Reproducing Kernel Hilbert Spaces in Probability and Statistics


Author: Alain Berlinet,Christine Thomas-Agnan
Publisher: Springer Science & Business Media
ISBN: 1441990968
Category: Business & Economics
Page: 355
View: 7888

Continue Reading →

The book covers theoretical questions including the latest extension of the formalism, and computational issues and focuses on some of the more fruitful and promising applications, including statistical signal processing, nonparametric curve estimation, random measures, limit theorems, learning theory and some applications at the fringe between Statistics and Approximation Theory. It is geared to graduate students in Statistics, Mathematics or Engineering, or to scientists with an equivalent level.

Reproducing Kernel Hilbert Spaces in Probability and Statistics


Author: Alain Berlinet,Christine Thomas-Agnan
Publisher: Springer
ISBN: 9781461347927
Category: Business & Economics
Page: 355
View: 7016

Continue Reading →

The book covers theoretical questions including the latest extension of the formalism, and computational issues and focuses on some of the more fruitful and promising applications, including statistical signal processing, nonparametric curve estimation, random measures, limit theorems, learning theory and some applications at the fringe between Statistics and Approximation Theory. It is geared to graduate students in Statistics, Mathematics or Engineering, or to scientists with an equivalent level.

Reproducing Kernel Hilbert Spaces in Probability and Statistics


Author: Alain Berlinet,Christine Thomas-Agnan
Publisher: Springer
ISBN: 1402076797
Category: Business & Economics
Page: 355
View: 2210

Continue Reading →

The book covers theoretical questions including the latest extension of the formalism, and computational issues and focuses on some of the more fruitful and promising applications, including statistical signal processing, nonparametric curve estimation, random measures, limit theorems, learning theory and some applications at the fringe between Statistics and Approximation Theory. It is geared to graduate students in Statistics, Mathematics or Engineering, or to scientists with an equivalent level.

An Introduction to the Theory of Reproducing Kernel Hilbert Spaces


Author: Vern I. Paulsen,Mrinal Raghupathi
Publisher: Cambridge University Press
ISBN: 1316558738
Category: Mathematics
Page: N.A
View: 2111

Continue Reading →

Reproducing kernel Hilbert spaces have developed into an important tool in many areas, especially statistics and machine learning, and they play a valuable role in complex analysis, probability, group representation theory, and the theory of integral operators. This unique text offers a unified overview of the topic, providing detailed examples of applications, as well as covering the fundamental underlying theory, including chapters on interpolation and approximation, Cholesky and Schur operations on kernels, and vector-valued spaces. Self-contained and accessibly written, with exercises at the end of each chapter, this unrivalled treatment of the topic serves as an ideal introduction for graduate students across mathematics, computer science, and engineering, as well as a useful reference for researchers working in functional analysis or its applications.

Theory of Reproducing Kernels and Applications


Author: Saburou Saitoh,Yoshihiro Sawano
Publisher: Springer
ISBN: 9811005303
Category: Mathematics
Page: 452
View: 5896

Continue Reading →

This book provides a large extension of the general theory of reproducing kernels published by N. Aronszajn in 1950, with many concrete applications.In Chapter 1, many concrete reproducing kernels are first introduced with detailed information. Chapter 2 presents a general and global theory of reproducing kernels with basic applications in a self-contained way. Many fundamental operations among reproducing kernel Hilbert spaces are dealt with. Chapter 2 is the heart of this book.Chapter 3 is devoted to the Tikhonov regularization using the theory of reproducing kernels with applications to numerical and practical solutions of bounded linear operator equations.In Chapter 4, the numerical real inversion formulas of the Laplace transform are presented by applying the Tikhonov regularization, where the reproducing kernels play a key role in the results.Chapter 5 deals with ordinary differential equations; Chapter 6 includes many concrete results for various fundamental partial differential equations. In Chapter 7, typical integral equations are presented with discretization methods. These chapters are applications of the general theories of Chapter 3 with the purpose of practical and numerical constructions of the solutions.In Chapter 8, hot topics on reproducing kernels are presented; namely, norm inequalities, convolution inequalities, inversion of an arbitrary matrix, representations of inverse mappings, identifications of nonlinear systems, sampling theory, statistical learning theory and membership problems. Relationships among eigen-functions, initial value problems for linear partial differential equations, and reproducing kernels are also presented. Further, new fundamental results on generalized reproducing kernels, generalized delta functions, generalized reproducing kernel Hilbert spaces, andas well, a general integral transform theory are introduced.In three Appendices, the deep theory of Akira Yamada discussing the equality problems in nonlinear norm inequalities, Yamada's unified and generalized inequalities for Opial's inequalities and the concrete and explicit integral representation of the implicit functions are presented.

