The Analysis and Deinterleaving of Periodic Point Processes

This paper addresses the problem of determining the underlying structure of a given point process. The point process data is a finite set of event times, given by a set of real numbers. We wish to determine if the process has been generated by one or more periodic processes and, if so, extract the period or periods from the data. If there are several periods, we also wish to deinterleave the data into separate periodic processes, each generated by a single period. We approach the problem by developing two algorithms. These algorithms are designed to work on all data sets, but, in particular, on extremely sparse data sets where other procedures do not work. The first algorithm works on sets of event times with only one underlying period, quickly producing an estimate of that period. The mathematical justification of the procedure involves number theory, including an interpretation of the Riemann zeta function as an asymptotic probability distribution. The second algorithm analyzes event times from multiple periodic processes. It relies on the first algorithm as an “engine” to create larger data sets, from which it produces estimates of underlying periods. This second procedure also relies on a mathematical justification, a result from both number theory and harmonic analysis—the equidistribution theorem of Weyl. We then deinterleave the data, breaking it down into components, each generated by a single period.