Bayesian Spatial Modeling of Complex and High Dimensional Data

The main objective of this dissertation is to apply Bayesian modeling to different complex and high-dimensional spatial data sets. I develop Bayesian hierarchical spatial models for both the observed location and the observation variable. Throughout this dissertation I execute the inference of the posterior distributions using Markov chain Monte Carlo by developing computational strategies that can reduce the computational cost.

I start with a "high level" image analysis by modeling the pixels with a Gaussian process and the objects with a marked-point process. The proposed method is an automatic image segmentation and classification procedure which simultaneously detects the boundaries and classifies the objects in the image into one of the predetermined shape families. Next, I move my attention to the piecewise non-stationary Gaussian process models and their computational challenges for very large data sets. I simultaneously model the non-stationarity and reduce the computational cost by using the innovative technique of full-scale approximation. I successfully demonstrate the proposed reduction technique to the Total Ozone Matrix Spectrometer (TOMS) data. Furthermore, I extend the reduction method for the non-stationary Gaussian process models to a dynamic partition of the space by using a modified Treed Gaussian Model. This modification is based on the use of a non-stationary function and the full-scale approximation. The proposed model can deal with piecewise non-stationary geostatistical data with unknown partitions. Finally, I apply the method to the TOMS data to explore the non-stationary nature of the data.