Linear estimation using covariance information in distributed parameter systems with non-independent uncertainty

Recursive algorithms for the linear least mean-squared one-stage prediction, filtering and fixed-point smoothing problems are obtained in distributed parameter systems with uncertain observations, when only the information on the first and second-order moments of the signal and observation noise is available. It is assumed that the variables describing the uncertainty in the observations are not necessarily independent. The algorithms for the different estimation problems are derived by using the orthogonal projection technique and the invariant imbedding method and they require that the covariance function of the signal can be expressed in a semi-degenerate kernel form.