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

Gaussian process (GP) models encounter computational difficulties with large spatial data sets, because the models’ computational complexity grows cubically with the sample size n. Although a full-scale approximation (FSA) using a block modulating function provides an effective way to approximate GP models, it has several shortcomings. These include a less smooth prediction surface on block boundaries and sensitivity to the knot set under small-scale data dependence. To address these issues, we propose a smoothed full-scale approximation (SFSA) method for analyzing large spatial data sets. The SFSA leads to a class of scalable GP models, with covariance functions that consist of two parts: a reduced-rank covariance function that captures large-scale spatial dependence, and a covariance that adjusts the local covariance approximation errors of the reduced-rank part, both within blocks and between neighboring blocks. This method reduces the prediction errors on block boundaries, and leads to inference and prediction results that are more robust under different dependence scales owing to the better approximation of the residual covariance. The proposed method provides a unified view of approximation methods for GP models, grouping several existing computational methods for large spatial data sets into one common framework. These methods include the predictive process, FSA, and nearest neighboring block GP methods, allowing efficient algorithms that provide robust and accurate model inferences and predictions for large spatial data sets within a unified framework. We illustrate the effectiveness of the SFSA approach using simulation studies and a total column ozone data set.