Covariance Functions on Spheres cross Time: Beyond Spatial Isotropy and Temporal Stationarity

Alessandra Fariñas (1) , Anne Estrade (2) , Emilio Porcu (3)

1 Deparamento de Matematica
2 MAP5 - UMR 8145 - Mathématiques Appliquées Paris 5
3 School of Mathematics and Statistics

Spectral representations uniquely define the covariance functions associated to random fields defined over spheres or spheres cross time. Covariance functions on spheres cross time are usually modelled under the assumptions of either spatial isotropy or axial symmetry, and the assumption of temporal stationarity. This paper goes beyond these assumptions. In particular, we consider the problem of spatially anisotropic covariance functions on spheres. The crux of our criterion is to escape from the addition theorem for spherical harmonics. We also challenge the problem of temporal nonstationarity in nonseparable space-time covariance functions, where space is the $n$-dimensional sphere embedded in the $(n+1)$-dimensional Euclidean space. We finally propose a simulation routine for the models proposed in this paper.

Mots clés: Spherical harmonics; Gegenbauer polynomial; Spectral representation; Space-time random field; Positive Definite

Domaines: Mathématiques [math]; Statistiques [stat]

Alessandra Fariñas, Anne Estrade, Emilio Porcu. Covariance Functions on Spheres cross Time: Beyond Spatial Isotropy and Temporal Stationarity. 2018. ⟨hal-01417668v3⟩