Le, A. (STAT) – Bayesian Nonparametric Analysis of Densities for Replicated Point Patterns

Many scientific applications produce repeated point pattern realizations across subjects, regions, or time. While such point patterns exhibit individual variation, we assume they arise from related point processes that share a common distributional structure. This dissertation develops a Bayesian nonparametric modeling framework built around an interpretable baseline. We work with Poisson processes, such that the point process stochastic mechanism is characterized by the total intensity and a density with compact support. Flexible, parsimonious weighted combinations of beta densities represent both the baseline and the replicate-specific densities. The weights corresponding to each replicate encode the features that characterize its density and are pooled hierarchically across replicates to estimate the shared baseline. Throughout, we illustrate the framework using bike-share demand at a Chicago Divvy station, where weekly demand shares a common daily pattern while its features evolve across weeks. We lay the foundation with a conditionally independent model, relating the replicates through a Dirichlet process prior centered on the shared baseline. The model admits a Pólya urn representation that yields fully conjugate posterior updates and partially parallel computation across replicates. Inference proceeds at both the shared and replicate-specific levels, and predictive inference on the density of demand for a new week follows readily through the baseline. Moving to a dynamic extension, we use the representation of the replicate-specific discrete random distributions to express structured dependence separately through the atoms and the weights. Stochastic processes on these components relate the locations of density features and their relative importance. The weights follow a geometric construction that keeps the model parsimonious. For the atoms, we develop a novel stochastic process with the baseline as its marginal distribution, so its interpretation is unchang

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