Modelling Non-Homogeneous Poisson Processes with Almost Periodic Intensity Functions

Abstract

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.

Super-resolution Estimation of Cyclic Arrival Rates

“…Thus windows with light spectral tails provide a solution for detecting weak frequency signals in the presence of strong ones. This addresses a point mentioned in passing on page 110 of [20]: Issues with the periodogram method arise when the dynamic range is large, even in the classical setting where the frequency gap is 1/o(T ). Our analysis provides a way for quantifying this for both the windowed and unwindowed periodograms when T is finite.…”

“…where the even number p of frequency components, the frequencies ν λ = {ν λ k } k in a pre-specified band [−B, +B], and the complex coefficients c λ = {c λ k } k are all unknown. Given the connections to Fourier series, this specification is very flexible and was introduced by Shao and Lii [19,20]. They resolve the estimation problem under the classical setting where the frequencies are assumed to be spaced more than order 1/T apart.…”

“…Under the classical setting where the frequency gap is assumed to be 1/o(T ), Shao and Lii [19,20] extend the periodogram method to the multiple frequencies setting. Their procedure corresponds to setting R = {ν : r ≤ |ν| ≤ B} in Algorithm 1 and running step 3 until p frequencies have been selected.…”

“…The lower bound of 4/T benchmarks the frequency gap employed in the super-resolution literature [10,21]. If instead the benchmark target is the classical setting in [19,20], then A1 may be relaxed to 6/T , see the remark following Proposition 3 below. Under A1, we must localize each ν λ k to within a neighbourhood of radius 2/T to avoid possible ambiguity from overlapping.…”

“…Remark. When p is known, no thresholding is necessary, and the asymptotic normality results in [20] for the classic periodogram can be extended to the windowed one. The details are provided in Appendix B.…”