UC Riverside

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 the rate of financial transactions or customer/phone calls arrivals. The concept of almost periodicity is described and the purely periodic process is just a special case of the almost periodic process. We consider a non-homogeneous Poisson process and model its rate of occurrence as the sum of sinusoidal functions plus a base line. Given the number of sinusoidal functions which is denoted as K, a set of simple and 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 have to 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 also discussed. The proposed model is a finite approximation of the almost periodic process in terms of a finite value of K. In practice, the value of K is usually unknown, and we suggest to use the model selection criteria to determine K. Two criteria AIC and BIC are reviewed and discussed in the frame work of our model. Simulation and real data examples are used to illustrate the theoretical results and the utility of the model.