Bayesian nonstationary spatial modeling for very large datasets

Abstract

With the proliferation of modern high‐resolution measuring instruments mounted on satellites, planes, ground‐based vehicles, and monitoring stations, a need has arisen for statistical methods suitable for the analysis of large spatial datasets observed on large spatial domains. Statistical analyses of such datasets provide two main challenges: first, traditional spatial‐statistical techniques are often unable to handle large numbers of observations in a computationally feasible way; second, for large and heterogeneous spatial domains, it is often not appropriate to assume that a process of interest is stationary over the entire domain. We address the first challenge by using a model combining a low‐rank component, which allows for flexible modeling of medium‐to‐long‐range dependence via a set of spatial basis functions, with a tapered remainder component, which allows for modeling of local dependence using a compactly supported covariance function. Addressing the second challenge, we propose two extensions to this model that result in increased flexibility: first, the model is parameterized on the basis of a nonstationary Matérn covariance, where the parameters vary smoothly across space; second, in our fully Bayesian model, all components and parameters are considered random, including the number, locations, and shapes of the basis functions used in the low‐rank component. Using simulated data and a real‐world dataset of high‐resolution soil measurements, we show that both extensions can result in substantial improvements over the current state‐of‐the‐art. Copyright © 2013 John Wiley & Sons, Ltd.

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