Coherence analysis of nonstationary time series: a linear filtering point of view

Coherence is a widely used measure for characterizing linear dependence between two time series. Classical books on time series analysis present coherence as “the frequency domain analogue of the autocorrelation function” which lacks intuitive appeal. The first goal of this paper is to present a more illuminating and yet still precise interpretation of coherence. Consider a filter whose power transfer function is concentrated on a particular frequency band Ω. We show that coherence at Ω is equivalent to the correlation between the two filtered time series. The second goal of this paper is to develop a novel adaptive statistical procedure for estimating coherence when the time series are non-stationary, that is, the nature of linear dependence between time series may evolve with time. The proposed method for estimating local coherence automatically selects, via repeated tests of homogeneity (in time) of coherence, the optimal width of the time window on which one computes the estimated local coherence. This approach is point-wise 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 evolve during an epileptic seizure