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A KERNEL APPROXIMATION TO THE KRIGING

PREDICTOR OF A SPATIAL PROCESS

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MICHAEL L. STEIN

*Department of Statistics, The University of Chicago, 5734 University Avenue,*

Chicago, IL 60637, U.S.A.
(Received September 6, 1988; revised February 13, 1990)

**Abstract.**
Suppose a two-dimensional spatial process
*z*(*x*) with generalized covariance function
*G*(*x*, *x*') \propto |*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.

*Key words and phrases*:
Thin plate spline, prediction of
random fields, Kelvin function, nonparametric regression.

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