Statistical modelling of time series using non-decimated wavelet representations

This article proposes the use of time-ordered non-decimated wavelet or nondecimated wavelet packet transforms to provide flexible representations of a time series (explanatory). The resulting representations are then used as variables in a statistical model to provide predictions of another (response) time series. The statistical model provides valuable information about which components in the explanatory time series drive the response time series. To represent our explanatory time series we use a collection of basis functions known as wavelet packets. Each wavelet packet component of the exploratory time series corresponds to a particular linear combination of the time series and its lagged versions (regressive models). The construction of the wavelet packet transform ensures that all possible regressive models on a grid are rapidly computed. Hence our model fully explores “model-space” which may be parametrised in terms of the time-frequency plane. Our modelling methodology is illustrated with examples from two different arenas: (a) a wind power example using a generalized linear model relating wind speeds at one weather station to a time-ordered non-decimated wavelet packet transform of wind speeds and wind directions at another station; (b) a biomedical example shows how infant sleep states can be successfully classified using the time-ordered non-decimated wavelet packet transform of heart rate and linear discriminant analysis.