Wavelet Analysis of Discrete Time Series

We give a brief review of some of the wavelet-based techniques currently available for the analysis of arbitrary-length discrete time series. We discuss the maximal overlap discrete wavelet packet transform (MODWPT), a non-decimated version of the usual discrete wavelet packet transform, and a special case, the maximal overlap discrete wavelet transform (MODWT). Using least-asymmetric or coiflet filters of the Daubechies class, the coefficients resulting from the MODWPT can be readily shifted to be aligned with events in the time series. We look at several aspects of denoising, and compare MODWT and cycle-spinning denoising. While the ordinary DWT basis provides a perfect decomposition of the autocovariance of a time series on a scale-by-scale basis, and is well-suited to decorrelating a stationary time series with ‘long-memory’ covariance structure, a time series with very different covariance structure can be decorrelated using a wavelet packet ‘best-basis’ determined by a series of white noise tests.