Wavelet spectra for multivariate point processes

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.

Automated summary

Wavelets provide flexibility for analyzing stochastic processes at different scales. In this article, we apply wavelets to multivariate point processes to detect and analyze unknown nonstationarity. We develop a temporally smoothed wavelet periodogram to ensure statistical tractability and demonstrate its equivalence to a multi-wavelet periodogram. Under the assumption of stationarity, the distribution of the temporally smoothed wavelet periodogram is shown to be asymptotically Wishart, with readily computable parameters. The distributional results also extend to wavelet coherence, a measure of inter-process correlation. We apply this statistical framework to construct a test for stationarity in multivariate point processes and successfully detect and characterize time-varying dependency patterns in neural spike-train data.