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

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

Kirchner, K. (TU Delft Analysis; TU Delft Delft Institute of Applied Mathematics)
Bolin, David (King Abdullah University of Science and Technology)

2022

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 ∂: 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 }nj =1, which accumulate at x*as n→∞(or, more generally, predicting φ(Z) based on {φj (Z)}nj =1 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, ∂), without any restrictive assumptions on ,∂ ∂ such as stationarity.We, for the first time, provide necessary and sufficient conditions on (m,∂) 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.

approximation in Hilbert spaces
Kriging
spatial statistics

http://resolver.tudelft.nl/uuid:89cb5114-d9d3-45a3-9166-77956807cd50

https://doi.org/10.1214/21-AOS2138

0090-5364

Annals of Statistics, 50 (2), 1038-1065

Institutional Repository

journal article

© 2022 K. Kirchner, David Bolin