# Wavelet Spectra for Multivariate Point Processes

## TL;DR

This paper introduces a wavelet-based statistical framework for analyzing multivariate point processes, enabling detection of non-stationarity and time-varying dependencies, with applications to neural spike train data.

## Contribution

It develops a temporally smoothed wavelet periodogram with asymptotic distributional properties and applies it to test stationarity and analyze dependencies in multivariate point processes.

## Key findings

- Wavelet periodogram is asymptotically Wishart distributed under stationarity.
- Method effectively detects non-stationarity in neural spike trains.
- Wavelet coherence measures inter-process correlation over time and scale.

## Abstract

Wavelets provide the flexibility to analyse stochastic processes at different scales. Here, we apply them to multivariate point processes as a means of detecting and analysing unknown non-stationarity, both within and across data streams. To provide statistical tractability, a temporally smoothed wavelet periodogram is developed and shown to be equivalent to a multi-wavelet periodogram. Under a stationary 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 methodology is applied to neural spike train data, where it is shown to detect and characterise time-varying dependency patterns.