Nonparametric Bayesian Modeling and Estimation of Spatial Correlation Functions for Global Data

We provide a nonparametric spectral approach to the modeling ofcorrelation functions on spheres. The sequence of Schoenberg coefficients and theirassociated covariance functions are treated as random rather than assuming aparametric form. We propose a stick-breaking representation for the spectrum, andshow that such a choice spans the support of the class of geodesically isotropiccovariance functions under uniform convergence. Further, we examine the firstorder properties of such representation, from which geometric properties can beinferred, in terms of H ̈older continuity, of the associated Gaussian random field.The properties of the posterior, in terms of existence, uniqueness, and Lipschitzcontinuity, are then inspected. Our findings are validated with MCMC simulationsand illustrated using a global data set on surface temperatures.