Bayesian Analysis of Spatial Log-Gaussian Cox Processes

Abstract

In this thesis we present methods and results that build upon the wealth of research into Bayesian inference of log-Gaussian Cox processes, extending the capabilities of the current computational algorithms, and in doing so, expanding the complexity of the models for spatial processes we can fit.

We look to the combination of the integrated nested Laplace approximation (INLA) and the Markov chain Monte Carlo (MCMC) algorithms, the INLA within MCMC (IwM) algorithm introduced by Gómez-Rubio & Rue in `Markov chain Monte Carlo with the Integrated Nested Laplace Approximation' (2018), which exploits the advantages of the separate algorithms while avoiding their pitfalls.

Initially, we look to optimise the efficiency of the IwM algorithm, specifically the INLA implementation, for aggregated point patterns by introducing a method for selecting the data aggregation resolution in combination with the mesh resolution for the SPDE approximation of the Gaussian process to optimise the accuracy of the algorithm while ensuring the algorithm is as efficient as possible.

We then define the Multivariate INLA within MCMC algorithm which involves the modification of the IwM algorithm in order to accept data from multiple disjoint study regions. This algorithm allows the shared data to influence the posterior samples for a particular subset of parameters that are pooled across the study regions. An application of this algorithm that we demonstrated in this thesis is the estimation of the contrasts in socio-economic variable effects on crime in US cities as well as measures of the uncertainty in these estimates.

The Multivariate INLA within MCMC in combination with the optimised selection of resolutions for the mesh and data aggregation will be shown to provide a flexible and efficient method for the inference of more complex spatial models that we could not perform within the INLA framework and with a smaller computational burden than an MCMC-based approach.

Date of Award	22 Jun 2022
Original language	English
Awarding Institution	University of Bath
Supervisor	Theresa Smith (Supervisor), Julian Faraway (Supervisor) & Nicole H. Augustin (Supervisor)