Consistent Classification of Nonstationary Time Series Using Stochastic Wavelet Representations

Abstract

Consider the situation when we have training data containing many time series having known group membership and testing data with unknown group membership. The goals are to find timescale features (using training data) that can best separate the groups, and to use these highly discriminant features to classify test data. We propose a method for classification using a bias-corrected nondecimated wavelet transform. Wavelets are ideal for identifying highly discriminant local time and scale features. The observed signals will be treated as realizations of locally stationary wavelet processes, under which we define and rigorously estimate the evolutionary wavelet spectrum (timescale decomposition of variance). The evolutionary wavelet spectrum, which contains the second-moment information on the signals, is used as the classification signature. For each test time series, we compute the empirical wavelet spectrum and its divergence from the wavelet spectrum of each group. The test time series is then assigned to the group to which it is the least dissimilar. Under the locally stationary wavelet framework, we rigorously demonstrate that the classification procedure is consistent (i.e., misclassification probability goes to zero at the rate that is inversely proportional to divergence between the evolutionary wavelet spectra). The method is illustrated using seismic signals (earthquake vs. explosion events) and is demonstrated to work very well in simulation studies.