Spatial regression is one of the central topics in spatial statistics. Based on the goals, interpretation or prediction, spatial regression models can be classified into two categories, linear mixed regression models and nonlinear regression models. This dissertation explored these models and their real world applications. New methods and models were proposed to overcome the challenges in practice. There are three major parts in the dissertation.

In the first part, nonlinear regression models were embedded into a multistage workflow to predict the spatial abundance of reef fish species in the Gulf of Mexico. There were two challenges, zero-inflated data and out of sample prediction. The methods and models in the workflow could effectively handle the zero-inflated sampling data without strong assumptions. Three strategies were proposed to solve the out of sample prediction problem. The results and discussions showed that the nonlinear prediction had the advantages of high accuracy, low bias and well-performed in multi-resolution.

In the second part, a two-stage spatial regression model was proposed for analyzing soil carbon stock (SOC) data. In the first stage, there was a spatial linear mixed model that captured the linear and stationary effects. In the second stage, a generalized additive model was used to explain the nonlinear and nonstationary effects. The results illustrated that the two-stage model had good interpretability in understanding the effect of covariates, meanwhile, it kept high prediction accuracy which is competitive to the popular machine learning models, like, random forest, xgboost and support vector machine.

A new nonlinear regression model, Gaussian process BART (Bayesian additive regression tree), was proposed in the third part. Combining advantages in both BART and Gaussian process, the model could capture the nonlinear effects of both observed and latent covariates. To develop the model, first, the traditional BART was generalized to accommodate correlated errors. Then, the failure of likelihood based Markov chain Monte Carlo (MCMC) in parameter estimating was discussed. Based on the idea of analysis of variation, back comparing and tuning range, were proposed to tackle this failure. Finally, effectiveness of the new model was examined by experiments on both simulation and real data.

The Gulf of Mexico (or “Gulf”) is of critical significance to the oil and gas industries’ offshore production, but the potential for accidental petrochemical influx into the Gulf due to such processes is high; two of the largest marine oil spills in history, Pemex's Ixtoc I spill (1979) and British Petroleum's (BP) Deepwater Horizon (2010), have occurred in the region. However, the Gulf is also of critical significance to thousands of unique species, many of which may be irreparably harmed by accidental petrochemical exposure. To better manage the conservation and recovery of marine species in the Gulf ecosystem, a Petrochemical Vulnerability Index was developed to determine the potential impact of a petrochemical influx on Gulf marine fishes, therein providing an objective framework with which to determine the best immediate and long term management strategies for resource managers and decision-makers. The resulting Petrochemical Vulnerability Index (PVI) was developed and applied to all bony fishes and shark/ray species in the Gulf of Mexico (1,670 spp), based on a theoretical petrochemical vulnerability framework developed by peer review. The PVI for fishes embodies three key facets of species vulnerability: likelihood of exposure, individual sensitivity, and population resilience, and comprised of 11 total metrics (Distribution, Longevity, Mobility, Habitat, Pre-Adult Stage Length, Pre-Adult Exposure; Increased Adult Sensitivity Due to UV Light, Increased Pre-Adult Sensitivity Due to UV Light; and Abundance, Reproductive Turnover Rate, Diet/Habitat Specialization). The resulting PVI can be used to guide attention to the species potentially most in need of immediate attention in the event of an oil spill or other petrochemical influx, as well as those species that may require intensive long-term recovery. The scored relative vulnerability rankings can also provide information on species that ought to be the focus of future toxicological research, by indicating which species lack toxicological data, and may potentially experience significant impacts.