The mathematical theory of ondelettes (wavelets) was developed by Yves Meyer and many collaborators about 10 years ago. It was designed for ap proximation of possibly irregular functions and surfaces and was successfully applied in data compression, turbulence analysis, image and signal process ing. Five years ago wavelet theory progressively appeared to be a power ful framework for nonparametric statistical problems. Efficient computa tional implementations are beginning to surface in this second lustrum of the nineties. This book brings together these three main streams of wavelet theory. It presents the theory, discusses approximations and gives a variety of statistical applications. It is the aim of this text to introduce the novice in this field into the various aspects of wavelets. Wavelets require a highly interactive computing interface. We present therefore all applications with software code from an interactive statistical computing environment. Readers interested in theory and construction of wavelets will find here in a condensed form results that are somewhat scattered around in the research literature. A practioner will be able to use wavelets via the available software code. We hope therefore to address both theory and practice with this book and thus help to construct bridges between the different groups of scientists. This te. xt grew out of a French-German cooperation (Seminaire Paris Berlin, Seminar Berlin-Paris). This seminar brings together theoretical and applied statisticians from Berlin and Paris. This work originates in the first of these seminars organized in Garchy, Burgundy in 1994.

Despite its short history, wavelet theory has found applications in a remarkable diversity of disciplines: mathematics, physics, numerical analysis, signal processing, probability theory and statistics. The abundance of intriguing and useful features enjoyed by wavelet and wavelet packed transforms has led to their application to a wide range of statistical and signal processing problems. On November 16-18, 1994, a conference on Wavelets and Statistics was held at Villard de Lans, France, organized by the Institute IMAG-LMC, Grenoble, France. The meeting was the 15th in the series of the Rencontres Pranco-Belges des 8tatisticiens and was attended by 74 mathematicians from 12 different countries. Following tradition, both theoretical statistical results and practical contributions of this active field of statistical research were presented. The editors and the local organizers hope that this volume reflects the broad spectrum of the conference. as it includes 21 articles contributed by specialists in various areas in this field. The material compiled is fairly wide in scope and ranges from the development of new tools for non parametric curve estimation to applied problems, such as detection of transients in signal processing and image segmentation. The articles are arranged in alphabetical order by author rather than subject matter. However, to help the reader, a subjective classification of the articles is provided at the end of the book. Several articles of this volume are directly or indirectly concerned with several as pects of wavelet-based function estimation and signal denoising.

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.

The last 15 years have seen an explosion of interest in wavelets with applications in fields such as image compression, turbulence, human vision, radar and earthquake prediction. Wavelets represent an area that combines signal in image processing, mathematics, physics and electrical engineering. As such, this title is intended for the wide audience that is interested in mastering the basic techniques in this subject area, such as decomposition and compression.

Wavelets from a Statistical Perspective offers a modern, 2nd generation look on wavelets, far beyond the rigid setting of the equispaced, dyadic wavelets in the early days. With the methods of this book, based on the lifting scheme, researchers can set up a wavelet or another multiresolution analysis adapted to their data, ranging from images to scattered data or other irregularly spaced observations. Whereas classical wavelets stand a bit apart from other nonparametric methods, this book adds a multiscale touch to your spline, kernel or local polynomial smoothing procedure, thereby extending its applicability to nonlinear, nonparametric processing for piecewise smooth data. One of the chapters of the book constructs B-spline wavelets on nonequispaced knots and multiscale local polynomial transforms. In another chapter, the link between wavelets and Fourier analysis, ubiquitous in the classical approach, is explained, but without being inevitable. In further chapters the discrete wavelet transform is contrasted with the continuous version, the nondecimated (or maximal overlap) transform taking an intermediate position. An important principle in designing a wavelet analysis through the lifting scheme is finding the right balance between bias and variance. Bias and variance also play a crucial role in the nonparametric smoothing in a wavelet framework, in finding well working thresholds or other smoothing parameters. The numerous illustrations can be reproduced with the online available, accompanying software. The software and the exercises can also be used as a starting point in the further exploration of the material.