Theoretical Foundations of Functional Data Analysis, with an Introduction to Linear Operators


Author: Tailen Hsing,Randall Eubank
Publisher: John Wiley & Sons
ISBN: 0470016914
Category: Mathematics
Page: 480
View: 771

Continue Reading →

Functional data is data in the form of curves that is becoming a popular method for interpreting scientific data. Statistical Analysis of Functional Data provides an authoritative account of function data analysis covering its foundations, theory, methodology, and practical implementation. It also contains examples taken from a wide range of disciplines, including finance, medicine, and psychology. The book includes a supporting Web site hosting the real data sets analyzed in the book and related software. Statistical researchers or practitioners analyzing functional data will find this book useful.

Support Vector Machines


Author: Ingo Steinwart,Andreas Christmann
Publisher: Springer Science & Business Media
ISBN: 0387772421
Category: Computers
Page: 601
View: 4319

Continue Reading →

Every mathematical discipline goes through three periods of development: the naive, the formal, and the critical. David Hilbert The goal of this book is to explain the principles that made support vector machines (SVMs) a successful modeling and prediction tool for a variety of applications. We try to achieve this by presenting the basic ideas of SVMs together with the latest developments and current research questions in a uni?ed style. In a nutshell, we identify at least three reasons for the success of SVMs: their ability to learn well with only a very small number of free parameters, their robustness against several types of model violations and outliers, and last but not least their computational e?ciency compared with several other methods. Although there are several roots and precursors of SVMs, these methods gained particular momentum during the last 15 years since Vapnik (1995, 1998) published his well-known textbooks on statistical learning theory with aspecialemphasisonsupportvectormachines. Sincethen,the?eldofmachine learninghaswitnessedintenseactivityinthestudyofSVMs,whichhasspread moreandmoretootherdisciplinessuchasstatisticsandmathematics. Thusit seems fair to say that several communities are currently working on support vector machines and on related kernel-based methods. Although there are many interactions between these communities, we think that there is still roomforadditionalfruitfulinteractionandwouldbegladifthistextbookwere found helpful in stimulating further research. Many of the results presented in this book have previously been scattered in the journal literature or are still under review. As a consequence, these results have been accessible only to a relativelysmallnumberofspecialists,sometimesprobablyonlytopeoplefrom one community but not the others.

Hilbert Space Methods in Probability and Statistical Inference


Author: Christopher G. Small,Don L. McLeish
Publisher: John Wiley & Sons
ISBN: 1118165535
Category: Mathematics
Page: 270
View: 7094

Continue Reading →

Explains how Hilbert space techniques cross the boundaries into the foundations of probability and statistics. Focuses on the theory of martingales stochastic integration, interpolation and density estimation. Includes a copious amount of problems and examples.

Smoothing Splines

Methods and Applications
Author: Yuedong Wang
Publisher: CRC Press
ISBN: 1420077562
Category: Mathematics
Page: 384
View: 1265

