Practical Bayesian modeling and inference for massive spatial data sets on modest computing environments^†

Abstract

With continued advances in Geographic Information Systems and related computational technologies, statisticians are often required to analyze very large spatial data sets. This has generated substantial interest over the last decade, already too vast to be summarized here, in scalable methodologies for analyzing large spatial data sets. Scalable spatial process models have been found especially attractive due to their richness and flexibility and, particularly so in the Bayesian paradigm, due to their presence in hierarchical model settings. However, the vast majority of research articles present in this domain have been geared toward innovative theory or more complex model development. Very limited attention has been accorded to approaches for easily implementable scalable hierarchical models for the practicing scientist or spatial analyst. This article devises massively scalable Bayesian approaches that can rapidly deliver inference on spatial process that are practically indistinguishable from inference obtained using more expensive alternatives.A key emphasis is on implementation within very standard (modest) computing environments (eg, a standard desktop or laptop) using easily available statistical software packages. Key insights are offered regarding assumptions and approximations concerning practical efficiency.

Modeling massive spatial datasets using a conjugate Bayesian linear modeling framework

“…The full latent NNGP model is 200 times faster than the full Gaussian process based model, while the conjugate latent NNGP model uses one tenth of the time required by the latent NNGP model to obtain similar inference on the regression coefficients and latent process. Further simulation experiments conducted by Zhang et al [15] also show that interpolation of the latent process is almost indistinguishable between the conjugate and full models.…”

“…whereX is theñ × p matrix of predictors observed at locations inL and˜ ∼ N (0,D τ ), where˜ is theñ × 1 vector with elements (˜ i ). The second equation in (15) expresses the relationship between the spatial effectsw across the unobserved locations inL and the spatial effects across the observed locations in L. Since there is one underlying random field over the entire domain, the covariance function for the random field specifies thẽ n × n coefficient matrix C. In particular, if w ∼ N (0,…”

“…Turning to a real example, we present a synopsis of the analysis in [15] of a spatial dataset from NASA comprising sea surface temperature (in degrees Centigrade) observations over 2,827,252 spatial locations of which approximately 90% (2,544,527) were used for model fitting and the rest were withheld for cross-validatory predictive assessment. Details of the dataset can be found in http://modis-atmos.gsfc.nasa.gov/index.html and details on the analysis can be found in [15]. The salient feature of the analysis is that a conjugate Bayesian framework for the NNGP model as in (12) was able to deliver full inference including the estimation of the spatial latent effects in about 2387 seconds.…”

Efficient inference of generalized spatial fusion models with flexible specification

“…An acceptable approach is to implement directly in Stan where it then does block updates on . This implementation was used by Zhang et al () and adopted for the fusion model in Wang et al (). However, this approach causes slow gradient evaluation due to custom NNGP log‐likelihood.…”

“…Moraga et al () used integrated nested Laplace approximations based on Rue et al (). Wang et al () adapted an implementation of the nearest neighbour Gaussian process (NNGP) (Datta et al, ; Zhang, Datta, & Banerjee, ) in the Stan modelling language.…”