A kernel approximation to the kriging predictor of a spatial process

Abstract

Suppose a two-dimensional spatial process z(x) with generalized covariance function G(x, x′) α |x − x′|2 log |x − x′| (Matheron, 1973, Adv. in Appl. Probab., 5, 439–468) is observed with error at a number of locations. This paper gives a kernel approximation to the optimal linear predictor, or kriging predictor, of z(x) under this model as the observations get increasingly dense. The approximation is in terms of a Kelvin function which itself can be easily approximated by series expansions. This generalized covariance function is of particular interest because the predictions it yields are identical to an order 2 thin plate smoothing spline. For moderate sample sizes, the kernel approximation is seen to work very well when the observations are on a square grid and fairly well when the observations come from a uniform random sample. This manuscript was prepared using computer facilities supported in part by National Science Foundation Grants No. DMS-8601732 and DMS-8404941 to the Department of Statistics at The University of Chicago.