Recursive structure of noncausal Gauss-Markov random fields

An approach is developed for noncausal Gauss-Markov random fields (GMRFs) that enables the use of recursive procedures while retaining the noncausality of the field. Recursive representations are established that are equivalent to the original field. This is achieved by first presenting a canonical representation for GMRFs that is based on the inverse of the covariance matrix, which is called the potential matrix. It is this matrix rather than the field covariance that reflects in a natural way the MRF structure. From its properties, two equivalent one-sided representations are derived, each of which is obtained as the successive iterates of a Riccati-type equation. For homogeneous fields, these unilateral descriptions are symmetrized versions of each other, the study of only one Riccati equation being required. It is proven that this Riccati equation converges at a geometric rate, therefore the one-sided representations are asymptotically invariant. These unilateral representations make it possible to process the fields with well-known recursive techniques such as Kalman-Bucy filters and two-point smoothers.<>