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Identification and isotropy characterization of deformed random fields through excursion sets

Part of: Geometric probability and stochastic geometry Inference from stochastic processes

Published online by Cambridge University Press:  16 November 2018

Julie Fournier
Julie Fournier*
Affiliation:
Université Paris Descartes and Sorbonne Université
*
* Postal address: MAP5 UMR CNRS 8145, Université Paris Descartes, 45 rue des Saints-Pères, 75006 Paris, France. Email address: julie.fournier@parisdescartes.fr
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Abstract

A deterministic application θ:ℝ2→ℝ2 deforms bijectively and regularly the plane and allows the construction of a deformed random field X∘θ:ℝ2→ℝ from a regular, stationary, and isotropic random field X:ℝ2→ℝ. The deformed field X∘θ is, in general, not isotropic (and not even stationary), however, we provide an explicit characterization of the deformations θ that preserve the isotropy. Further assuming that X is Gaussian, we introduce a weak form of isotropy of the field X∘θ, defined by an invariance property of the mean Euler characteristic of some of its excursion sets. We prove that deformed fields satisfying this property are strictly isotropic. In addition, we are able to identify θ, assuming that the mean Euler characteristic of excursion sets of X∘θ over some basic domain is known.

Keywords

Random fieldspace deformationexcursion setEuler characteristicisotropy

MSC classification

Primary: 62M30: Spatial processes
Secondary: 62M40: Random fields; image analysis 60D05: Geometric probability and stochastic geometry
Type
Original Article
Information
Advances in Applied Probability , Volume 50 , Issue 3 , September 2018 , pp. 706 - 725
Copyright
Copyright © Applied Probability Trust 2018 

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