Kristian C. Lum
Assistant Research ProfessorVirginia Tech
Bayesian Spatial Quantile Regression
Spatial quantile regression is the combination of two separate and individually well developed ideas that, to date, has barely been explored. Quantile regression seeks to model each quantile of an outcome distribution, whether separately or jointly, conditional upon covariates. Spatial methods have been developed for instances when spatial dependence ought to be incorporated into the model, whether to adjust for the decreased effective sample size that comes with highly correlated data or to allow the ability to create a model-based spatial surface that interpolates between the data collected. Combining the spatial methods with quantile regression, this dissertation proposes and studies the properties of several process models for quantile regression that incorporate spatial dependence. In each chapter, we present an application for the model presented therein. In all cases, we are able to achieve improved check loss by incorporating a spatial component into the model. In Chapter 1, the introduction, we motivate this work by exploring several examples that demonstrate the utility of both quantile regression and spatial models separately. In Chapter 2, we present the asymmetric Laplace process (ALP), a process model suitable for quantile regression. We derive several covariance properties of various specifications of this model and discuss the advantages and disadvantages of each option. As an example, we apply this model to real estate data. In Chapter 3, we extend the ALP to accommodate large data sets by incorporating a predictive process covariance structure and sampling scheme into the ALP. By doing so, we create the asymmetric Laplace predictive process (ALPP), which we apply to a data set of approximately 3,000 births in the state of North Carolina in the year 2000. Here, interest lies primarily on the relationship between various maternal covariates and the lower tails of the distribution of birth weights. In Chapter 4, we again extend the ALP, this time to incorporate a temporal component. We discuss several ways in which both continuous and discrete time can be included in the model. We further develop and outline the details of a discrete time spatial dynamic model. We apply this model to a data set of spatially and temporally indexed temperatures, given elevation. In Chapter 5, we propose an alternative to the ALP, which re-scales a Gaussian process using two separate scale parameters. We investigate the properties of this double normal process (DNP), and present a simulation example to illustrate the utility (and disutility) of this model.