Consistent classification of non-stationary time series using stochastic wavelet representations

A method is proposed for classifying an observed non-stationary time series using a bias-corrected non-decimated wavelet transform. Wavelets are ideal for identifying highly discriminant local time and scale features. We view the observed signals as realizations of locally stationary wavelet (LSW) processes. The LSW model provides a time-scale decomposition of the signals under which we can define and rigorously estimate the evolutionary wavelet spectrum. The evolutionary spectrum, which contains the second-moment information on the signals, is used as the classification signature. For each time series to be classified, we compute the empirical wavelet spectrum and the divergence from the wavelet spectrum of each group. It is then assigned it to the group to which it is the least dissimilar. Under the LSW framework, we rigorously demonstrate that the classification procedure is consistent, i.e., misclassification probability goes to zero at the rate that is proportional to divergence between the true spectra. The method is illustrated using seismic signals and is demonstrated to work very well in simulation studies.