Periodic point processes: Theory and application

We address the problems of extracting information generated by one dimensional periodic point processes. These problems arise in numerous situations, from astronomy and biomedical applications to reliability and quality control and signal processing. We divide our analysis into two cases, namely single and then multiple source(s). We wish to extract the fundamental period of the generator(s), and, in the second case, to deinterleave the processes. We present two algorithms, designed to work on all one dimensional periodic processes, but in particular on sparse datasets where other procedures break down. The first algorithm works on data from single period processes, computing an estimate of the underlying period. It is extremely computationally efficient and straightforward, and works on all single period processes, but in particular on sparse datasets where others break down. Its justification, however, rests on some deep mathematics, including a probabilistic interpretation of the Riemann zeta function. We then build upon this procedure to analyze data from multiple periodic processes. This second procedure relies on the Riemann zeta function, Weyl's equidistribution theorem, and Wiener's periodogram.