Book Details
| Author | Wolfgang Härdle, Gerard Kerkyacharian, Dominique Picard, Alexander Tsybakov |
| Publisher | Springer New York |
| Year | 1998 |
| Language | English |
| Pages | 265 (XVIII, 265) |
| Size | 1.54 MB |
| Extension | |
| Series | Lecture Notes in Statistics (Volume 129) |
| ISBN | 978-0-387-98453-7 (print) / 978-1-4612-2222-4 (eBook) |
Summary
Wavelets, Approximation, and Statistical Applications by Wolfgang Härdle, Gerard Kerkyacharian, Dominique Picard, and Alexander Tsybakov is a comprehensive introduction to the theory and application of wavelets in statistics and approximation. The book is part of the Lecture Notes in Statistics series (Volume 129) and was published by Springer in 1998.
The mathematical theory of wavelets was developed by Yves Meyer and his collaborators about ten years prior to the book's publication. Wavelets were originally designed for the approximation of irregular functions and surfaces, and they have since found successful applications in data compression, turbulence analysis, image processing, and signal processing[reference:0][reference:1]. This book brings together the three main streams of wavelet theory: the mathematical foundations, the approximation properties, and the statistical applications[reference:2].
The text begins with an introduction to wavelets and their basic properties, including the Haar basis wavelet system and the concept of multiresolution analysis[reference:3]. It then covers essential background material from Fourier analysis, which is crucial for understanding the construction and properties of wavelet bases[reference:4]. The authors present the basic relations of wavelet theory and guide the reader through the construction of various wavelet bases, including compactly supported wavelets[reference:5].
One of the key strengths of this book is its focus on practical applications. The authors recognize that wavelets require a highly interactive computing interface, and they provide software code from an interactive statistical computing environment to accompany the theoretical discussions[reference:6]. This makes the book valuable for practitioners who want to apply wavelet methods to real-world problems.
The book addresses both theory and practice, aiming to build bridges between different groups of scientists—from theoreticians to applied statisticians[reference:7]. It grew out of a French-German cooperation (Séminaire Paris-Berlin) that brought together theoretical and applied statisticians from Berlin and Paris[reference:8]. The material originated from the first of these seminars, organized in Garchy, Burgundy, in 1994[reference:9].
Readers will learn about the fundamental concepts of wavelet theory, including the Haar basis, multiresolution analysis, Fourier analysis, and the construction of wavelet bases. They will also gain an understanding of how wavelets can be used for nonparametric statistical problems, such as density estimation and regression. The book is suitable for graduate students, researchers, and practitioners in statistics, mathematics, engineering, and related fields who want to understand and apply wavelet methods.
Key Features
- Comprehensive coverage of wavelet theory: From the Haar basis and multiresolution analysis to compactly supported wavelets.
- Integration of approximation theory: Explains how wavelets provide optimal approximations for irregular functions and surfaces.
- Statistical applications: Covers nonparametric regression, density estimation, and other statistical problems using wavelet methods.
- Practical software code: Includes code from an interactive statistical computing environment, enabling hands-on implementation.
- Clear and accessible presentation: Designed to introduce novices to the field while providing depth for advanced readers.
- Rich set of examples and applications: Illustrates the use of wavelets in data compression, turbulence analysis, and image/signal processing.
- Strong theoretical foundations: Includes essential background from Fourier analysis and functional analysis.
- Part of a respected series: Published in Springer's Lecture Notes in Statistics, known for high-quality monographs.
- Collaborative authorship: Written by leading experts in statistics and wavelet theory from France and Germany.
- Bridges theory and practice: Aims to connect theoreticians and practitioners, fostering interdisciplinary work.
About the Authors
Wolfgang Härdle is a German statistician and Professor at the Faculty of Economic Sciences at Humboldt University of Berlin, where he holds the Ladislaus von Bortkiewicz Chair of Statistics[reference:10][reference:11]. He is a leading figure in computational statistics, machine learning, and quantitative finance. His research focuses on dimension reduction techniques, computational statistics, and digital finance[reference:12]. He has published widely and is the author of several influential books, including Applied Multivariate Statistical Analysis.
Gerard Kerkyacharian is a French statistician affiliated with the Laboratoire de Probabilités et Modèles Aléatoires (LPMA) at Université Paris Diderot and Université Pierre et Marie Curie[reference:13]. His research interests include mathematical statistics, nonparametric estimation, and the statistical applications of wavelets. He has made significant contributions to the theory of thresholding, maxisets, and concentration inequalities[reference:14].
Dominique Picard is a French mathematician and professor at the Laboratoire de Probabilités et Modèles Aléatoires of Paris Diderot University[reference:15]. She earned her Ph.D. in mathematical statistics from Université Paris-Sud in 1983[reference:16]. Her research focuses on mathematical statistics, with special interest in high-dimensional problems, structural change detection, and geometrical aspects of statistical experiments[reference:17]. She is known for her work on the statistical applications of wavelets[reference:18].
Alexander Tsybakov is a professor and Head of the Statistics Department at CREST-ENSAE Paris, and also a professor at Sorbonne University[reference:19]. He has been a leading figure in nonparametric estimation, statistical learning theory, and high-dimensional statistics. He is the author of several books and over 150 journal articles[reference:20]. His research has had a profound impact on the fields of mathematical statistics and machine learning.
Related Books
- An Introduction to Wavelets – Charles K. Chui
- Wavelets and Subband Coding – Martin Vetterli, Jelena Kovačević
- Ten Lectures on Wavelets – Ingrid Daubechies
- A Wavelet Tour of Signal Processing – Stéphane Mallat
- Wavelet Methods in Statistics with R – Guy Nason
- Nonparametric Curve Estimation – Murray Rosenblatt
- Introduction to Nonparametric Estimation – Alexandre B. Tsybakov
- Spline Models for Observational Data – Grace Wahba
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Frequently Asked Questions
1. What is the difficulty level of this book?
This book is suitable for graduate students, researchers, and practitioners in statistics, mathematics, and engineering. It assumes a background in calculus, linear algebra, and basic probability/statistics. While the material is advanced, the authors present it in a clear and accessible manner, making it a good entry point for novices interested in wavelets.
2. Who is the target audience for this book?
The book is aimed at statisticians, mathematicians, engineers, and data scientists who want to understand the theory and applications of wavelets. It is particularly useful for those working in nonparametric statistics, signal processing, image analysis, and data compression.
3. What are the prerequisites for reading this book?
Readers should have a solid foundation in calculus, linear algebra, and basic probability/statistics. Familiarity with Fourier analysis is helpful but not strictly required, as the book provides a review of essential facts from Fourier analysis.
4. What topics are covered in the book?
The book covers the Haar basis, multiresolution analysis, Fourier analysis, basic relations of wavelet theory, construction of wavelet bases, compactly supported wavelets, and statistical applications such as nonparametric regression and density estimation.
5. Does the book include software code?
Yes, the authors provide software code from an interactive statistical computing environment, allowing readers to implement wavelet methods in practice.
6. How is this book different from other wavelet books?
This book uniquely integrates the three main streams of wavelet theory: mathematical foundations, approximation theory, and statistical applications. It emphasizes both theory and practice, making it valuable for both theoreticians and practitioners.
7. Is this book suitable for self-study?
Yes, the book is designed to be self-contained and includes numerous examples and code snippets. It can be used for self-study by motivated readers with the appropriate background.
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