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2020 Reduction problems and deformation approaches to nonstationary covariance functions over spheres
Emilio Porcu, Rachid Senoussi, Enner Mendoza, Moreno Bevilacqua
Electron. J. Statist. 14(1): 890-916 (2020). DOI: 10.1214/19-EJS1670


The paper considers reduction problems and deformation approaches for nonstationary covariance functions on the $(d-1)$-dimensional spheres, $\mathbb{S}^{d-1}$, embedded in the $d$-dimensional Euclidean space. Given a covariance function $C$ on $\mathbb{S}^{d-1}$, we chase a pair $(R,\Psi)$, for a function $R:[-1,+1]\to \mathbb{R}$ and a smooth bijection $\Psi$, such that $C$ can be reduced to a geodesically isotropic one: $C(\mathbf{x},\mathbf{y})=R(\langle \Psi (\mathbf{x}),\Psi (\mathbf{y})\rangle )$, with $\langle \cdot ,\cdot \rangle $ 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 $\mathbb{R}^{3}$. 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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Emilio Porcu. Rachid Senoussi. Enner Mendoza. Moreno Bevilacqua. "Reduction problems and deformation approaches to nonstationary covariance functions over spheres." Electron. J. Statist. 14 (1) 890 - 916, 2020.


Received: 1 October 2018; Published: 2020
First available in Project Euclid: 12 February 2020

zbMATH: 07200220
MathSciNet: MR4062758
Digital Object Identifier: 10.1214/19-EJS1670

Keywords: covariance function , nonstationarity , reducibility problem , spheres


Vol.14 • No. 1 • 2020
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