On Bayesian Methods for Spatial Point Processes

Spatial point pattern data are routinely encountered. A flexible regression model for the underlying intensity is essential to characterizing and understanding the pattern. Spatial point processes are a widely used to model for such data. Additional measurements are often available along with spatial points, which are called marks. Such data can be modeled using marked spatial point processes. The first part of this dissertation focuses on the heterogeneity of point processes. We propose a Bayesian semiparametric model where the observed points follow a spatial Poisson process with an intensity function which adjusts a nonparametric baseline intensity with multiplicative covariate effects. The baseline intensity is approached with a powered Chinese restaurant process (PCRP) prior. The parametric regression part allows for variable selection through the spike-slab prior on the regression coefficients. An efficient Markov chain Monte Carlo (MCMC) algorithm is developed. The performance of the methods is validated in an extensive simulation study and the Beilschmiedia pendula trees data. Spatial smoothness is often observed in some environmental spatial point pattern data, and the PCRP may have lower efficiency for such data since it allows more flexibility without any spatial constraint. Distance dependent Chinese restaurant process (ddCRP) can be easily realized by introducing a decay function to Chinese restaurant process. The second part of this dissertation introduces the ddCRP model with Bayesian inference methods, whose performance is illustrated using simulation study. In the third part, we investigate the marked spatial point process, which is motivated by the basketball shot data. We develop a Bayesian joint model of the mark and the intensity, where the intensity is incorporated in the mark’s model as a covariate. An MCMC algorithm is developed to draw posterior samples from this model. Two Bayesian model comparison criteria, the modified Deviance Information Criterion and the modified Logarithm of the Pseudo-Marginal Likelihood, are developed to assess the fitness of different models focusing on the mark. Simulation study and application to NBA basketball shot data are conducted to show the performance of proposed methods