Abstract
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.
Highlights
Introduction and statement of the problemPositive definite functions are fundamental to mathematics, probability and statistics
The advent of massive data sets distributed over the whole planet Earth has motivated several scientists to study modeling strategies for random fields defined over the sphere of R3 representing our planet
The increasing interest in modeling stochastic processes over spheres or spheres cross time with an explicit covariance function is reflected in works in areas as diverse as mathematical analysis, spatial and space-time statistics, and we refer the reader to the recent reviews by Gneiting (2013), Jeong et al (2017), Porcu et al (2018) and Porcu et al (2019) for a comprehensive account
Summary
Positive definite functions are fundamental to mathematics, probability and statistics Their use has become ubiquitous in many areas of applied sciences, and we refer the reader to Porcu et al (2018) and Porcu et al (2019) for recent overviews as well as for collections of open problems and statistical implications. The increasing interest in modeling stochastic processes over spheres or spheres cross time with an explicit covariance function is reflected in works in areas as diverse as mathematical analysis, spatial and space-time statistics, and we refer the reader to the recent reviews by Gneiting (2013), Jeong et al (2017), Porcu et al (2018) and Porcu et al (2019) for a comprehensive account. Characterization of covariance functions being geodesically isotropic on Sd−1 has been available thanks to Schoenberg (1942).
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.