The Rational SPDE Approach for Gaussian Random Fields With General Smoothness

• Published in: Journal of computational and graphical statistics Vol. 29; no. 2; pp. 274 - 285
• Main Authors: Bolin, David, Kirchner, Kristin
• Format: Journal Article
• Language: English
• Published: Alexandria Taylor & Francis 02.04.2020; Taylor & Francis Ltd
• Subjects: Approximation; approximations; Fields (mathematics); Fractional operators; markov random-fields; Matematik; Matern covariances; Mathematical models; Mathematical sciences; Mathematics; Matérn covariances; Nonstationary Gaussian fields; Operators (mathematics); Parameters; Partial differential equations; Smoothness; Spatial statistics; Statistical inference; Stochastic partial differential equations; Stochastic processes; White noise
• ISSN: 1061-8600; 1537-2715; 1537-2715
• DOI: 10.1080/10618600.2019.1665537

A popular approach for modeling and inference in spatial statistics is to represent Gaussian random fields as solutions to stochastic partial differential equations (SPDEs) of the form , where is Gaussian white noise, L is a second-order differential operator, and is a parameter that determines the smoothness of u. However, this approach has been limited to the case , which excludes several important models and makes it necessary to keep β fixed during inference. We propose a new method, the rational SPDE approach, which in spatial dimension is applicable for any , and thus remedies the mentioned limitation. The presented scheme combines a finite element discretization with a rational approximation of the function to approximate u. For the resulting approximation, an explicit rate of convergence to u in mean-square sense is derived. Furthermore, we show that our method has the same computational benefits as in the restricted case . Several numerical experiments and a statistical application are used to illustrate the accuracy of the method, and to show that it facilitates likelihood-based inference for all model parameters including β. Supplementary materials for this article are available online.