14. Wavelets

Yuri Golubev, Wolfgang Härdle , Zdenek Hlávka , Sigbert Klinke , Michael H. Neumann and Stefan Sperlich
28 July 2004

Wavelets are a powerful statistical tool which can be used for a wide range of applications, namely

One of the main advantages of wavelets is that they offer a simultaneous localization in time and frequency domain. The second main advantage of wavelets is that, using fast wavelet transform, it is computationally very fast.

You can learn more about wavelets in the following overview. We advise you to consult e.g. Daubechies (1992), Kaiser (1995) or Härdle et al. (1998) if you wish to study wavelets in more detail.

Overview

Wavelet expansion of a certain function $ f$ is a special case of an orthonormal series expansion. Orthonormal series expansions of functions, or more generally transformations in the frequency domain, have several important applications. First, this is simply an alternative way of describing a signal. Such a description in the frequency domain often provides a more parsimonious representation than the usual one on a grid in the time domain. Orthonormal series expansions can also serve as a basis for nonparametric estimators and much more.

You can recall the classical Fourier series expansion of a periodic function. A nonperiodic function in $ L_2(R)$ can be described by the Fourier integral transform which is represented by an appropriately weighted integral over the harmonic functions.

A first step towards a time-scale representation of a function with a time-varying behavior in the frequency domain is given by the windowed Fourier transform, which approximately amounts to a local Fourier expansion of $ f$. To attain time localization, one multiplies $ f$ with a time window $ g$ and performs the usual Fourier integral transform. For the windowed Fourier transform,

$\displaystyle \mathcal{F}(\omega,t)=\int g(u-t)f(u)e^{-2\pi i\omega u}du;
$

the inverse transform exists which means that no information is lost by this transformation. The main drawback of this transform is that the same time window is used over the whole frequency domain. Thus, for signals with a high power at high frequencies, this window will turn out to be unnecessarily large whereas it is too small for signals with dominating contributions from low frequencies.

In contrast, the wavelet transform provides a decomposition into components from different scales whose degree of localization is connected to the size of the scale window. This is achieved by translations and dilations of a single function, the so-called wavelet. It provides an amount of simultaneous localization in time and scale domain which is in accordance to the so-called uncertainty principle. This principle basically says that the best possible amount of localization in one domain is inversely proportional to the size of the localization window in the other domain. Note that we do not use the term ``frequency'' in connection with wavelet decompositions. The use of this term would be adequate in connection with functions that show an oscillating behavior. For wavelets it is more convenient to use the term ``scale'' to describe such phenomena.

A function $ f$ in $ L_2(R)$ can be represented by its continuous wavelet transform

$\displaystyle \mathcal{F}(\omega,t)=\int f(s)\psi\left(\frac{s-t}{\omega}\right)
\omega^{-1/2}ds.
$

One can also obtain an invertible transform by a simple dyadic translation and dilation scheme which is based on functions $ \psi_{j,k}(x)=2^{j/2}\psi(s^jx-k)$, where $ j,k\in Z$. There exist functions $ \psi$ such that $ \{\psi_{j,k};j,k\in Z\}$ forms a complete orthonormal basis of $ L_2(R)$. A particular construction of such a function is described by Daubechies (1988). For a function $ f\in L_2(R)$ we have the homogeneous wavelet expansion

$\displaystyle f=\sum_{j=-\infty}^\infty \sum_{k=-\infty}^\infty
\langle f,\psi_{j,k} \rangle \psi_{j,k},
$

where $ \langle f,\psi_{j,k} \rangle
=\int f(t)\psi_{j,k}(t)dt$. If we truncate the above expansion at a certain resolution scale $ j=J-1$, we obtain the projection of $ f$ on some linear space, say $ V_j$. This space can be alternatively spanned by translates of an accordingly dilated version of a so-called scaling function or father wavelet $ \varphi$. Let $ \varphi_{l,k}(x)=2^{l/2}\varphi(2^lx-k)$. Now we can also expand $ f$ in an inhomogeneous wavelet series,

$\displaystyle f=\sum_{k=-\infty}^\infty \langle f,\varphi_{l,k}\rangle\varphi_{...
...um_{j=l}^\infty \sum_{k=-\infty}^\infty
\langle f,\psi_{j,k}\rangle\psi_{j,k}.
$

There exist wavelets with several additional interesting properties. A frequently used tool are wavelets with compact support, which were first developed by Daubechies (1988). Moreover, one can construct wavelets of any given degree of smoothness and with an arbitrary number of vanishing moments. Finally, there exist boundary corrected bases which are appropriate for a wavelet analysis on compact intervals. This turns out to be quite a technical matter and you can consult the literature Daubechies (1992), Kaiser (1995), Härdle et al. (1998), if you are interested in more details.

The principle of estimation of a function in the $ L_2(R)$ space by wavelets, the underlying multiresolution analysis (MRA), and the meaning of father and mother wavelets, will be explained also in the context of function approximation in Section 14.3.