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

Abstract

Self-exciting point processes allow modeling thetemporal location of an event of interest, consideringthe history provided by previously observed events.This family of point processes is commonly used inseveral areas such as criminology, economics, or seis-mology, among others. The standard formulation of theself-exciting process implies assuming that the under-lying stochastic process is dependent on its previoushistory over the entire period under analysis. In thispaper, we consider the possibility of modeling a pointpattern through a point process whose structure is notnecessarily of self-exciting type at every instant or tem-poral interval. Specifically, we propose a mixture pointprocess model that allows the point process to be eitherself-exciting or homogeneous Poisson, depending on theinstant within the study period. The performance of thismodel is evaluated both through a simulation study anda case study. The results indicate that the model is ableto detect the presence of instants in time, referred to aschange points, where the nature of the process varies.