Necessary and sufficient conditions for asymptotically optimal linear prediction of random fields on compact metric spaces

Abstract

Optimal linear prediction (aka. kriging) of a random field {Z(x)}x∈X indexed by a compact metric space (X,dX) can be obtained if the mean value function m:X→R and the covariance function ϱ:X×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 {xj}j=1n, which accumulate at x∗ as n→∞ (or, more generally, predicting φ(Z) based on {φj(Z)}j=1n for linear functionals φ,φ1,…,φn). Our main result characterizes the asymptotic performance of linear predictors (as n increases) based on an incorrect second-order structure (m˜,ϱ˜), without any restrictive assumptions on ϱ, ϱ˜ such as stationarity. We, for the first time, provide necessary and sufficient conditions on (m˜,ϱ˜) for asymptotic optimality of the corresponding linear predictor holding uniformly with respect to φ. These general results are illustrated by weakly stationary random fields on X⊂Rd with Matérn or periodic covariance functions, and on the sphere X= S2 for the case of two isotropic covariance functions.