With continued advances in Geographic Information Systems and related computational technologies, researchers in diverse fields like forestry, environmental health, climate sciences etc. have growing interests in analyzing large scale data sets measured at a substantial number of geographic locations. Geostatistical models used to capture the space varying relationships in such data are often accompanied by onerous computations which prohibit the analysis of large scale spatial data sets. Less burdensome alternatives proposed recently for analyzing massive spatial datasets often lead to inaccurate inference or require slow sampling process. Bayesian inference, while attractive for accommodating uncertainties through their hierarchical structures, can become computationally onerous for modeling massive spatial data sets because of their reliance on iterative estimation algorithms. My dissertation research aims at developing computationally scalable Bayesian geostatistical models that provide valid inference through highly accelerated sampling process. We also study the asymptotic properties of estimators in spatial analysis.
In Chapter 2 and 3, we develop conjugate Bayesian frameworks for analyzing univariate and multivariate spatial data. We propose a conjugate latent Nearest-Neighbor Gaussian Process (NNGP) model in Chapter 2, which uses analytically tractable posterior distributions to obtain posterior inferences, including the large dimensional latent process. In Chapter 3, we focus on building conjugate Bayesian frameworks for analyzing multivariate spatial data. We utilize Matrix-Normal Inverse-Wishart(MNIW) prior to propose conjugate Bayesian frameworks and algorithms that can incorporate a family of scalable spatial modeling methodologies.
In Chapter 4, we pursue general Bayesian modeling methodologies beyond a conjugate Bayesian hierarchical modeling. We build scalable versions of a hierarchical linear model of coregionalization (LMC) and spatial factor models, and propose a highly accelerated block update MCMC algorithm. Using the proposed Bayesian LMC model, we extend scalable modeling strategies for a single process into multivariate process cases.
All proposed frameworks are tested on simulated data and fit to real data sets with observed locations numbering in the millions. Our contribution is to offer practicing scientists and spatial analysts practical and flexible scalable hierarchical models for analyzing massive spatial data sets.
In Chapter 5, we investigate the asymptotic properties of the estimators in spatial analysis. We formally establish results on the identifiability and consistency of the nugget in spatial models based upon the Gaussian process within the framework of in-fill asymptotics, i.e. the sample size increases within a sampling domain that is bounded. We establish the identifiability of parameters in the Matern covariance function and the consistency of their maximum likelihood estimators in the presence of discontinuities due to the nugget.