Date of Award

12-2013

Degree Name

Doctor of Philosophy

Department

Statistics

First Advisor

Dr. Rajib Paul

Second Advisor

Dr. Magdalena Niewiadomska-Bugaj

Third Advisor

Dr. Joseph McKean

Fourth Advisor

Dr. Kathleen Baker

Abstract

One of the most common goals of geostatistical analysis is that of spatial prediction, in other words: filling in the blank areas of the map. There are two popular methods for accomplishing spatial prediction. Either kriging, or Bayesian hierarchical models. Both methods require the inverse of the spatial covariance matrix of the data. As the sample size, n, becomes large, both of these methods become impractical. Reduced rank spatial models (RRSM) allow prediction on massive datasets without compromising the complexity of the spatial process. This dissertation focuses on RRSMs, particularly situations where the data follow non-Gaussian distributions.

The manner in which data can be non-Gaussian varies, and we address multiple such situations. We begin by developing multivariate log-normal kriging and block kriging equations, and explain how to implement them for data that are compositional. We also propose a robust non-parametric approach to parameter estimation for kriging.

Moving towards hierarchical models, we develop an empirical Bayes method to estimate parameters for heavy-tailed distributions. After, we turn to the problem of knot selection. The topic is under-represented in literature despite its importance to reduced rank spatial models.

Finally, we explore fully Bayesian models for non-Gaussian data based on scale mixtures of Gaussians. The method is flexible enough to model a variety of distributional forms. Furthermore, we discuss how our models can be extended to consider dependencies in space as well as time.

Access Setting

Dissertation-Open Access

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