Modelling non-homogeneous Poisson processes with almost periodic intensity functions

We propose a model for the analysis of non-stationary point processes with almost periodic rate of occurrence. The model deals with the arrivals of events which are unequally spaced and show a pattern of periodicity or almost periodicity, such as stock transactions and earthquakes. We model the rate of occurrence of a non-homogeneous Poisson process as the sum of sinusoidal functions plus a baseline. Consistent estimates of frequencies, phases and amplitudes which form the sinusoidal functions are constructed mainly by the Bartlett periodogram. The estimates are shown to be asymptotically normally distributed. Computational issues are discussed and it is shown that the frequency estimates must be resolved with order o(T-1) to guarantee the asymptotic unbiasedness and consistency of the estimates of phases and amplitudes, where T is the length of the observation period. The prediction of the next occurrence is carried out and the mean-squared prediction error is calculated by Monte Carlo integration. Simulation and real data examples are used to illustrate the theoretical results and the utility of the model.