Extrapolation of stationary random fields via level sets

Abstract

In this paper, we use the concept of excursion sets for the extrapolation of stationary random fields. Doing so, we define excursion sets for the field and its linear predictor, and then minimize the expected volume of the symmetric difference of these sets under the condition that the univariate distributions of the predictor and of the field itself coincide. We illustrate the new approach on Gaussian random fields.

Prediction of random variables by excursion metric projections

“…Unfortunately, the analytic form of Λ g can be found only in specific cases when the pre-knowledge of the distribution of (X, X n ) is available, such as in the Gaussian or α-stable case, cf. [3]. Should our prediction be law-preserving (i.e., X λ d = X), it holds E F X = G, and the optimization problem G(X, X λ ) → min λ∈Λ with Λ = Λ g rewrites using (8) as…”

“…The question of uniqueness of a solution λ to problems ( 12), ( 13) and ( 16) cannot be resolved in such generality. As illustrated in the Gaussian case [3], it will require further specification of the dependence structure of observations X j within the set X n , of the statistic g and parameter set Λ. This will be done for some specific classes of random variables in forthcoming research.…”

“…In the case of a Gaussian random process X, the exact solution of minimization problem (13) is given in by [3,Theorem 3.5]. For numerical illustrations, we take X with standard normal marginal distribution and covariance function C(t) = e −|t|/2 , t ∈ R.…”

“…. , t n in a compact window W ⊂ R. As proposed in [3], the linear predictor Y t = n j=1 λ j Y tj , t ∈ {t 1 , . .…”