Wavelet Analysis and Synthesis of Stationary Long-Memory Processes
Emma J. McCoy, Andrew T. Walden · Journal of Computational and Graphical Statistics · 1996
The discrete wavelet transform (DWT) can be interpreted as a filtering of a time series by a set of octave band filters such that the width of each band as a proportion of its center frequency is constant. A long-memory process having a power spectrum that plots as a straight line on log-frequency/log-power scales over many octaves of frequency is intrinsically related to such a structure. As an example of such processes, we focus on one class of discrete-time, stationary, long-memory processes, the fractionally differenced Gaussian white noise processes (fdGn). We show how the DWT breaks down a fdGn, and show the exact correlation structure of the resulting coefficients for different wavelets (Daubechies' minimum-phase and least-asymmetric and Haar). The DWT is an impressive “whitening filter.” A discrete wavelet-based scheme for simulating fdGn's is discussed and is shown to be equivalent to a spectral decomposition of the covariance matrix of the process; however, it can be carried out using only information on the nature of the spectrum of the process—that is, time-domain information is not required. It produces results comparable with the exact Hosking method. We then show that, using wavelet methods, the spectral slope parameter d can be estimated as well, or better, than when using the best Fourier-based method known to us, namely regression on multitaper spectral ordinates. Since wavelet analysis and synthesis methods can be applied to a much wider variety of empirical or theoretical long-memory processes, wavelet methods could prove a valuable tool in the future in the analysis and synthesis of stochastic processes.