The circular Matérn covariance function and its link to Markov random fields on the circle

The connection between Gaussian random fields and Markov random fields has been well-established in Euclidean spaces, with Matérn covariance functions playing a pivotal role. In this paper, we explore the extension of this link to circular spaces and uncover different results. It is known that Matérn covariance functions are not always positive definite on the circle; however, the circular Matérn covariance functions are shown to be valid on the circle and are the focus of this paper. For these circular Matérn random fields on the circle, we show that the corresponding Markov random fields can be obtained explicitly on equidistance grids. Consequently, the equivalence between the circular Matérn random fields and Markov random fields is then exact and this marks a departure from the Euclidean space counterpart, where only approximations are achieved. Moreover, the key motivation in Euclidean spaces for establishing such link relies on the assumption that the corresponding Markov random field is sparse. We show that such sparsity does not hold in general on the circle. In addition, for the sparse Markov random field on the circle, we derive its corresponding Gaussian random field.