Continue Reading →

A general class of powerful and flexible modeling techniques, spline smoothing has attracted a great deal of research attention in recent years and has been widely used in many application areas, from medicine to economics. Smoothing Splines: Methods and Applications covers basic smoothing spline models, including polynomial, periodic, spherical, thin-plate, L-, and partial splines, as well as more advanced models, such as smoothing spline ANOVA, extended and generalized smoothing spline ANOVA, vector spline, nonparametric nonlinear regression, semiparametric regression, and semiparametric mixed-effects models. It also presents methods for model selection and inference. The book provides unified frameworks for estimation, inference, and software implementation by using the general forms of nonparametric/semiparametric, linear/nonlinear, and fixed/mixed smoothing spline models. The theory of reproducing kernel Hilbert space (RKHS) is used to present various smoothing spline models in a unified fashion. Although this approach can be technical and difficult, the author makes the advanced smoothing spline methodology based on RKHS accessible to practitioners and students. He offers a gentle introduction to RKHS, keeps theory at a minimum level, and explains how RKHS can be used to construct spline models. Smoothing Splines offers a balanced mix of methodology, computation, implementation, software, and applications. It uses R to perform all data analyses and includes a host of real data examples from astronomy, economics, medicine, and meteorology. The codes for all examples, along with related developments, can be found on the book’s web page.

Hilbert Space Methods in Signal Processing


Author: Rodney A. Kennedy,Parastoo Sadeghi
Publisher: Cambridge University Press
ISBN: 1107010039
Category: Mathematics
Page: 420
View: 1894

Continue Reading →

An accessible introduction to Hilbert spaces, combining the theory with applications of Hilbert methods in signal processing.

Spline Models for Observational Data


Author: Grace Wahba
Publisher: SIAM
ISBN: 0898712440
Category: Mathematics
Page: 169
View: 1801

Continue Reading →

This book serves well as an introduction into the more theoretical aspects of the use of spline models. It develops a theory and practice for the estimation of functions from noisy data on functionals. The simplest example is the estimation of a smooth curve, given noisy observations on a finite number of its values. Convergence properties, data based smoothing parameter selection, confidence intervals, and numerical methods are established which are appropriate to a number of problems within this framework. Methods for including side conditions and other prior information in solving ill posed inverse problems are provided. Data which involves samples of random variables with Gaussian, Poisson, binomial, and other distributions are treated in a unified optimization context. Experimental design questions, i.e., which functionals should be observed, are studied in a general context. Extensions to distributed parameter system identification problems are made by considering implicitly defined functionals.

Learning with Kernels

Support Vector Machines, Regularization, Optimization, and Beyond
Author: Bernhard Schölkopf,Alexander J. Smola
Publisher: MIT Press
ISBN: 9780262194754
Category: Computers
Page: 626
View: 7864

Continue Reading →

A comprehensive introduction to Support Vector Machines and related kernel methods.

Modern Mathematics and Applications in Computer Graphics and Vision


Author: Hongyu Guo
Publisher: World Scientific Publishing Company
ISBN: 9814449350
Category: Computers
Page: 524
View: 9124

Continue Reading →

This book presents a concise exposition of modern mathematical concepts, models and methods with applications in computer graphics, vision and machine learning. The compendium is organized in four parts — Algebra, Geometry, Topology, and Applications. One of the features is a unique treatment of tensor and manifold topics to make them easier for the students. All proofs are omitted to give an emphasis on the exposition of the concepts. Effort is made to help students to build intuition and avoid parrot-like learning. There is minimal inter-chapter dependency. Each chapter can be used as an independent crash course and the reader can start reading from any chapter — almost. This book is intended for upper level undergraduate students, graduate students and researchers in computer graphics, geometric modeling, computer vision, pattern recognition and machine learning. It can be used as a reference book, or a textbook for a selected topics course with the instructor's choice of any of the topics.

Kernel-based Approximation Methods using MATLAB


Author: Gregory Fasshauer,Michael McCourt
Publisher: World Scientific Publishing Company
ISBN: 9814630152
Category: Mathematics
Page: 536
View: 4038

Continue Reading →

In an attempt to introduce application scientists and graduate students to the exciting topic of positive definite kernels and radial basis functions, this book presents modern theoretical results on kernel-based approximation methods and demonstrates their implementation in various settings. The authors explore the historical context of this fascinating topic and explain recent advances as strategies to address long-standing problems. Examples are drawn from fields as diverse as function approximation, spatial statistics, boundary value problems, machine learning, surrogate modeling and finance. Researchers from those and other fields can recreate the results within using the documented MATLAB code, also available through the online library. This combination of a strong theoretical foundation and accessible experimentation empowers readers to use positive definite kernels on their own problems of interest.

