Wavelets from a Statistical Perspective

Maarten Jansen

• Publisher: Chapman and Hall/CRC
• Year: 2022
• ISBN:
  9781000564174
• Edition: 1
• Topic: Economía
• Subtopic: Estadísticas para los negocios y la economía

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.

1Wavelets: nonlinear processing in multiscale sparsity sparsity

DOI: 10.1201/9781003265375-1

1.1 Compressing big data

In a scientist’s effort to explain a phenomenon through a model, a major challenge lies in filling in the model parameters. In the setting of a simple and known model and without measurement errors, model parameters are easily found by solving small linear systems applied to the observed response. In the real world, where measurement errors are ubiquitous, parameter values can be learnt or estimated from the observations. As data are big these days, there is a good chance that there are more observations to learn from than parameters to be estimated. The linear system for the model parameters is then overdetermined, leading to least squares solutions or other regression techniques.

And yet, not only do we have big data, but the models tend to become high dimensional as well. In many applications, the model itself has to be learnt from the data. This may lead to model or variable selection methods. Another option to start with is to realise that the data may well be less complicated than they appear to be. In that case, a data transformation reorganises the representation of the data so that the available information appears in a more concentrated way. The wavelet transform is a typical example of such an operation, and images are a popular illustration of what a wavelet transform can do. Figure 1.1 shows an grey scale image of $1024\times 1417$ pixels¹. Each of the 1,451,008 pixels represents an observation, but also a parameter to store on your computer. The pixel representation is not sparse, or economical, as typically adjacent pixels have similar grey values. Figure 1.2 displays another image, which is the wavelet transform of the image in Figure 1.1. Obviously, the wavelet representation is for internal use only. Whenever the image needs to be displayed on a screen or in a book which is not about wavelets, we first apply the inverse wavelet transform to get back Figure 1.1. There are many wavelet transforms to choose from, but the one used in this case is quite popular for image processing. While its details will be developed later, we mention here for further reference, it is known as th...

Table of contents

• Cover Page
• Half-Title Page
• Title Page
• Copyright Page
• Table of Contents
• List of recurrent symbols
• Introduction
• 1 Wavelets: nonlinear processing in multiscale sparsity
• 2 Wavelet building blocks
• 3 Using lifting for the design of a wavelet transform
• 4 Wavelet transforms from factored refinement schemes
• 5 Dyadic wavelets
• 6 Dyadic wavelet design in the frequency domain
• 7 Design of dyadic wavelets
• 8 Approximation in a wavelet basis
• 9 Overcomplete wavelet transforms
• 10 Two-dimensional wavelet transforms
• 11 The multiscale local polynomial transform
• 12 Estimation in a wavelet basis
• Outlook
• References
• Subject index