Wavelets are transforming current thinking in a wide range of fields by allowing for intermittent information and non- homogeneous behaviour. This book examines their increasing use and potential in many areas, including physical systems, turbulence, statistics, mechanical engineering, neural networks, physiology, vision engineering, signal processing, economics and astronomy. It is a must for specialists and non specialists alike.

Marking a distinct departure from the perspectives of frame theory and discrete transforms, this book provides a comprehensive mathematical and algorithmic introduction to wavelet theory. As such, it can be used as either a textbook or reference guide. As a textbook for graduate mathematics students and beginning researchers, it offers detailed information on the basic theory of framelets and wavelets, complemented by self-contained elementary proofs, illustrative examples/figures, and supplementary exercises. Further, as an advanced reference guide for experienced researchers and practitioners in mathematics, physics, and engineering, the book addresses in detail a wide range of basic and advanced topics (such as multiwavelets/multiframelets in Sobolev spaces and directional framelets) in wavelet theory, together with systematic mathematical analysis, concrete algorithms, and recent developments in and applications of framelets and wavelets. Lastly, the book can also be used to teach on or study selected special topics in approximation theory, Fourier analysis, applied harmonic analysis, functional analysis, and wavelet-based signal/image processing.

These notes represent our summary of much of the recent research that has been done in recent years on approximations and bounds that have been developed for compound distributions and related quantities which are of interest in insurance and other areas of application in applied probability. The basic technique employed in the derivation of many bounds is induc tive, an approach that is motivated by arguments used by Sparre-Andersen (1957) in connection with a renewal risk model in insurance. This technique is both simple and powerful, and yields quite general results. The bounds themselves are motivated by the classical Lundberg exponential bounds which apply to ruin probabilities, and the connection to compound dis tributions is through the interpretation of the ruin probability as the tail probability of a compound geometric distribution. The initial exponential bounds were given in Willmot and Lin (1994), followed by the nonexpo nential generalization in Willmot (1994). Other related work on approximations for compound distributions and applications to various problems in insurance in particular and applied probability in general is also discussed in subsequent chapters. The results obtained or the arguments employed in these situations are similar to those for the compound distributions, and thus we felt it useful to include them in the notes. In many cases we have included exact results, since these are useful in conjunction with the bounds and approximations developed.

An Introduction to Wavelets and Other Filtering Methods in Finance and Economics presents a unified view of filtering techniques with a special focus on wavelet analysis in finance and economics. It emphasizes the methods and explanations of the theory that underlies them. It also concentrates on exactly what wavelet analysis (and filtering methods in general) can reveal about a time series. It offers testing issues which can be performed with wavelets in conjunction with the multi-resolution analysis. The descriptive focus of the book avoids proofs and provides easy access to a wide spectrum of parametric and nonparametric filtering methods. Examples and empirical applications will show readers the capabilities, advantages, and disadvantages of each method. - The first book to present a unified view of filtering techniques - Concentrates on exactly what wavelets analysis and filtering methods in general can reveal about a time series - Provides easy access to a wide spectrum of parametric and non-parametric filtering methods

The Handbook of Computational Statistics - Concepts and Methods (second edition) is a revision of the first edition published in 2004, and contains additional comments and updated information on the existing chapters, as well as three new chapters addressing recent work in the field of computational statistics. This new edition is divided into 4 parts in the same way as the first edition. It begins with "How Computational Statistics became the backbone of modern data science" (Ch.1): an overview of the field of Computational Statistics, how it emerged as a separate discipline, and how its own development mirrored that of hardware and software, including a discussion of current active research. The second part (Chs. 2 - 15) presents several topics in the supporting field of statistical computing. Emphasis is placed on the need for fast and accurate numerical algorithms, and some of the basic methodologies for transformation, database handling, high-dimensional data and graphics treatment are discussed. The third part (Chs. 16 - 33) focuses on statistical methodology. Special attention is given to smoothing, iterative procedures, simulation and visualization of multivariate data. Lastly, a set of selected applications (Chs. 34 - 38) like Bioinformatics, Medical Imaging, Finance, Econometrics and Network Intrusion Detection highlight the usefulness of computational statistics in real-world applications.