with chapters on nonlinear regression, classification, and ensemble methods. The final chapters focus on clustering, dimension reduction, variable selection, and multiple comparisons. All these topics have undergone extraordinarily rapid development in recent years and this treatment offers a modern perspective emphasizing the most recent contributions. The presentation of foundational results is detailed and includes many accessible proofs not readily available outside original sources. While the orientation is conceptual and theoretical, the main points are regularly reinforced by computational comparisons.
Intended primarily as a graduate level textbook for statistics, computer science, and electrical engineering students, this book assumes only a strong foundation in undergraduate statistics and mathematics, and facility with using R packages. The text has a wide variety of problems, many of an exploratory nature. There are numerous computed examples, complete with code, so that further computations can be carried out readily. The book also serves as a handbook for researchers who want a conceptual overview of the central topics in data mining and machine learning.
Variability, information, prediction.- Kernel smoothing.- Spline smoothing.- New wave nonparametrics.- Supervised learning: Partition methods.- Alternative nonparametrics.- Computational comparisons.- Unsupervised learning: Clustering.- Learning in high dimensions.- Variable selection.- Multiple testing.
Series: Statistics and Computing
2009, XXII, 727 p., Hardcover
ISBN: 978-0-387-98143-7
Due: September 2009
Computational inference has taken its place alongside asymptotic inference and exact techniques in the standard collection of statistical methods. Computational inference is based on an approach to statistical methods that uses modern computational power to simulate distributional properties of estimators and test statistics. This book describes computationally-intensive statistical methods in a unified presentation, emphasizing techniques, such as the PDF decomposition, that arise in a wide range of methods.
The book assumes an intermediate background in mathematics, computing, and applied and theoretical statistics. The first part of the book, consisting of a single long chapter, reviews this background material while introducing computationally-intensive exploratory data analysis and computational inference.
The six chapters in the second part of the book are on statistical computing. This part describes arithmetic in digital computers and how the nature of digital computations affects algorithms used in statistical methods. Building on the first chapters on numerical computations and algorithm design, the following chapters cover the main areas of statistical numerical analysis, that is, approximation of functions, numerical quadrature, numerical linear algebra, solution of nonlinear equations, optimization, and random number generation.
The third and fourth parts of the book cover methods of computational statistics, including Monte Carlo methods, randomization and cross validation, the bootstrap, probability density estimation, and statistical learning.
The book includes a large number of exercises with some solutions provided in an appendix.
Mathematical and statistical preliminaries.- Computer storage and arithmetic.- Algorithms and programming.- Approximation of functions and numerical quadrature.- Numerical linear algebra.- Solution of nonlinear equations and optimization.- Generation of random numbers.- Graphical methods in computational statistics.- Tools for identification of structure in data.- Estimation of functions.- Monte Carlo methods for statistical inference.- Data randomization, partitioning, and augmentation.- Bootstrap methods.- Estimation of probability density functions using parametric models.- Nonparametric estimation of probability density functions.- Statistical learning and data mining.- Statistical models of dependencies.
Series: Springer Texts in Statistics
2009, Approx. 315 p., Hardcover
ISBN: 978-1-4419-0161-3
Due: August 2009
The purpose of this book is to provide the reader with a solid background and understanding of the basic results and methods in probability theory before entering into more advanced courses. The first six chapters focus on some central areas of what might be called pure probability theory: multivariate random variables, conditioning, transforms, order variables, the multivariate normal distribution, and convergence. A final chapter is devoted to the Poisson process as a means both to introduce stochastic processes and to apply many of the techniques introduced earlier in the text.
Students are assumed to have taken a first course in probability, though no knowledge of measure theory is assumed. Throughout, the presentation is thorough and includes many examples that are discussed in detail. Thus, students considering more advanced research in probability theory will benefit from this wide-ranging survey of the subject that provides them with a foretaste of the subject's many treasures.
The present second edition offers updated content, one hundred additional problems for solution, and a new chapter that provides an outlook on further areas and topics, such as stable distributions and domains of attraction, extreme value theory and records, and martingales. The main idea is that this chapter may serve as an appetizer to the more advanced theory.
Introduction.- Multivariate Random Variables.- Conditioning.- Transforms.- Order statistics.- The multivariate normal distribution.- Convergence.- An outlook into further topics.- The Poisson Process.
