Wavelet packet modelling of nonstationary time series

This article models the relationship between a response time series, fYtgt2Zand an explanatory time series fXtgt2Z. We hope that any model we choose might be interesting in its own right but we shall also be interested in using it to predict future values of Yt from future values of Xt. When both time series fall into the class of ARMA type models then it is appropriate to use \transfer function" models (see, e.g. Priestley 1981, chap. 9). However, our modelling methodology can be used when either or both time series are not stationary although it is intended for series that exhibit patches of stationarity (where some stationarity regimes appear more than once in the observed series) or are locally stationary (e.g. fall into the class of oscillatory processes, see Priestley (1981, chap. 11), locally stationary (Fourier) processes, see Dahlhaus (1997) or locally stationary wavelet processes, see Nason, von Sachs and Kroisandt 2000) The models that we build rst express Xt in terms of (non-decimated) wavelet packets which analyse Xt at di erent scales, frequencies and locations. Then standard statistical modelling is used to relate the response Yt to the nondecimated wavelet packet transform (NWPT) of Xt. The selected model often reveals valuable information about which types of oscillatory behaviour in Xt in uence Yt and also supplies a method to predict future values of Yt from future values of Xt. We do not (yet) have a theoretical formulation of our modelling procedure Our aim is to introduce the method and show that it can produce interesting and veri able results on real time series. Recently, Walden and Contreras Cristan (1998) used the NWPT in the analysis of a single non-stationary series of hourly averaged Southern Hemisphere solar magnetic eld magnitude observations. Our work di ers in that we relate a time series, Yt, to the NWPT of another series Xt. Ramsey and Lamport (1998) have carried our similar analyses to us but using standard decimated wavelets