A Class of Spatially Correlated Self-Exciting Models

The statistical modeling of multivariate count data observed on a space-time lattice has generally focused on using a hierarchical modeling approach where space-time correlation structure is placed on a continuous, latent, process. The count distribution is then assumed to be conditionally independent given the latent process. However, in many real-world applications, especially in the modeling of criminal or terrorism data, the conditional independence between the count distributions is inappropriate. In this manuscript we propose a class of models that capture spatial variation and also account for the possibility of data model dependence. The resulting model allows both data model dependence, or self-excitation, as well as spatial dependence in a latent structure. We demonstrate how second-order properties can be used to characterize the spatio-temporal process and how misspecificaiton of error may inflate self-excitation in a model. Finally, we give an algorithm for efficient Bayesian inference for the model demonstrating its use in capturing the spatio-temporal structure of burglaries in Chicago from 2010-2015.