UC Santa Cruz

This work provides a Bayesian nonparametric modeling framework for spatial point processes to account for the irregular domain over which the resulting point pattern occurs in the model formulation while balancing flexible inference with efficient implementation. We start with models for the spatial Poisson process, which assumes independence among points given the number of occurrences, and progress to models for Hawkes processes over space and space-time that capture the self-triggering behaviors and relax the independence assumption. We develop nonparametric Bayesian modeling approaches for Poisson processes using weighted combinations of structured beta densities to represent the point process intensity function. For a regular spatial domain, i.e., the unit square, the model construction implies a Bernstein-Dirichlet prior for the Poisson process density, which supports flexible inference about point process functionals with theoretical guarantees. The key contribution is two classes of flexible and computationally efficient models for spatial Poisson process intensities over irregular domains. We address the choice or estimation of the number of beta basis densities and develop methods for prior specification. For the spatial Hawkes process, we develop a semi-parametric modeling approach, leveraging its clustering representation defined as the superposition of an immigrant Poisson process and several offspring Poisson clustering processes centered on parent points generated by earlier generations. We apply the model for the Poisson process developed earlier to the latent immigrant Poisson process and complete the hierarchical model for the spatial Hawkes process with parametric formulations for the offspring Poisson processes and a model for the latent branching structure that specifies lineage among points. Finally, we develop a nonparametric model for the spatial offspring Poisson process under the assumption of spatial isotropy, which reduces modeling for the spatial offspring density to that for the spatial offspring-parent distance density. Such construction allows the model to be free from the implied tail behavior constraints imposed by existing parametric options for the offspring density kernel. We incorporate such a method to model for space-time Hawkes processes. For all methods developed in the dissertation, we design posterior simulation algorithms for full inference on key point process functionals and model checking techniques to examine the model fit. Model capacity is demonstrated with numerous simulation studies, and we focus on real data examples using crime point patterns from the city of Boston.