Hierarchical statistical modeling of big spatial datasets using the exponential family of distributions

Big spatial datasets are very common in scientific problems, such as those involving remote sensing of the earth by satellites, climate-model output, small-area samples from national surveys, and so forth. In this article, our interest lies primarily in very large, non-Gaussian datasets. We consider a hierarchical statistical model consisting of a conditional exponential family model for the data and an underlying (hidden) geostatistical process for some transformation of the (conditional) mean of the data model. Within this hierarchical model, dimension reduction is achieved by modeling the geostatistical process as a linear combination of a fixed number of spatial basis functions, which results in substantial computational speedups. These models do not rely on specifying a spatial-weights matrix, and no assumptions of homogeneity, stationarity, or isotropy are made. Our approach to inference using these models is empirical-Bayesian in nature. We develop maximum likelihood (ML) estimates of the unknown parameters using Laplace approximations in an expectation-maximization (EM) algorithm. We illustrate the performance of the resulting empirical hierarchical model using a simulation study. We also apply our methodology to analyze a remote sensing dataset of aerosol optical depth.