Bayesian Modeling and Computation for Complex Spatial Point Patterns

This thesis focuses on solving some problems associated with complex spatial point patterns from each modeling and computational perspective. Chapter 1 reviews spatial point patterns and introduces repulsive point processes. We also discuss some potential problems for spatial point patterns, to which this thesis contributes.

In Chapter 2, we begin with modeling the space and time dependence of crime

events in San Francisco. We imagine event times as wrapped around a circle of circumference 24 hours, then introduce the circular dependence of crime times within a log Gaussian Cox process (LGCP). To construct a valid space and circular time LGCP, we propose valid separable and nonseparable space and circular time covariance matrices.

We also compare the proposed models with a nonhomogeneous

Poisson process (NHPP) through the model validation strategy for Cox processes. Our proposed models show better fitting and capture the space and circular time dependence of the intensity surface for crime events.

In Chapter 3, we propose a new Bayesian inference scheme for univariate and multivariate LGCPs. Although a LGCP is a flexible class which can incorporate space and spatio-temporal clustering dependence, Bayesian inference for LGCP is notoriously computationally tough because sampling of high dimensional Gaussian processes

(GP) and their hyperparameters involves high correlation and requires some matrix factorization of high dimensional covariance functions. We tackle with this problem by considering a separable inference scheme for GP and hyperparameters. Our approach utilizes the pseudo-marginal Markov chain Monte Carlo (MCMC) and

estimate the approximate marginal posterior distribution of parameters. Given approximate posterior samples of parameters, the efficient sampling of high dimensional GP is available. We demonstrate the performance of our algorithm by comparing with preceding MCMC algorithms and show the better performance with respect to

the computational time and the accurate parameter recovery for simulated datasets.

In Chapter 4, we develop an approximate Bayesian computation (ABC) scheme for two types of repulsive point processes, Gibbs point processes and determinantal point processes, for which the exact Bayesian inference is unavailable or involves poor mixing. Although these processes have been investigated in mathematics and physics communities, the exact inference of these processes is generally unavailable. Furthermore, the straightforward model comparison between both types of processes has not been investigated though both processes might show different repulsive patterns.

We propose an ABC algorithm for these processes, this approach enables us to compare both processes under a unied approximation strategy. We demonstrate the recovery of parameters and true model classes through the simulation study. The result suggests the true model can be recovered for the relatively large number of points.

In Chapter 5, we propose some models for origin-destination point patterns motivated from two car theft datasets. Both datasets have theft (origin) and corresponding recovered (destination) location information. First one is theft and recovered locations in Neza region in Mexico. Although this dataset has some covariate information,

but some of recovered points are located outside Neza region and some of them are missing. The other one is theft and recovered locations in Belo Horizonte in Brazil. Although all events have origin and destination information within Belo Horizonte, covariate information is unavailable. We suggest four modeling directions for both datasets, different ways of incorporating the spatial dependence between origin and destination pairs. The proposed models are compared with independent

models and show their superior performance.