Locally Stationary Wavelet Packet Processes: Basis Selection and Model Fitting

Abstract

For non-stationary time series, the fixed Fourier basis is no longer canonical. Rather than limit our basis choice to wavelet or Fourier functions, we propose the use of a library of non-decimated wavelet packets from which we select a suitable basis (frame). Non-decimated packets are preferred to decimated basis libraries so as to prevent information ‘loss’ at scales coarser than the finest. This article introduces a new class of locally stationary wavelet packet processes and a method to fit these to time series. We also provide new material on the boundedness of the inverse of the inner product operator of autocorrelation wavelet packet functions. We demonstrate the effectiveness of our modelling and basis selection on simulated series and Standard and Poor's 500 index series.