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

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 Ho spacing diaeresis lder 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.

Automated summary

This study introduces a nonparametric spectral approach to modeling correlation functions on spheres, treating the sequence of Schoenberg coefficients and their associated covariance functions as random. The stick-breaking representation for the spectrum is proposed and shown to span the support of geodesically isotropic covariance functions under uniform convergence. The properties of the posterior, regarding existence, uniqueness, and Lipschitz continuity, are examined, and validated using MCMC simulations with global surface temperature data.