A Primer on Reproducing Kernel Hilbert Spaces


Author: Jonathan H. Manton,Pierre-Olivier Amblard
Publisher: N.A
ISBN: 9781680830927
Category: Technology & Engineering
Page: 138
View: 2652

Continue Reading →

Hilbert space theory is an invaluable mathematical tool in numerous signal processing and systems theory applications. Hilbert spaces satisfying certain additional properties are known as Reproducing Kernel Hilbert Spaces (RKHSs). This primer gives a gentle and novel introduction to RKHS theory. It also presents several classical applications. It concludes by focusing on recent developments in the machine learning literature concerning embeddings of random variables. Parenthetical remarks are used to provide greater technical detail, which some readers may welcome, but they may be ignored without compromising the cohesion of the primer. Proofs are there for those wishing to gain experience at working with RKHSs; simple proofs are preferred to short, clever, but otherwise uninformative proofs. Italicised comments appearing in proofs provide intuition or orientation or both. A Primer on Reproducing Kernel Hilbert Spaces empowers readers to recognize when and how RKHS theory can profit them in their own work.

Numerical Analysis for Statisticians


Author: Kenneth Lange
Publisher: Springer Science & Business Media
ISBN: 1441959459
Category: Business & Economics
Page: 600
View: 9695

Continue Reading →

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.

Smoothing Spline ANOVA Models


Author: Chong Gu
Publisher: Springer Science & Business Media
ISBN: 1475736835
Category: Mathematics
Page: 290
View: 3446

Continue Reading →

Smoothing methods are an active area of research. In this book, the author presents a comprehensive treatment of penalty smoothing under a unified framework. Methods are developed for (i) regression with Gaussian and non-Gaussian responses as well as with censored life time data; (ii) density and conditional density estimation under a variety of sampling schemes; and (iii) hazard rate estimation with censored life time data and covariates. Extensive discussions are devoted to model construction, smoothing parameter selection, computation, and asymptotic convergence. Most of the computational and data analytical tools discussed in the book are implemented in R, an open-source clone of the popular S/S- PLUS language.

Learning Theory

An Approximation Theory Viewpoint
Author: Felipe Cucker,Ding Xuan Zhou
Publisher: Cambridge University Press
ISBN: 1139462865
Category: Computers
Page: N.A
View: 3467

Continue Reading →

The goal of learning theory is to approximate a function from sample values. To attain this goal learning theory draws on a variety of diverse subjects, specifically statistics, approximation theory, and algorithmics. Ideas from all these areas blended to form a subject whose many successful applications have triggered a rapid growth during the last two decades. This is the first book to give a general overview of the theoretical foundations of the subject emphasizing the approximation theory, while still giving a balanced overview. It is based on courses taught by the authors, and is reasonably self-contained so will appeal to a broad spectrum of researchers in learning theory and adjacent fields. It will also serve as an introduction for graduate students and others entering the field, who wish to see how the problems raised in learning theory relate to other disciplines.

Digital Nets and Sequences

Discrepancy Theory and Quasi–Monte Carlo Integration
Author: Josef Dick,Friedrich Pillichshammer
Publisher: Cambridge University Press
ISBN: 1139490052
Category: Computers
Page: N.A
View: 2476

Continue Reading →

Indispensable for students, invaluable for researchers, this comprehensive treatment of contemporary quasi–Monte Carlo methods, digital nets and sequences, and discrepancy theory starts from scratch with detailed explanations of the basic concepts and then advances to current methods used in research. As deterministic versions of the Monte Carlo method, quasi–Monte Carlo rules have increased in popularity, with many fruitful applications in mathematical practice. These rules require nodes with good uniform distribution properties, and digital nets and sequences in the sense of Niederreiter are known to be excellent candidates. Besides the classical theory, the book contains chapters on reproducing kernel Hilbert spaces and weighted integration, duality theory for digital nets, polynomial lattice rules, the newest constructions by Niederreiter and Xing and many more. The authors present an accessible introduction to the subject based mainly on material taught in undergraduate courses with numerous examples, exercises and illustrations.