Title: The Analysis of Periodic Point Processes

Speakers: Stephen D. Casey, Thomas J. Casey

Date: Tuesday March 16, 2021 

Abstract: Point processes are an important component of data analysis, from the queuing theory used to analyze customer arrival times in business to the occurrences of radar pulses in signal analysis to the analysis of neuron firing rates in computational neuroscience. Our talk addresses the problems of extracting information from periodic point processes. We divide our analysis into two cases – periodic processes created by a single source, and those processes created by several sources. We wish to extract the fundamental period of the generators, and, in the second case, to deinterleave the processes. We first present very efficient algorithm for extracting the fundamental period from a set of sparse and noisy observations of a single source periodic process. The procedure is straightforward and converges quickly. Its use is justified by a probabilistic interpretation of the Riemann zeta function. We then build upon this procedure to deinterleave and then analyze data from multiple source periodic processes. This second process relies both on the probabilistic interpretation of the Riemann zeta function, the equidistribution theorem of Weyl, and Wiener’s periodogram. Both procedures are general and very efficient and can be applied to the analysis of all periodic processes. The focus of this talk is to describe the algorithms and provide analysis as to why they work. Both algorithms rely on the structure of randomness, which tells us that random data can settle into a structure based on the set from which the data is extracted. We close by demonstrating simulations of the procedures.