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Interpolation of spatial data : some theory for kriging / Michael L. Stein. p. cm. -(Springer series in statistics) Inc1udes bibliographical references and index.

Latin hypercube sampling (McKay, Conover, and Beckman 1979) is a method of sampling that can be used to produce input values for estimation of expectations of functions of output variables. The asymptotic variance of such an estimate is obtained. The estimate is also shown to be asymptotically normal. Asymptotically, the variance is less than that obtained using simple random sampling, with the degree of variance reduction depending on the degree of additivity in the function being integrated. A method for producing Latin hypercube samples when the components of the input variables are statistically dependent is also described. These techniques are applied to a simulation of the performance of a printer actuator.

Likelihood methods are often difficult to use with large, irregularly sited spatial data sets, owing to the computational burden. Even for Gaussian models, exact calculations of the likelihood for "n" observations require "O"("n"-super-3) operations. Since any joint density can be written as a product of conditional densities based on some ordering of the observations, one way to lessen the computations is to condition on only some of the 'past' observations when computing the conditional densities. We show how this approach can be adapted to approximate the restricted likelihood and we demonstrate how an estimating equations approach allows us to judge the efficacy of the resulting approximation. Previous work has suggested conditioning on those past observations that are closest to the observation whose conditional density we are approximating. Through theoretical, numerical and practical examples, we show that there can often be considerable benefit in conditioning on some distant observations as well. Copyright 2004 Royal Statistical Society.

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