Space–time covariance functions based on linear response theory and the turning bands method

Abstract

The generation of non-separable, physically motivated covariance functions is a theme of ongoing research interest, given that only a few classes of such functions are available. We construct a non-separable space–time covariance function based on a diffusive Langevin equation. We employ ideas from statistical mechanics to express the response of an equilibrium (i.e., time independent) random field to a driving noise process by means of a linear, diffusive relaxation mechanism. The equilibrium field is assumed to follow an exponential joint probability density which is determined by a spatial local interaction model. We then use linear response theory to express the temporal evolution of the random field around the equilibrium state in terms of a Langevin equation. The latter yields an equation of motion for the space–time covariance function, which can be solved explicitly at certain limits. We use the explicit covariance model obtained in one spatial dimension and time. By means of the turning bands transform, we derive a non-separable space–time covariance function in three space dimensions and time. We investigate the mathematical properties of this space–time covariance function, and we use it to model a dataset of daily ozone concentration values from the conterminous USA.