Necessary and sufficient conditions for asymptotically optimal linear prediction of random fields on compact metric spaces
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Optimal linear prediction (also known as kriging) of a random field {Z(x)}_x∈X indexed by a compact metric space (X,d_X) can be obtained if the mean value function mX→R and the covariance function ϱX×X→R of Z are known. We consider the problem of predicting the value of Z(x^*) at some location x^*∈X based on observations at locations {x_j}_j=1^n which accumulate at x^* as n→∞. Our main result characterizes the asymptotic performance of linear predictors (as n increases) based on an incorrect second order structure (m̃,ϱ̃). We, for the first time, provide necessary and sufficient conditions on (m̃,ϱ̃) for asymptotic optimality of the corresponding linear predictor, without any restrictive assumptions on ϱ, ϱ̃ such as stationarity. These general results are illustrated by an example on the sphere S^2 for the case of two isotropic covariance functions.
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