Methods for spatial hierarchical generalized linear mixed models

Abstract

Spatial statistics is a growing branch of statistics. It deals with data having a spatial component. This component gives a structural relationship to locations both near each other and further in distance. The spatial nature of this data can be difficult to analyze. Recent advancements in spatial statistics have developed a parametric approach to analyzing non-Guassian spatial data by the use of hierarchical generalized linear mixed models (HGLMM). This parametric approach allows HGLMM to be estimated without the use of posterior distributions and Monte Carlo Markov Chains (MCMC), but, rather integrating out the unobserved latent variables. This framework has facilitated new developments in understanding spatial statistics, without the issues MCMC presents. Given model development, spatial models can be modeled more quickly and accurately, even if the data is large and sparse. The focus of this dissertation is on the development of the hierarchical generalized linear mixed model for spatial data.

We begin in Chapter 2, by developing a framework for modeling correlated non-Gaussian data using a parametric approach to a HGLMM. We start the dissertation by outlining a Laplace approximation with Newton-Raphson to marginally estimate the covariance parameters by integrating out the fixed and latent variables. We then outline procedures for estimating fixed effect and latent variables at observed locations, while predicting latent variables at unobserved locations. Simulations are used to evaluate estimations. Two data examples are used to illustrate the utility of the HGLMM for spatial data, using data of baseball statistics and moose counts.

In Chapter 3, we develop conditional simulation methods based on the HGLMM, to predict unobserved values, and functions of unobserved values, within a domain. We evaluate the performance of our predictions using simulated values. We also employ a case study to estimate the total number of moose in a given location.

In Chapter 4, we develop the HGLMM for use on data with a large sample size. We show how the estimation of spatial parameters can be computationally more efficient with the use of reduced rank models and Sherman Morrison Woodbury. We then use these models to estimate covariates, and predict values at unobserved location. We conclude this work, in Chapter 5, by simulating a variety of different scenarios to further understand the effectiveness of the HGLMM.

The methods developed in this work serve to advance the modeling of spatial statistics. The benefits of understanding quick and efficient ways to model spatial data are countless in many disciplines, including ecology, geology, and epidemiology.