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# Wavelets

Statistical and Numerical Computing

1. Donald B. Percival

Published Online: 15 SEP 2006

DOI: 10.1002/9780470057339.vaw006

## Encyclopedia of Environmetrics

#### How to Cite

Percival, D. B. 2006. Wavelets. Encyclopedia of Environmetrics. 6.

#### Author Information

1. University of Washington, WA, USA

#### Publication History

1. Published Online: 15 SEP 2006

## SEARCH

### Abstract

Wavelets are a special class of functions (or sequences) that are widely used for analyzing time- series. Just as Fourier analysis is based upon the notion of representing (or re-expressing) a time series as a linear combination of sinusoids, the idea underlying wavelet analysis is to represent the series as a linear combination of wavelets. In Fourier analysis, each sinusoid is associated with a particular frequency f, so what frequencies are important in a particular time series can be deduced by studying the magnitudes of the coefficients of the various sinusoids in the linear combination. In contrast, each wavelet is associated with two independent variables, namely, time t and scale τ, because each wavelet is essentially nonzero only inside a particular interval of times, namely, [t – τ, t + τ]. Within that interval, the wavelet spends roughly an equal amount of time above and below zero, so it appears to be a ‘small wave’ centered at time t and having a width of 2τ. We can thus learn how a time series varies on particular scales across time if it is re-expressed using wavelets.