Estimation of Local Anisotropy Based on Level Sets

TL;DR

This paper develops estimators for local anisotropy in Gaussian fields based on level set observations, providing consistency and asymptotic normality results to test isotropy versus anisotropy.

Contribution

It introduces new estimators for the affinity parameters of Gaussian fields using level set functionals, with proofs of their consistency and asymptotic properties.

Findings

Estimators are almost surely consistent.

Asymptotic normality of estimators is established.

Confidence intervals for anisotropy parameters are derived.

Abstract

Consider an affine Gaussian field X : R 2 $\rightarrow$ R, that is a process equal in law to Z(At), where Z is isotropic and A : R2 $\rightarrow$ R2 is a self-adjoint definite positive matrix. Denote 0 < $\lambda$ = $\lambda$\_2 / $\lambda$\_1 \le 1 the ratio of the eigenvalues of A. This paper is aimed at testing the null hypothesis '' X is isotropic'' versus the alternative '' X is affine''. Roughly speaking, this amounts to testing '' $\lambda$ = 1 '' versus '' $\lambda$ < 1 ''. By setting level u in R, this is implemented by the partial observations of process X through some particular level functionals viewed over a square T, which grows to R2. This leads us to provide estimators for the affinity parameters that are shown to be almost surely consistent. Their asymptotic normality provide confidence intervals for parameters. This paper offered an important opportunity to study general level functionals near the level u, part of the difficulties arises from the fact that the topology of level set CT,X (u) = {t $\in$ T : X(t) = u} can be irregular, even if the trajectories of X are regular. A significant part of the paper is dedicated to show the L2-continuity in the level u of these general functionals.