Estimating Time-Evolving Partial Coherence between Signals via Multivariate Locally Stationary Wavelet Processes

We consider the problem of estimating time-localized cross-dependence in a collection of nonstationary signals. To this end, we develop the multivariate locally stationary wavelet framework, which provides a time-scale decomposition of the signals and, thus, naturally captures the time-evolving scale-specific cross-dependence between components of the signals. Under the proposed model, we rigorously define and estimate two forms of cross-dependence measures: wavelet coherence and wavelet partial coherence. These dependence measures differ in a subtle but important way. The former is a broad measure of dependence, which may include indirect associations, i.e., dependence between a pair of signals that is driven by another signal. Conversely, wavelet partial coherence measures direct linear association between a pair of signals, i.e., it removes the linear effect of other observed signals. Our time-scale wavelet partial coherence estimation scheme thus provides a mechanism for identifying hidden dynamic relationships within a network of nonstationary signals, as we demonstrate on electroencephalograms recorded in a visual-motor experiment.