The statistical analysis of space-time point processes

Abstract

A space-time point process is a stochastic process having as realizations points with random coordinates in both space and time. We define a general class of space-time point processes which we term {\em analytic}. These are point processes that have only finite numbers of points in finite time intervals, absolutely continuous joint-occurrence distributions, and for which points do not occur with certainty in finite time intervals. Analytic point processes possess an intensity determined by the past of the point process. As a class, analytic point processes remain closed under randomization by a parameter. The problem we consider is that of estimating a random parameter of an observed space-time point process. This parameter may be drawn from a function space and can, therefore, model a random variable, random process, or random field that influences the space-time point process. Feedback interactions between the point process and the randomizing parameter are included. The conditional probability measure of the parameter given the observed space-time point process is a sufficient statistic for forming estimates satisfying a wide variety of performance criteria. A general representation for this conditional measure is developed, and applications to filtering, smoothing, prediction, and hypothesis testing are given.