Nonparametric Statistics with Applications to Science and Engineering with R, 2nd Edition

14Wavelets

It is error only, and not truth, that shrinks from inquiry.

Thomas Paine (1737–1809)

14.1 Introduction to Wavelets

Wavelet‐based procedures are now indispensable in many areas of modern statistics, for example, in regression, density and function estimation, factor analysis, modeling and forecasting of time series, functional data analysis, and data mining and classification, with ranges of application areas in science and engineering. Wavelets owe their initial popularity in statistics to shrinkage, a simple and yet powerful procedure efficient for many nonparametric statistical models.

Wavelets are functions that satisfy certain requirements. The name wavelet comes from the requirement that they integrate to zero, “waving” above and below the x‐axis. The diminutive in wavelet suggests its good localization. Other requirements are technical and needed mostly to ensure quick and easy calculation of the direct and inverse wavelet transform.

There are many kinds of wavelets. One may choose between smooth wavelets, compactly supported wavelets, wavelets with simple mathematical expressions, or wavelets with short associated filters. The simplest is the Haar wavelet, and we discuss it as an introductory example in Section 14.2.1. Examples of some wavelets (from Daubechies' family) are given in Figure 14.1. Note that scaling and wavelet functions in panels (a, b) in Figure 14.1 (Daubechies 4) are supported on a short interval (of length 3) but are not smooth; the ...