A link between Gaussian random fields and Markov random fields on the circle

Oregon Convention Center 

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 Matern covariance functions plan an essential role. We explore the extension of this link to circular spaces and uncover different results. It is known that Matern covariance functions are not always positive definite on the circle; however, the circular Matern covariance functions are shown to be valid on the circle and are the focus of this research. We show that the equivalence between the circular Matern random fields and Markov random fields is 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 assumptions that the corresponding Markov random field is sparse. We show that such sparsity does not hold in general on the circle. Additionally, we formally define the white noise space and its associated Brownian bridge on the circle for the stochastic differential equation used in this research.

Conditional

autoregressive model

Circulant matrix 

Section on Statistics and the Environment