Regularity and approximation of Gaussian random fields evolving temporally over compact two-point homogeneous spaces

Abstract

We consider Gaussian random fields on the product of a compact two-point homogeneous space cross the time, which are space isotropic and time stationary. We study regularity properties of these random fields in terms of function spaces whose elements have different smoothness in the space and time domain. Namely, we express the norm of the corresponding covariance kernel functions in terms of the summability of the associated spectral coefficients. Furthermore, we define an approximation method based on the truncation of the expansion related to the spectral representation of a given random field. The accuracy of this approximation is measured in the $$L^p$$ sense. Finally, we model a space–time dataset of ozone concentrations in Mexico City using a seasonal temporal covariance function constructed through an expansion of Jacobi polynomials. We find that we need relatively few Jacobi polynomials to get the best fit to the data in terms of the deviance information criterion. We discuss the characteristics of this model, including seasonality, decay and approximate conditional independencies.