Abstract
We study multivariate Gaussian random fields defined over d-dimensional spheres. First, we provide a nonparametric Bayesian framework for modeling and inference on matrix-valued covariance functions. We determine the support (under the topology of uniform convergence) of the proposed random matrices, which cover the whole class of matrix-valued geodesically isotropic covariance functions on spheres. We provide a thorough inspection of the properties of the proposed model in terms of (a) first moments, (b) posterior distributions, and (c) Lipschitz continuities. We then provide an approximation method for multivariate fields on the sphere for which measures of Lp accuracy are established. Our findings are supported through simulation studies that show the rate of convergence when truncating a spectral expansion of a multivariate random field at a finite order. To illustrate the modeling framework developed in this paper, we consider a bivariate spatial data set of two 2019 NCEP/NCAR Flux Reanalyses.
Highlights
The paper deals with multivariate Gaussian random fields defined over the d-dimensional unit sphere Sd = {x ∈ Rd+1, x = 1} embedded in Rd+1, having a specified matrix-valued covariance function, see Ma
The benefit of adopting a nonparametric Bayesian framework are that (a) we focus on Gaussian random fields, so that the matrix-valued covariance function becomes crucial for modeling, inference, and prediction, and (b) we can exploit the simple spectral representation for multivariate covariance functions on spheres (Hannan, 2009; Yaglom, 1987)
We focus our work on developing approaches for nonparametric Bayesian modeling and fast simulation for multivariate random fields over spheres
Summary
The paper deals with multivariate (or vector-valued) Gaussian random fields defined over the d-dimensional unit sphere Sd = {x ∈ Rd+1, x = 1} embedded in Rd+1, having a specified matrix-valued covariance function, see Ma. There has been some recent work on nonparametric Bayesian inference for scalar-valued random fields and their covariance functions on spheres (Porcu et al, 2019) and more general manifolds (Castillo, Kerkyacharian and Picard, 2014). Fast simulation of multivariate Gaussian random fields over spheres has only been considered to a limited extent, and the reader is referred to the HEALPix software (Gorski et al, 2005), and recent contributions by Emery and Porcu (2019) and Alegrıa, Emery and Lantuejoul (2020).
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