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 ${\{\mathit{Z}(\mathit{x})\}}_{\mathit{x}\in \mathcal{X}}$ indexed by a compact metric space $(\mathcal{X},{\mathit{d}}_{\mathcal{X}})$ can be obtained if the mean value function $\mathit{m}:\mathcal{X}\to \mathbb{R}$ and the covariance function $\mathit{\varrho }:\mathcal{X}\times \mathcal{X}\to \mathbb{R}$ of Z are known. We consider the problem of predicting the value of $\mathit{Z}({\mathit{x}}^{\ast })$ at some location ${\mathit{x}}^{\ast }\in \mathcal{X}$ based on observations at locations ${\{{\mathit{x}}_{\mathit{j}}\}}_{\mathit{j}=1}^{\mathit{n}}$, which accumulate at ${\mathit{x}}^{\ast }$ as $\mathit{n}\to \infty$ (or, more generally, predicting $\mathit{\phi }(\mathit{Z})$ based on ${\{{\mathit{\phi }}_{\mathit{j}}(\mathit{Z})\}}_{\mathit{j}=1}^{\mathit{n}}$ for linear functionals $\mathit{\phi },{\mathit{\phi }}_1,\dots ,{\mathit{\phi }}_{\mathit{n}}$). Our main result characterizes the asymptotic performance of linear predictors (as n increases) based on an incorrect second-order structure $(\tilde{\mathit{m}},\tilde{\mathit{\varrho }})$, without any restrictive assumptions on ϱ, $\tilde{\mathit{\varrho }}$ such as stationarity. We, for the first time, provide necessary and sufficient conditions on $(\tilde{\mathit{m}},\tilde{\mathit{\varrho }})$ 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 $\mathcal{X}\subset {\mathbb{R}}^{\mathit{d}}$ with Matérn or periodic covariance functions, and on the sphere $\mathcal{X}={\mathbb{S}}^2$ for the case of two isotropic covariance functions.