Series: Universitext
2009, XX, 305 p. 17 illus., Softcover
ISBN: 978-0-387-89487-4
Due: September 2009
The first edition of Stochastic Partial Differential Equations: A Modeling, White Noise Functional Approach, gave a comprehensive introduction to SPDEs. In this, the second edition, the authors build on the theory of SPDEs driven by space-time Brownian motion, or more generally, space-time Levy process noise, and introduce new applications of the field.
Because the authors allow the noise to be in both space and time, the solutions to SPDEs are usually of the distribution type, rather than a classical random field. To make this study rigorous and as general as possible, the discussion of SPDEs is therefore placed in the context of Hida white noise theory. The key connection between white noise theory and SPDEs is that integration with respect to Brownian random fields can be expressed as integration with respect to the Lebesgue measure of the Wick product of the integrand with Brownian white noise, and similarly with Levy process.
The first part of the book deals with the classical Brownian motion case. The second extends it to the Levy white noise case. For SPDEs of the Wick type, a general solution method is given by means of the Hermite transform, which turns a given SPDE into a parameterized family of deterministic PDEs. Applications of this theory are emphasized throughout. The stochastic pressure equation for fluid flow in porous media is treated, as are applications to finance.
Preface to the Second Edition.- Preface to the First Edition.- Introduction.- Framework.- Applications to stochastic ordinary differential equations.- Stochastic partial differential equations driven by Brownian white noise.- Stochastic partial differential equations driven by Levy white noise.- Appendix A. The Bochner-Minlos theorem.- Appendix B. Stochastic calculus based on Brownian motion.- Appendix C. Properties of Hermite polynomials.- Appendix D. Independence of bases in Wick products.- Appendix E. Stochastic calculus based on Levy processes- References.- List of frequently used notation and symbols.- Index.
Series: Algorithms and Computation in Mathematics , Vol. 24
2009, Approx. 660 p., Hardcover
ISBN: 978-3-642-01286-0
Due: September 2009
The book provides a self-contained account of the formal theory of general, i.e. also under- and overdetermined, systems of differential equations which in its central notion of involution combines geometric, algebraic, homological and combinatorial ideas. It presents for the first time in book form the theory of Pommaret bases, a special kind of Grobner bases closely related to Koszul homology, and contains an extensive discussion of the existence and uniqueness of solutions of formally well-posed initial value problems and a novel presentation of Vessiot's dual version of the Cartan-Kahler theory. Special emphasis is put on a constructive approach leading to effective algorithms.
1 Introduction.- 2 Formal Geometry of Differential Equations.- 3 Involution I: Algebraic Theory.- 4 Completion to Involution.- 5 Structure Analysis of Polynomial Modules.- 6 Involution II: Homological Theory.- 7 Involution III: Differential Theory.- 8 The Size of the Formal Solution Space.- 9 Existence and Uniqueness of Solutions.- 10 Linear Differential Equations.- A Miscellaneous.- B Algebra.- C Differential Geometry.- References.- Glossary.- Index.
Series: Oberwolfach Seminars , Vol. 42
2009, Approx. 180 p., Softcover
ISBN: 978-3-0346-0068-2
Due: November 2009
First textbook which treats certain recent developments in the solution of variational inequalities, like a posteriori error analysis, adaptive discretisations including convergence analysis, and fast algebraic solvers
Combines all important aspects (analysis, discretisation, multigrid) into an exposition on the graduate or even undergraduate level, assuming some basic knowledge on partial differential equations and finite elements
Variational inequalities provide the mathematical framework for a variety of nonlinear and non-smooth phenomena in science and engineering. Efficient and reliable solution procedures are closely linked to the mathematics of function spaces and partial differential equations. The main subjects of this book are a posteriori error analysis and convergence of adaptive finite element methods, the treatment of multigrid methods and multilevel methods with solution-dependent multilevel bases. The discussion is centered around some model obstacle and contact problems and based upon introductory sections on the mathematics of variational inequalities and on finite element discretisation including short MATLAB programs
1.Variational Inequalities: An Introduction.- 2. Finite Element Discretisation of Variational Inequalities.- 3. Multigrid Methods for Variational Inequalities and Contact Problems in Linear Elasticity.