Point process modeling through a mixture of homogeneous and self‐exciting processes

Abstract

Self‐exciting point processes allow modeling the temporal location of an event of interest, considering the history provided by previously observed events. This family of point processes is commonly used in several areas such as criminology, economics, or seismology, among others. The standard formulation of the self‐exciting process implies assuming that the underlying stochastic process is dependent on its previous history over the entire period under analysis. In this paper, we consider the possibility of modeling a point pattern through a point process whose structure is not necessarily of self‐exciting type at every instant or temporal interval. Specifically, we propose a mixture point process model that allows the point process to be either self‐exciting or homogeneous Poisson, depending on the instant within the study period. The performance of this model is evaluated both through a simulation study and a case study. The results indicate that the model is able to detect the presence of instants in time, referred to as change points, where the nature of the process varies.