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Reduction problems and deformation approaches to nonstationary covariance functions over spheres

Abstract : The paper considers reduction problems and deformation approaches for nonstationary covariance functions on the (d - 1)-dimensional spheres, Sd-1, embedded in the d-dimensional Euclidean space. Given a covariance function C on Sd-1, we chase a pair (R, Psi), for a function R:[-1,+1] -> R and a smooth bijection Psi, such that C can be reduced to a geodesically isotropic one: C(x, y) = R(), with (., .) denoting the dot product. The problem finds motivation in recent statistical literature devoted to the analysis of global phenomena, defined typically over the sphere of R-3. The application domains considered in the manuscript makes the problem mathematically challenging. We show the uniqueness of the representation in the reduction problem. Then, under some regularity assumptions, we provide an inversion formula to recover the bijection Psi, when it exists, for a given C. We also give sufficient conditions for reducibility.
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https://hal.inrae.fr/hal-03165726
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Submitted on : Wednesday, March 10, 2021 - 7:56:10 PM
Last modification on : Tuesday, June 1, 2021 - 10:48:04 PM

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Emilio Porcu, Rachid Senoussi, Enner Mendoza, Moreno Bevilacqua. Reduction problems and deformation approaches to nonstationary covariance functions over spheres. Electronic Journal of Statistics , Shaker Heights, OH : Institute of Mathematical Statistics, 2020, 14 (1), pp.890-916. ⟨10.1214/19-EJS1670⟩. ⟨hal-03165726⟩

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