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

3046 

Contributed Abstract 

Paper 

Chunfeng Huang (1), Ao Li (1), Nicholas Bussberg (2), Haimeng Zhang (3)

(1) Indiana University, N/A, (2) Elon University, N/A, (3) University of North Carolina at Greensboro, N/A

Ao Li  
Indiana University

Nicholas Bussberg  
Elon University

Haimeng Zhang  
University of North Carolina at Greensboro

Chunfeng Huang  
Indiana University

Chunfeng Huang  
Indiana University

The connection between Gaussian random fields and Markov random fields has been well-established in Euclidean spaces, with Mat\'ern 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 Mat\'ern covariance functions are not always positive definite on the circle; however, the circular Mat\'ern covariance functions are shown to be valid on the circle and are the focus of this paper. For these circular Mat\'ern 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 Mat\'ern 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 ran

Conditional |autoregressive model |Circulant matrix| | |

Section on Statistics and the Environment

Spatio-temporal statistics