Evolutionary Coherence of Nonstationary Signals

Coherence is a widely used measure for characterizing linear dependence between a pair of signals. For nonstationary signals, the autospectrum, cross spectrum, and coherence between signals may evolve over time. A standard approach is to divide the signals into overlapping blocks of fixed width and then smooth (over frequency) the periodogram matrix at each time block. In this paper, a consistent estimation procedure is developed using time-localized linear filtering. The proposed method automatically selects, via repeated tests of homogeneity, the optimal window width for estimating local coherence. It is pointwise adaptive in the sense that the width of the optimal interval is allowed to change across time. Under the locally stationary process framework, we develop a central limit theorem on the Fisher-z transform of our time-localized band coherence. We apply our method to a pair of highly dynamic brain waves signals whose coherence is shown to evolve during an epileptic seizure.