Environmental datasets such as those from remote-sensing platforms and sensor networks are often spatial, temporal, and very large or even massive. Analyzing large spatial or spatio-temporal datasets can be challenging and dimension reduction is usually necessary. In this work, we exploit the Spatial Random Effects (SRE) model with a fixed number of known but not necessarily orthogonal (multi-resolutional) spatial basis functions. The SRE model allows a flexible family of nonstationary covariance functions and the fixed number of basis functions results in dimension reduction and thus efficient computation. We propose priors on the parameters of the SRE model in a fully Bayesian framework. These priors are based on the covariance matrix parameterized in terms of Givens angles and eigenvalues, and they recognize the multi-resolutional nature of the basis functions. We compare this Givens-angle prior to other methods in a simulation study, to show its advantages and apply it to a large remote-sensing spatial dataset. We also apply the SRE model with the Givens-angle prior in a Bayesian meta analysis, where outputs from six different regional climate outputs (RCMs) are combined to construct a consensus climate signal with “votes” from each RCM.
Moreover, we extend the SRE model to the Spatio-Temporal Random Effects (STRE) model for massive spatio-temporal datasets. We explicitly model the measurement error, the non-dynamic fine-scale variation, the dynamic spatial variation, and the trend. The optimal spatio-temporal predictions are derived efficiently through the fixed-rank model and a rapid recursive updating procedure through the Kalman filter. Formulas for optimal smoothing, filtering, and forecasting are derived. The improvement of combining past and current data using the methodology called Fixed Rank Filtering (FRF) to predict the current hidden process of interest, is illustrated with a simulation experiment. The methodology is also applied to a large spatio-temporal remote-sensing dataset.