Hierarchical Nearest-Neighbor Gaussian Process Models for Large Geostatistical Datasets

Abstract

Spatial process models for analyzing geostatistical data entail computations that become prohibitive as the number of spatial locations become large. This article develops a class of highly scalable nearest-neighbor Gaussian process (NNGP) models to provide fully model-based inference for large geostatistical datasets. We establish that the NNGP is a well-defined spatial process providing legitimate finite-dimensional Gaussian densities with sparse precision matrices. We embed the NNGP as a sparsity-inducing prior within a rich hierarchical modeling framework and outline how computationally efficient Markov chain Monte Carlo (MCMC) algorithms can be executed without storing or decomposing large matrices. The floating point operations (flops) per iteration of this algorithm is linear in the number of spatial locations, thereby rendering substantial scalability. We illustrate the computational and inferential benefits of the NNGP over competing methods using simulation studies and also analyze forest biomass from a massive U.S. Forest Inventory dataset at a scale that precludes alternative dimension-reducing methods. Supplementary materials for this article are available online.

Mapping local and global variability in plant trait distributions

“…Gaussian processes (GPs) are widely used for modeling unknown spatial surfaces such as w(s), due to their convenient formulation as a multivariate Gaussian prior for the spatial random effect, unparalleled predictive performance (53), and ease of generating uncertainty-quantified predictions at unobserved locations. We use the computationally effective nearest-neighbor GP (27), which nicely embeds into the Bayesian hierarchical setup as a prior for w(s) in the second stage of the model specification. All technical specifications of the Bayesian spatial model are provided in SI Appendix, section S1.…”

“…More critically, the first two global methodologies emphasized estimating a single trait value per PFT at every location, whereas both ground-based (5,14) and remotely sensed (26) observations suggest that at ecosystem or landscape scales traits would be better represented by distributions. Here, we use an updated version of the largest global database of plant traits (14) coupled with modern Bayesian spatial statistical modeling techniques (27) to capture local and global variability in plant traits. This combination allows the representation of trait variation both within pixels on a gridded land surface and across global environmental gradients.…”

laGP: Large-Scale Spatial Modeling via Local Approximate Gaussian Processes inR

“…Ignoring them in order to work with much smaller, n-sized, matrices will bring a big computational savings with little impact on accuracy. This is a sensible idea: It can be shown to induce a valid stochastic process (Datta, Banerjee, Finley, and Gelfand 2016); when n 1000 the method is fast and accurate, and as n grows the predictions increasingly resemble their full N -data counterparts; and, for smaller n, V n (x) is organically inflated relative to V N (x), acknowledging greater uncertainty in approximation. The simplest version of such a scheme would be via nearest neighbors (NN): X n (x) comprised of closest elements of X N to x.…”

Smoothed Full-Scale Approximation of Gaussian Process Models for Computation of Large Spatial Datasets

“…Sparse approximation techniques either shrink the covariance of distant pairs of spatial locations to zero for yielding a sparse covariance matrix (Furrer et al (2006) ;Kaufman et al (2008)), or assume Gaussian-Markov property of the spatial random field for yielding a sparse precision matrix (Rue and Tjelmeland (2002) ;Lindgren et al (2011)). Another way for inducing sparsity of the precision matrix is to use conditional likelihoods (e.g., Vecchia (1988)), and most recently, Datta et al (2016) proposed a nearest-neighbor GP (NNGP); a new permutation and grouping method for improving the performance of NNGP can be found in Guinness (2016). Most recent "hybrid" methods extending the low-rank models include Nychka et al (2015); Katzfuss (2017); Ma and Kang (2017).…”

“…We first extend the nearest-neighbor GP models developed by Datta et al (2016) to construct a nearest neighboring block GP model. We then propose to apply it to approximate the residual covariance and combine this approximated residual covariance with a reduced-rank predictive process.…”