The field of spatial and spatio-temporal statistics is increasingly faced with the challenge of very large datasets. Examples include data obtained from remote sensing satellites, global weather stations, outputs from climate models and medical imagery. The classical approach to spatial and spatio-temporal modeling is extremely computationally expensive when the datasets are large. Dimension-reduced modeling approach has proved to be effective in such situations. In this thesis I focus on the problem of modeling two spatio-temporal processes where the primary goal is to predict one process from the other and where the datasets for both processes are large.
I outline a general dimension-reduced Bayesian hierarchical approach to modeling of two spatio-temporal processes. The spatial structures of both processes are modeled in terms of a low number of basis vectors, hence reducing the spatial dimension of the problem. The temporal evolution of the spatio-temporal processes is then modeled through the coefficients (i.e. amplitudes) of the basis functions. I demonstrate that known multivariate statistical methods, Maximum Covariance Analysis (MCA) and Canonical Correlation Analysis (CCA), can be used to obtain basis vectors for dimension-reduced modeling of two spatio-temporal processes. Furthermore, I present a new method of obtaining data-dependent basis vectors that is geared to the goal of predicting one process from the other. The new basis vectors are called Maximum Covariance Patterns (MCPs) and an orthogonal version is called Orthogonal Maximum Covariance Patterns (OMCPs). I apply these methods to a statistical downscaling example, where surface temperatures on a coarse grid over the Antarctic are downscaled onto a finer grid.