Wavelet Analysis and its Statistical Applications

Abstract

In recent years there has been a considerable development in the use of wavelet methods in statistics. As a result, we are now at the stage where it is reasonable to consider such methods to be another standard tool of the applied statistician rather than a research novelty. With that in mind, this paper gives a relatively accessible introduction to standard wavelet analysis and provides a review of some common uses of wavelet methods in statistical applications. It is primarily orientated towards the general statistical audience who may be involved in analysing data where the use of wavelets might be effective, rather than to researchers who are already familiar with the ®eld. Given that objective, we do not emphasize mathematical generality or rigour in our exposition of wavelets and we restrict our discussion to the more frequently employed wavelet methods in statistics. We provide extensive references where the ideas and concepts discussed can be followed up in greater detail and generality if required. The paper ®rst establishes some necessary basic mathematical background and terminology relating to wavelets. It then reviews the more well-established applications of wavelets in statistics including their use in nonparametric regression, density estimation, inverse problems, changepoint problems and in some specialized aspects of time series analysis. Possible extensions to the uses of wavelets in statistics are then considered. The paper concludes with a brief reference to readily available software packages for wavelet analysis.

Filters, Waves and Spectra

“…Wavelets analysis has many practical applications, of which the statistical applications are touched on in the paper of Abramovich et al (2000). From the present point of view, wavelets analysis can be regarded as a synthesis of the discrete-time and the continuous-time analyses of signals.…”

Multivariate Analysis: Overview

“…Such data matrices bring their own problems and objectives, and this has led to many advances in computationally intensive multivariate analysis. Some specific techniques include: curve fitting through localized nonparametric smoothing and recursive partitioning, a flexible version of which is MARS ('multivariate adaptive regression splines') [12] ; the use of wavelets for function estimation [1] ; the analysis of data that themselves come in the form of functions [23] ; the detection of pattern in two-way arrays by using biclustering [21] ; and the decomposition of multivariate spatial and temporal data into meaningful components [3] . While much of the impetus for these developments has come from the scientific area, many of the resultant techniques are, of course, applicable to the large data sets that are becoming more prevalent in behavioral science also.…”

An Efficient Wavelet Based Feature Reduction and Classification Technique for the Diagnosis of Dementia

“…Since wavelet transform [35][36][37][38][39][40][41][42][43] has been successful in signal processing application a continuity relationship connecting (i-1), i and (i+1) feature sets is preferred, as it exists in case of signals. The localization property of wavelet transform has the capability of extracting the finer details from the spatial signals.…”