Fast and exact synthesis of some operator scaling Gaussian random fields

Hermine Biermé (1) , Céline Lacaux (2, 3)

1 LMA [Poitiers] - Laboratoire de mathématiques et applications [UMR 7348]
2 LMA - EA2151 Laboratoire de Mathématiques d'Avignon
3 BIGS - Biology, genetics and statistics

Operator scaling Gaussian random fields, as anisotropic generalizations of self-similar fields, know an increasing interest for theoretical studies in the literature. However, up to now, they were only defined through stochastic integrals, without explicit covariance functions. In this paper we exhibit explicit covariance functions, as anisotropic generalizations of fractional Brownian fields ones, and define corresponding Operator scaling Gaussian random fields. This allows us to propose a fast and exact method of simulation based on the circulant embedding matrix method, following ideas of Stein [34] for fractional Brownian surfaces syntheses. This is a first piece of work to popularize these models in anisotropic spatial data modeling.

Mots clés: Fractional Brownian fields; Simulation; Covariance; Anisotropy; Operator scaling; Self-similarity; Gaussian field

Domaines: Probabilités [math.PR]

Hermine Biermé, Céline Lacaux. Fast and exact synthesis of some operator scaling Gaussian random fields. Applied and Computational Harmonic Analysis, 2020, 48 (1), pp.293-320. ⟨10.1016/j.acha.2018.05.004⟩. ⟨hal-01485685⟩