Spatio-Temporal Data Analysis at Scale Using Models Based on Gaussian Processes
Gaussian processes are the most commonly used statistical model for spatial and spatio-temporal processes that vary continuously. They are broadly applicable in the physical sciences and engineering and are also frequently used to approximate the output of complex computer models, deterministic or stochastic. We undertook research related to theory, computation, and applications of Gaussian processes as well as some work on estimating extremes of distributions for which a Gaussian process assumption might be inappropriate. Our theoretical contributions include the development of new classes of spatial-temporal covariance functions with desirable properties and new results showing that certain covariance models lead to predictions with undesirable properties. To understand how Gaussian process models behave when applied to deterministic computer models, we derived what we believe to be the first significant results on the large sample properties of estimators of parameters of Gaussian processes when the actual process is a simple deterministic function. Finally, we investigated some theoretical issues related to maxima of observations with varying upper bounds and found that, depending on the circumstances, standard large sample results for maxima may or may not hold. Our computational innovations include methods for analyzing large spatial datasets when observations fall on a partially observed grid and methods for estimating parameters of a Gaussian process model from observations taken by a polar-orbiting satellite. In our application of Gaussian process models to deterministic computer experiments, we carried out some matrix computations that would have been infeasible using even extended precision arithmetic by focusing on special cases in which all elements of the matrices under study are rational and using exact arithmetic. The applications we studied include total column ozone as measured from a polar-orbiting satellite, sea surface temperatures over the Pacific Ocean, and annual temperature extremes at a site in New York City. In each of these applications, our theoretical and computational innovations were directly motivated by the challenges posed by analyzing these and similar types of data.