Prediction of stationary Gaussian random fields with incomplete quarterplane past

Let {X-m,X-n : (m, /1) is an element of Z(2)} be a stationary Gaussian random field. Consider the problem of predicting X-0,X-0 based on the quarterplane Q = ((m, n) : m >= 0, n >= 0} \ {(0, 0)}, but with finitely many observations missing. Two solutions are presented. The first solution expresses the best predictor in terms of the moving average coefficients of {X-m,X-n}, under the assumption that the spectral density function has a strongly outer factorization. The second solution expresses the prediction error variance in terms of the autoregressive coefficients of {X-m,X-n}; it requires the reciprocal of the density function to have a strongly outer factorization, and relies on a modified duality argument. These solutions are extended by allowing the quarterplane past to be replaced with a much broader class of parameter sets. This enables the solution, for example, of the quarterplane interpolation problem. (C) 2015 Elsevier Inc. All rights reserved.