The Circular Matern Covariance Function and its Link to Markov Random Fields on the Circle

TL;DR

This paper extends the connection between Gaussian and Markov random fields to circular domains using a novel circular Matérn covariance function, providing explicit formulas and addressing non-ergodicity issues.

Contribution

It introduces a circular Matérn covariance function, linking Gaussian and Markov random fields on circles with explicit formulas and theoretical insights.

Findings

Circular Matérn covariance is the stationary solution of a stochastic differential equation on the circle.

Explicit covariance formulas for the conditional autoregressive model on the circle.

Explanation of non-ergodicity and mean estimator inconsistency on circles.

Abstract

The link between Gaussian random fields and Markov random fields is well established based on a stochastic partial differential equation in Euclidean spaces, where the Mat\'ern covariance functions are essential. However, the Mat\'ern covariance functions are not always positive definite on circles and spheres. In this manuscript, we focus on the extension of this link to circles, and show that the link between Gaussian random fields and Markov random fields on circles is valid based on the circular Mat\'ern covariance function instead. First, we show that this circular Mat\'ern function is the covariance of the stationary solution to the stochastic differential equation on the circle with a formally defined white noise space measure. Then, for the corresponding conditional autoregressive model, we derive a closed form formula for its covariance function. Together with a closed form formula for the circular Mat\'ern covariance function, the link between these two random fields can be established explicitly. Additionally, it is known that the estimator of the mean is not consistent on circles, we provide an equivalent Gaussian measure explanation for this non-ergodicity issue.