SummaryWith scientific data available at geocoded locations, investigators are increasingly turning to spatial process models for carrying out statistical inference. Over the last decade, hierarchical models implemented through Markov chain Monte Carlo methods have become especially popular for spatial modelling, given their flexibility and power to fit models that would be infeasible with classical methods as well as their avoidance of possibly inappropriate asymptotics. However, 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 are collected at frequent time points and spatiotemporal process models are used. With regard to this challenge, our contribution is to work with what we call 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, thereby reducing the computational burden. Hence, we achieve the flexibility to accommodate non-stationary, non-Gaussian, possibly multivariate, possibly spatiotemporal processes in the context of large data sets. We discuss attractive theoretical properties of these predictive processes. We also provide a computational template encompassing these diverse settings. Finally, we illustrate the approach with simulated and real data sets.

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.

The Gaussian process is an indispensable tool for spatial data analysts. The onset of the “big data” era, however, has lead to the traditional Gaussian process being computationally infeasible for modern spatial data. As such, various alternatives to the full Gaussian process that are more amenable to handling big spatial data have been proposed. These modern methods often exploit low-rank structures and/or multi-core and multi-threaded computing environments to facilitate computation. This study provides, first, an introductory overview of several methods for analyzing large spatial data. Second, this study describes the results of a predictive competition among the described methods as implemented by different groups with strong expertise in the methodology. Specifically, each research group was provided with two training datasets (one simulated and one observed) along with a set of prediction locations. Each group then wrote their own implementation of their method to produce predictions at the given location and each was subsequently run on a common computing environment. The methods were then compared in terms of various predictive diagnostics. Supplementary materials regarding implementation details of the methods and code are available for this article online. Electronic Supplementary Material Supplementary materials for this article are available at 10.1007/s13253-018-00348-w.

Advances in Geographical Information Systems (GIS) and Global Positioning Systems (GPS) enable accurate geocoding of locations where scientific data are collected. This has encouraged collection of large spatial datasets in many fields and has generated considerable interest in statistical modeling for location-referenced spatial data. The setting where the number of locations yielding observations is too large to fit the desired hierarchical spatial random effects models using Markov chain Monte Carlo methods is considered. This problem is exacerbated in spatial-temporal and multivariate settings where many observations occur at each location. The recently proposed predictive process, motivated by kriging ideas, aims to maintain the richness of desired hierarchical spatial modeling specifications in the presence of large datasets. A shortcoming of the original formulation of the predictive process is that it induces a positive bias in the non-spatial error term of the models. A modified predictive process is proposed to address this problem. The predictive process approach is knot-based leading to questions regarding knot design. An algorithm is designed to achieve approximately optimal spatial placement of knots. Detailed illustrations of the modified predictive process using multivariate spatial regression with both a simulated and a real dataset are offered.

This article introduces yaImpute, an R package for nearest neighbor search and imputation. Although nearest neighbor imputation is used in a host of disciplines, the methods implemented in the yaImpute package are tailored to imputation-based forest attribute estimation and mapping. The impetus to writing the yaImpute is a growing interest in nearest neighbor imputation methods for spatially explicit forest inventory, and a need within this research community for software that facilitates comparison among different nearest neighbor search algorithms and subsequent imputation techniques. yaImpute provides directives for defining the search space, subsequent distance calculation, and imputation rules for a given number of nearest neighbors. Further, the package offers a suite of diagnostics for comparison among results generated from different imputation analyses and a set of functions for mapping imputation results.