Autocorrelation Functions for Point-Process Time Series

Abstract

This article introduces autocorrelograms for time series of point processes. Such time series usually arise when a longer temporal or spatio-temporal point process is sliced into smaller time units; for example, when an annual process is sliced into 365 daily replications. We assume the point processes follow a doubly-stochastic Poisson model with log-Gaussian intensity functions. The proposed autocorrelograms are computationally simple and based on binning. The asymptotic distribution of the autocorrelations is established. The ability of the method to detect the patterns of common autoregressive and moving-average time series models is shown by simulation. Two examples of application to temporal and spatial point-process time series are shown, pertaining bike demand in the Divvy bike-sharing system of Chicago and street theft in Chicago, respectively.