Wavelet spectra for multivariate point processes

Abstract

Wavelets provide the flexibility for analysing stochastic processes at different scales. In this article we apply them to multivariate point processes as a means of detecting and analysing unknown nonstationarity, both within and across component processes. To provide statistical tractability, a temporally smoothed wavelet periodogram is developed and shown to be equivalent to a multi-wavelet periodogram. Under a stationarity assumption, the distribution of the temporally smoothed wavelet periodogram is demonstrated to be asymptotically Wishart, with the centrality matrix and degrees of freedom readily computable from the multi-wavelet formulation. Distributional results extend to wavelet coherence, a time-scale measure of inter-process correlation. This statistical framework is used to construct a test for stationarity in multivariate point processes. The methods are applied to neural spike-train data, where it is shown to detect and characterize time-varying dependency patterns.