Dimension Reduction Approaches for Analyzing Large Spatial Datasets
by
Alan E. Gelfand
Duke University

Fitting hierarchical spatial models often involves expensive matrix decompositions whose computational complexity increases in cubic order with the number of spatial locations, rendering such models infeasible for large spatial data sets.This computational burden is exacerbated in multivariate settings with several spatially dependent response variables. It is also aggravated when data is collected at frequent time points and spatiotemporal process models are used.

Dimension reduction approaches provide one strategy for addressing this problem. Here, we argue for the use of predictive process models for spatial and spatiotemporal data. Every spatial (or spatiotemporal) process induces a predictive process model (in fact, arbitrarily many of them). The latter models project process realizations of the former to a lower-dimensional subspace; we achieve the flexibility to accommodate nonstationary, non-Gaussian, possibly multivariate, possibly spatiotemporal processes in the context of large datasets.

We discuss attractive theoretical properties of these predictive processes as well as a computationally feasible template encompassing these diverse settings. Finally, we illustrate with spatial modelling of forest biomass where interest lies in detecting how biomass changes across the landscape (as a continuous surface) and how homogeneous it is across the region. We employ point-referenced biomass data observed at 9, 500 locations obtained from the USDA Forest Service Forest Inventory and Analysis.

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Date received: June 17, 2007