High-Dimensional Bayesian Geostatistics

Abstract

With the growing capabilities of Geographic Information Systems (GIS) and user-friendly software, statisticians today routinely encounter geographically referenced data containing observations from a large number of spatial locations and time points. Over the last decade, hierarchical spatiotemporal process models have become widely deployed statistical tools for researchers to better understand the complex nature of spatial and temporal variability. However, fitting hierarchical spatiotemporal models often involves expensive matrix computations with complexity increasing in cubic order for the number of spatial locations and temporal points. This renders such models unfeasible for large data sets. This article offers a focused review of two methods for constructing well-defined highly scalable spatiotemporal stochastic processes. Both these processes can be used as "priors" for spatiotemporal random fields. The first approach constructs a low-rank process operating on a lower-dimensional subspace. The second approach constructs a Nearest-Neighbor Gaussian Process (NNGP) that ensures sparse precision matrices for its finite realizations. Both processes can be exploited as a scalable prior embedded within a rich hierarchical modeling framework to deliver full Bayesian inference. These approaches can be described as model-based solutions for big spatiotemporal datasets. The models ensure that the algorithmic complexity has ~ n floating point operations (flops), where n the number of spatial locations (per iteration). We compare these methods and provide some insight into their methodological underpinnings.

Keywords: Bayesian statistics; Gaussian process; Nearest Neighbor Gaussian process (NNGP); low rank Gaussian process; predictive process; sparse Gaussian process; spatiotemporal statistics.

Figure 1

95% credible intervals for the nugget for 40 different low-rank radial-basis models with knots varying between 5 and 200 in steps of 5. The horizontal line at τ² = 5 denotes the true value of τ² with which the data was simulated.

Figure 2

Comparing estimates of a simulated random field using a full Gaussian Process (Full GP) and a Gaussian Predictive process (PPGP) with 64 knots. The oversmoothing by the low-rank predictive process is evident.

Figure 3

Sparsity using directed acyclic graphs.

Figure 4

Structure of the factors making up the sparse ${\stackrel{\sim }{K}}_{\theta }^{-1}$ matrix.

Figure 5

95% credible intervals for the effective spatial range from an NNGP model with m = 10 and a full GP model fitted to 10 different simulated datasets with true effective range fixed at values between 0.1 and 1.0 in increments of 0.1.