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1998 | Buch

Wavelets Kapitel lesen Erstes Kapitel lesen

Wavelets, Approximation, and Statistical Applications

verfasst von: Wolfgang Härdle, Gerard Kerkyacharian, Dominique Picard, Alexander Tsybakov

Verlag: Springer New York

Buchreihe : Lecture Notes in Statistics

Enthalten in: Professional Book Archive

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Über dieses Buch

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.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Wavelets
Abstract
A wavelet is, as the name suggests, a small wave. Many statistical phenomena have wavelet structure. Often small bursts of high frequency wavelets are followed by lower frequency waves or vice versa. The theory of wavelet reconstruction helps to localize and identify such accumulations of small waves and helps thus to better understand reasons for these phenomena. Wavelet theory is different from Fourier analysis and spectral theory since it is based on a local frequency representation.
Wolfgang Härdle, Gerard Kerkyacharian, Dominique Picard, Alexander Tsybakov
Chapter 2. The Haar basis wavelet system
Abstract
The Haar basis is known since 1910. Here we consider the Haar basis on the real line IR and describe some of its properties which are useful for the construction of general wavelet systems. Let L2 (IR) be the space of all complex valued functions f on IR such that their L2-norm is finite:
$$ \left\| {f\left\| {2 = \left( {\int_{ - \infty }^\infty {\left| {f(x)} \right|^2 dx} } \right)} \right.} \right.^{\frac{1}{2}} \langle \infty .$$
Wolfgang Härdle, Gerard Kerkyacharian, Dominique Picard, Alexander Tsybakov
Chapter 3. The idea of multiresolution analysis
Abstract
The Haar system is not very convenient for approximation of smooth functions. In fact, any Haar approximation is a discontinuous function. One can show that even if the function f is very smooth, the Haar coefficients still decrease slowly. We therefore aim to construct wavelets that have better approximation properties.
Wolfgang Härdle, Gerard Kerkyacharian, Dominique Picard, Alexander Tsybakov
Chapter 4. Some facts from Fourier analysis
Abstract
This small chapter is here to summarize the classical facts of Fourier analysis that will be used in the sequel. We omit the proofs (except for the Poisson summation formula). They can be found in standard textbooks on the subject, for instance in Katznelson (1976), Stein & Weiss (1971).
Wolfgang Härdle, Gerard Kerkyacharian, Dominique Picard, Alexander Tsybakov
Chapter 5. Basic relations of wavelet theory
Abstract
Let us formulate in the exact form the conditions on the functions φ and ψ which guarantee that the wavelet expansion (3.5) holds. This formulation is connected with the following questions.
Wolfgang Härdle, Gerard Kerkyacharian, Dominique Picard, Alexander Tsybakov
Chapter 6. Construction of wavelet bases
Abstract
In Chapter 5 we derived general conditions on the functions φ and ψ that guarantee the wavelet expansion (3.5). It was shown that to find an appropriate pair (φ,ψ) it suffices, in fact, to find a father wavelet φ. Then one can derive a mother wavelet ψ), given φ. In this chapter we discuss two concrete approaches to the construction of father wavelets. The first approach is starting from Riesz bases, and the second approach is starting from a function m0. For more details on wavelet basis construction we refer to Daubechies (1992),Chui(1992a, 1992b), Meyer (1993), Young (1993), Cohen & Ryan (1995), Holschneider (1995), Kahane & Lemarié-Rieusset (1995), Kaiser (1995).
Wolfgang Härdle, Gerard Kerkyacharian, Dominique Picard, Alexander Tsybakov
Chapter 7. Compactly supported wavelets
Abstract
The original construction of compactly supported wavelets is due to Daubechies (1988). Here we sketch the main points of Daubechies’ theory. We are interested to find the exact form of functions m0(ξ), which are trigonometric polynomials, and produce father φ and mother ψ with compact supports such that, in addition, the moments of φ and ψ of order from 1 to n vanish. This property is necessary to guarantee good approximation properties of the corresponding wavelet expansions, see Chapter 8.
Wolfgang Härdle, Gerard Kerkyacharian, Dominique Picard, Alexander Tsybakov
Chapter 8. Wavelets and Approximation
Abstract
In this chapter we study the approximation properties of wavelet expansions on the Sobolev spaces. We specify how fast does the wavelet expansion converge to the true function f , if f belongs to some Sobolev space. This study is continued in Chapter 9 where we consider the approximation on the Besov spaces and show that it has an intrinsic relation to wavelet expansions. The presentation in this chapter and in Chapter 9 is more formal than in the previous ones. It is designed for the mathematically oriented reader who is interested in a deeper theoretical insight into the properties of wavelet bases.
Wolfgang Härdle, Gerard Kerkyacharian, Dominique Picard, Alexander Tsybakov
Chapter 9. Wavelets and Besov Spaces
Abstract
This chapter is devoted to approximation theorems in Besov spaces. The advantage of Besov spaces as compared to the Sobolev spaces is that they are much more general tool in describing the smoothness properties of functions. We show that Besov spaces admit a characterization in terms of wavelet coefficients, which is not the case for Sobolev spaces. Thus the Besov spaces are intrinsically connected to the analysis of curves via wavelet techniques. The results of Chapter 8 are substantially used throughout. General references about Besov spaces are Nikol‘skii (1975), Peetre (1975), Besov, Il‘in & Nikol‘skii (1978), Bergh & Löfström (1976), Triebel (1992), DeVore & Lorentz (1993).
Wolfgang Härdle, Gerard Kerkyacharian, Dominique Picard, Alexander Tsybakov
Chapter 10. Statistical estimation using wavelets
Abstract
In Chapters 3, 5, 6 and 7 we discussed techniques to construct functions φ and ψ (father and mother wavelets), such that the wavelet expansion (3.5) holds for any function f in L2(IR). This expansion is a special kind of orthogonal series. It is “special”, since unlike the usual Fourier series, the approximation is both in frequency and space. In this chapter we consider the problem of nonparametric statistical estimation of a function f in L2(IR) by wavelet methods. We study the density estimation and nonparametric regression settings. We also present empirical results of wavelet smoothing.
Wolfgang Härdle, Gerard Kerkyacharian, Dominique Picard, Alexander Tsybakov
Chapter 11. Wavelet thresholding and adaptation
Abstract
This chapter treats in more detail the adaptivity property of nonlinear (thresholded) wavelet estimates. We first introduce different modifications and generalizations of soft and hard thresholding. Then we develop the notion of adaptive estimators and present the results about adaptivity of wavelet thresholding for density estimation problems. Finally, we consider the data-driven methods of selecting the wavelet basis, the threshold value and the initial resolution level, based on Stein’s principle. We finish by a discussion of oracle inequalities and miscellaneous related topics.
Wolfgang Härdle, Gerard Kerkyacharian, Dominique Picard, Alexander Tsybakov
Chapter 12. Computational aspects and statistical software implementations
Abstract
In this chapter we discuss how to compute the wavelet estimators and give a brief overview of the statistical wavelets software.
Wolfgang Härdle, Gerard Kerkyacharian, Dominique Picard, Alexander Tsybakov
Backmatter
Metadaten
Titel
Wavelets, Approximation, and Statistical Applications
verfasst von
Wolfgang Härdle
Gerard Kerkyacharian
Dominique Picard
Alexander Tsybakov
Copyright-Jahr
1998
Verlag
Springer New York
Electronic ISBN
978-1-4612-2222-4
Print ISBN
978-0-387-98453-7
DOI
https://doi.org/10.1007/978-1-4612-2222-4