ABSTRACT
A point process is a stochastic process whose realizations are countable sets of points, sometimes called events to distinguish them from arbitrary locations, on some relevant domain. In the current context, each event is a location in a planar region, $R$, with an associated time of occurrence in an interval $[a, b]$, hence a spatio-temporal point process. To fix ideas, the chapter first reviews some results on purely spatial point process theory and models. Later sections cover moment properties, parametric models, and inference. In the modeling section, a fundamental distinction is made between Cox processes, whose behavior is governed by the evolution of an unobserved, real-valued spatio-temporal process, and conditional intensity models, for which there is a direct dependence between past events and current behavior of the process. In the inference section, the focus primarily is on likelihood-based methods and describe two applications. One application uses a Cox process model for real-time surveillance of non-specific gastro-intestinal illness, where the goal is rapid identification of anomalous patterns of incidence. The other uses a conditional intensity model to describe the revolution of the 2001 UK foot-and-mouth epidemic, where the goal is to estimate parameters that describe the transmission mechanism.
