High-Dimensional Bayesian Geostatistics

With the growing capabilities of Geographic Information Systems(GIS) and user-friendly software, statisticians today routinely encounter geographicallyreferenced data containing observations from a large number of spatial locationsand time points. Over the last decade, hierarchical spatiotemporal processmodels have become widely deployed statistical tools for researchers to better understandthe complex nature of spatial and temporal variability. However, fittinghierarchical spatiotemporal models often involves expensive matrix computationswith complexity increasing in cubic order for the number of spatial locations andtemporal points. This renders such models unfeasible for large data sets. Thisarticle offers a focused review of two methods for constructing well-defined highlyscalable spatiotemporal stochastic processes. Both these processes can be used as“priors” for spatiotemporal random fields. The first approach constructs a lowrankprocess operating on a lower-dimensional subspace. The second approachconstructs a Nearest-Neighbor Gaussian Process (NNGP) that ensures sparseprecision matrices for its finite realizations. Both processes can be exploited asa scalable prior embedded within a rich hierarchical modeling framework to deliverfull Bayesian inference. These approaches can be described as model-basedsolutions for big spatiotemporal datasets. The models ensure that the algorithmiccomplexity has ∼ n floating point operations (flops), where n the number of spatiallocations (per iteration). We compare these methods and provide some insightinto their methodological underpinnings