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Multi-Resolution Spatial Random-Effects Models for Irregularly Spaced Data

The spatial random-effects model is flexible in modeling spatial covariance functions, and is computationally efficient for spatial prediction via fixed rank kriging. However, the success of this model depends on an appropriate set of basis functions. In this research, we propose a class of basis functions extracted from thin-plate splines. These functions are ordered in terms of their degrees of smoothness with a higher-order function corresponding to larger-scale features and a lower-order one corresponding to smaller-scale details, leading to a parsimonious representation for a nonstationary spatial covariance function. Consequently, only a small to moderate number of functions are needed in a spatial random-effects model. The proposed class of basis functions has several advantages over commonly used ones. First, we do not need to concern about the allocation of the basis functions, but simply select the total number of functions corresponding to a resolution. Second, only a small number of basis functions is usually required, which facilitates computation. Third, estimation variability of model parameters can be considerably reduced, and hence more precise covariance function estimates can be obtained. Fourth, the proposed basis functions depend only on the data locations but not the measurements taken at those locations, and are applicable regardless of whether the data locations are sparse or irregularly spaced. In addition, we derive a simple close-form expression for the maximum likelihood estimates of model parameters in the spatial random-effects model. Some numerical examples are provided to demonstrate the effectiveness of the proposed method.