Essential Wavelets for Statistical Applications and Data Analysis

ISBN-13: 9781461268765 / Angielski / Miękka / 2012 / 206 str.

Todd Ogden; Todd Ogdenglish

Opis

I once heard the book by Meyer (1993) described as a "vulgarization" of wavelets. While this is true in one sense of the word, that of making a sub- ject popular (Meyer's book is one of the early works written with the non- specialist in mind), the implication seems to be that such an attempt some- how cheapens or coarsens the subject. I have to disagree that popularity goes hand-in-hand with debasement. is certainly a beautiful theory underlying wavelet analysis, there is While there plenty of beauty left over for the applications of wavelet methods. This book is also written for the non-specialist, and therefore its main thrust is toward wavelet applications. Enough theory is given to help the reader gain a basic understanding of how wavelets work in practice, but much of the theory can be presented using only a basic level of mathematics. Only one theorem is for- mally stated in this book, with only one proof. And these are only included to introduce some key concepts in a natural way.

Recenzje

"An accessible introductory survey of new wavelet-analysis tools and the way they can be applied to fundamental data-analysis problems... [The author] gives only the necessary mathematics for a good understanding of wavelets and...how to apply them. A variety of problems in statistics is discussed in a nontheoretical style. The author also reviews some of the ways wavelets have been applied in various fields and considers how specific properties of wavelets in these fields can be exploited in statistical analysis. For many of the statistical problems mentioned in the book, more than one methodology is discussed. Moreover, the author discusses the relative advantages and disadvantages of each competing method in order to guide the analyst in choosing the best method suited for his situation." -Metrika

"The clear and intuitive presentation makes the book ideal for a broad audience." -Zentralblatt Math

"The material is organized and presented in a way which makes the book suitable for self-education... The reader must only be familiar with a basic knowledge of calculus, linear algebra, and basic statistical theory.... Introducing the concepts in an accessible and intuitive form, accompanied by a lot of illustrative examples, graphics, and applications, the work is a useful book not only for graduates and professionals in statistics, but for all scientists and engineers who use data analysis methods." -Mathematica Tome

"The book is...an introduction to [wavelets'] successful applications in statistics and data analysis. Only a limited knowledge of calculus, linear algebra, and elementary statistics is assumed. The book is thus accessible to advanced undergraduate students and graduate students as well as to applied statisticians and engineers concerned with data treatment and analysis. An appendix on vector spaces, a glossary of terms and notation, an index, and a long list of references close the book, which will certainly be appreciated by a large circle of readers for its easy style, many included examples, and elucidating discussion of problems presented." -Applications of Mathematics

"The present book can be certainly recommended to those interested in statistical applications.... [The] approach needed for statistical applications is provided here quite well requiring only basics of statistical theory and familiarity with calculus and linear algebra.... This reader finds Chapters 7 and 8 especially useful for statisticians. What are statisticians doing with wavelets currently is discussed here." -The Journal of the Indian Institute of Science

"The aim of the book is to present an accessible introductory survey of new tools of wavelet analysis and how they can be applied to basic data analysis problems such as signal processing, image analysis, data compression, etc. The author shows how these problems can be solved by fast algorithms which are of a simple form. Solutions of many practical examples are presented.... The book is ideal for a braod audience including advanced undergraduate students and graduates.... Also, professionals in statistics, researchers, and engineers who use methods of data analysis and their applications in statistics will learn about new wavelet methods." -Mathematica Bohemica 

"Accessible, clearly presented background material. Examples presented throughout the book. Variety of statistical applications. Intuitive style of presentation. Website for the book with additional resources includes S-plus software code functions for graphics used in the book." -L'enseignement Mathématique

"As an accessible work that explains the decomposition and reconstruction of wavelet algorithms as applied to statistical data, Ogden's book is one of the growing number on wavelets. But this book is in a class of its own: it concentrates solely on the application of wavelets to statistics and data analysis. Ogden gives a brief introduction to basic theory, and then provides generous examples of wavelets and how they are constructed. He briefly compares Fourier methods, then discusses statistical applications, such as density estimation, estimation of regression functions through the application of the convolution-type kernel function, statistical testing, and Bayesian methods.... The book could conveniently be used by scientists who wish to apply wavelet methods to data analysis and by graduate students as a supplementary text. Recommended. Graduates through faculty and professionals."   -Choice

Spis treści

1 Wavelets: A Brief Introduction.- 1.1 The Discrete Fourier Transform.- 1.2 The Haar System.- Multiresolution Analysis.- The Wavelet Representation.- Goals of Multiresolution Analysis.- 1.3 Smoother Wavelet Bases.- 2 Basic Smoothing Techniques.- 2.1 Density Estimation.- Histograms.- Kernel Estimation.- Orthogonal Series Estimation.- 2.2 Estimation of a Regression Function.- Kernel Regression.- Orthogonal Series Estimation.- 2.3 Kernel Representation of Orthogonal Series Estimators.- 3 Elementary Statistical Applications.- 3.1 Density Estimation.- Haar-Based Histograms.- Estimation with Smoother Wavelets.- 3.2 Nonparametric Regression.- 4 Wavelet Features and Examples.- 4.1 Wavelet Decomposition and Reconstruction.- Two-Scale Relationships.- The Decomposition Algorithm.- The Reconstruction Algorithm.- 4.2 The Filter Representation.- 4.3 Time-Frequency Localization.- The Continuous Fourier Transform.- The Windowed Fourier Transform.- The Continuous Wavelet Transform.- 4.4 Examples of Wavelets and Their Constructions.- Orthogonal Wavelets.- Biorthogonal Wavelets.- Semiorthogonal Wavelets.- 5 Wavelet-based Diagnostics.- 5.1 Multiresolution Plots.- 5.2 Time-Scale Plots.- 5.3 Plotting Wavelet Coefficients.- 5.4 Other Plots for Data Analysis.- 6 Some Practical Issues.- 6.1 The Discrete Fourier Transform of Data.- The Fourier Transform of Sampled Signals.- The Fast Fourier Transform.- 6.2 The Wavelet Transform of Data.- 6.3 Wavelets on an Interval.- Periodic Boundary Handling.- Symmetric and Antisymmetric Boundary Handling.- Meyer Boundary Wavelets.- Orthogonal Wavelets on the Interval.- 6.4 When the Sample Size is Not a Power of Two.- 7 Other Applications.- 7.1 Selective Wavelet Reconstruction.- Wavelet Thresholding.- Spatial Adaptivity.- Global Thresholding.- Estimation of the Noise Level.- 7.2 More Density Estimation.- 7.3 Spectral Density Estimation.- 7.4 Detections of Jumps and Cusps.- 8 Data Adaptive Wavelet Thresholding.- 8.1 SURE Thresholding.- 8.2 Threshold Selection by Hypothesis Testing.- Recursive Testing.- Minimizing False Discovery.- 8.3 Cross-Validation Methods.- 8.4 Bayesian Methods.- 9 Generalizations and Extensions.- 9.1 Two-Dimensional Wavelets.- 9.2 Wavelet Packets.- Wavelet Packet Functions.- The Best Basis Algorithm.- 9.3 Translation Invariant Wavelet Smoothing.- References.- Glossary of Notation.- Glossary of Terms.