Self-Exciting Spatio-Temporal Statistical Models for Count Data with Applications to Modeling the Spread of Violenc

In this dissertation we provide statistical models and inferential techniques for analyzing the number of violent or criminal events as they evolve over space and time. Our research focuses on a class of models we refer to as self-exciting spatio-temporal models. These are a class of parametric models that allow for dependence in a latent structure as well as dependence in the data model combining what is sometimes referred to as observation driven and parameter driven models. This class of models arise from straight-forward assumptions on how violence or crime evolves over space and time and has use in the statistical modeling of situations where there is an expected repeat or near-repeat victimization. In Chapter 2 we present the spatially correlated self-exciting model and the reaction-diffusion self-exciting model to analyze the number of violent events in different regions in Iraq. We also demonstrate how Laplace approximations can be used to conduct efficient Bayesian inference. We further show how the choice of the latent structure matters in this problem. In Chapter 3 we generalize the spatially correlated self-exciting model and show how it extends the classic integer generalized auto-regressive conditionally heteroskedastic, or INGARCH, model to account for spatial correlation and improves the second order properties of the INGARCH model. We refer to this new class of models as the spatially correlated INGARCH, or SPINGARCH, model. We show how the spatially correlated self-exciting model is similar to the SPINGARCH(0,1) model. Finally in Chapter 4 we present a fast extended Laplace approximation algorithm for fitting the SPINGARCH(0,1) model demonstrating empirically how the extended Laplace approximation method reduces a bias that exists in the Laplace approximation method while performing much quicker than Markov Chain Monte Carlo approaches.