Wavelets for Regression and Other Statistical Problems
Guy P. Nason, Bernard W. Silverman · Wiley series in probability and statistics · 2000
This paper provides an introductory review, largely based on Nason and Silverman (1994), on the use of the discrete wavelet transform (DWT) for statistical purposes, and a guide to a publicly available package of routines, Nason (1993), for the statistical language S. The package is called wavethresh and is available from the StatLib archive. The paper also includes discussion of two subsequent developments of wavelet regression methodology using material drawn from Nason (1996) and Johnstone and Silverman (1997). A gentle introduction to wavelet methods is provided by Strang (1993). For a more detailed discussion the reader is referred, for example, to Daubechies (1992) and Chui (1992). The statistical aspects of the package are mainly due to Donoho and Johnstone (1994). In this paper we concentrate on the discrete wavelet transform. This is based on filtering ideas that have been discussed extensively in the engineering literature. Vaidyanathan (1990) and Vetterli and Herley (1992), provide detailed surveys and numerous references. Some other specific references are mentioned in Section 6 below. We do not claim that wavelets are useful in all statistical curve and surface estimation problems. The general aim of this review is to widen interest in, and access to, wavelet methods so that they can be tried and tested in practice, and a mature view thereby obtained of their usefulness and potential. Standard linear regression techniques formulate a model of the response in terms of some explanatory variables. If a polynomial regression is appropriate then orthogonal polynomials may be useful. Although the resultant variables may be more complicated, the regression coefficients of the polynomial-variables are independent. This independence is desirable, especially when the...