Spatiotemporal Gaussian random fields using stochastic partial differential equations

Abstract

In various scientific disciplines, measurement data is collected across space and over time with the aim of inferring information regarding an underlying stochastic spatiotemporal phenomenon. The high computational costs of current methods render this task intractable for the large datasets typically encountered in spatiotemporal statistics.

We propose a model based on Gaussian random fields, defined as the solutions to a class of space-time fractional stochastic partial differential equations (SPDEs) driven by Gaussian noise. The tunable parameters characterizing this class are expected to admit intuitive interpretations, in which case an efficient numerical approximation scheme for solutions to these SPDEs yields a powerful spatiotemporal model which is usable in practice.

The study of this class of SPDEs is the focus of this work. We define weak and mild solution concepts and show their equivalence under lenient conditions on the differential operators involved. Subsequently we establish results linking the well-posedness, spatiotemporal regularity and asymptotic covariance structure of the SPDEs to conditions on the model parameters, confirming their interpretability. Lastly, we specialize the SPDEs to a class of fractional ordinary differential equations in time, and describe a method to compute numerical approximations of their solutions. Future work is needed to analyze this scheme and generalize it to the original class of SPDEs.