  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.
