Wavelet Spectra for Multivariate Point Processes
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Wavelets provide the flexibility to detect and analyse unknown non-stationarity in stochastic processes. Here, we apply them to multivariate point processes as a means of characterising correlation structure within and between multiple event data streams. To provide statistical tractability, a temporally smoothed wavelet periodogram is developed for both real- and complex-valued wavelets, and shown to be equivalent to a multi-wavelet periodogram. Under certain regularity assumptions, the wavelet transform of a point process is shown to be asymptotically normal. The temporally smoothed wavelet periodogram is then shown to be asymptotically Wishart distributed with tractable centrality matrix and degrees of freedom computable from the multi-wavelet formulation. Distributional results extend to wavelet coherence; a time-scale measure of inter-process correlation. The presented theory and methodology are verified through simulation and applied to neural spike train data.
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