Abstract
We provide a nonparametric spectral approach to the modeling of correlation functions on spheres. The sequence of Schoenberg coefficients and their associated covariance functions are treated as random rather than assuming a parametric form. We propose a stick-breaking representation for the spectrum, and show that such a choice spans the support of the class of geodesically isotropic covariance functions under uniform convergence. Further, we examine the first order properties of such representation, from which geometric properties can be inferred, in terms of Hölder continuity, of the associated Gaussian random field. The properties of the posterior, in terms of existence, uniqueness, and Lipschitz continuity, are then inspected. Our findings are validated with MCMC simulations and illustrated using a global data set on surface temperatures.
Highlights
There has been an increasing interest in the modeling, inference and prediction of random processes defined continuously over a large portion or the entire planet Earth
This paper deals with nonparametric Bayesian modeling, inference and prediction, for Gaussian random fields defined continuously over the two-dimensional sphere embedded in the three dimensional Euclidean space
Spectral representations for positive definite functions on spheres, being the equivalent of Bochner and Schoenberg’s theorems in Euclidean spaces are available thanks to Schoenberg (1942), who shows that a mapping ψ : [0, π] → R belongs to the class Ψd if and only if it can be uniquely written as ψ(θ) = bn,dc(nd−1)/2(cos θ), θ ∈ [0, π], (4)
Summary
There has been an increasing interest in the modeling, inference and prediction of random processes defined continuously over a large portion or the entire planet Earth. Global data sets have become ubiquitous, and along with their increasing complexity, new scientific questions and challenges have emerged involving several branches of applied mathematics, statistics, computer sciences and machine learning
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