Jason A. Duan
Associate ProfessorUniversity of Texas, McCombs School of Business
Bayesian Nonparametric and Differential Equation Approaches
The present thesis addresses three important issues in modelling spatio-temporal data: (i) develop a flexible nonparametric Bayesian methodology for spatial random effect models; (ii) extend the current Bayesian nonparametric approach to model discrete spatial data; and (iii) construct a spatio-temporal point process that incorporates established scientific models into a Bayesian hierarchical model. The spatial Dirichlet process(SDP) is the first attempt to introduce a nonparametric model for a neither Gaussian nor stationary spatial process. The SDP arises as a probability weighted collection of random surfaces. This can be unattractive for modelling, hence inferential purposes since it insists that a process realization is one of these surfaces. In Chapter 2, we introduce a generalized spatial Dirichlet process(GSDP) model for the spatial effects that allows different surface selection at different sites. Moreover, we can specify the model to preserve the property that the marginal distribution of the effect at each site still comes from a Dirichlet process. The development is offered constructively, providing a multivariate extension of the stick-breaking representation of the weights. We then introduce mixing using this generalized spatial Dirichlet process (GSDP). We illustrate the fitting of this novel model with a simulated data set and demonstrate how to embed the GSDP within a dynamic linear model. In Chapter 3, we extend the SDP to a generalized linear model(GLM) setting by proposing a Bayesian nonparametric spatial approach to analyze disease mapping data. We develop a hierarchical specification using random effects modelled with a spatial Dirichlet process prior. We introduce a dynamic formulation for the spatial random effects to apply the model to spatio-temporal settings. Chapter 4 introduces a novel structured model for spatio-temporal point processes. We formulate a dynamic Cox process model where the evolution of latent intensity is governed by deterministic and stochastic differential equations describing the population growth mechanisms. We construct a Bayesian hierarchical model based on this point process and propose a process convolution approximation for statistical inference. We address the Bayesian estimation and space-time prediction issues and illustrate with simulated and real house construction data examples.