Multiple-window wavelet transform and local scaling exponent estimation

We propose here a multiple-window wavelet transform for the purpose of identifying non-stationary self-similar structures in random processes and estimating the time-varying scaling exponent H(t) that controls the local regularity and correlation of the process. More specifically, our final aim is to be able to track even rapidly varying trajectories (t, H(t)). The solution described here combines analysis obtained from scalograms computed with a set of multi-windows designed so as to satisfy to a decorrelation condition. We derive here the statistics for the estimate of H(t), compare it against numerical simulations and show that we obtain a substantial reduction of variance in estimation, without introducing bias.