Resolution Adaptive Fixed Rank Kriging

The spatial random effects model is flexible in modeling spatial covariance functions and is computationally efficient for spatial prediction via fixed rank kriging (FRK). However, the model depends on a class of basis functions, which if not selected properly, may result in unstable or undesirable results. Additionally, the maximum likelihood (ML) estimates of the model parameters are commonly computed using an expectation-maximization (EM) algorithm, which further limits its applicability when a large number of basis functions are required. In this research, we propose a class of basis functions extracted from thin-plate splines. The functions are ordered in terms of their degrees of smoothness with higher-order functions corresponding to larger-scale features and lower-order ones corresponding to smaller-scale details, leading to a parsimonious representation of a (nonstationary) spatial covariance function with the number of basis functions playing the role of spatial resolution. The proposed class of basis functions avoids the difficult knot-allocation or scale-selection problem. In addition, we show that ML estimates of the random effects covariance matrix can be expressed in simple closed forms, and hence the resulting FRK can accommodate a much larger number of basis functions without numerical difficulties. Finally, we propose to select the number of basis functions using Akaike’s information criterion, which also possesses a simple closed-form expression. The whole procedure, involving no additional tuning parameter, is efficient to compute, easy to program, automatic to implement, and applicable to massive amounts of spatial data even when they are sparsely and irregularly located. Proofs of the theorems and an R package autoFRK are provided in supplementary materials available online.