Extendability tests for multidimensional covariance sequences

The problem of extendibility of multidimensional covariance sequences and the equivalent problem of the existence of maximum entropy (ME) spectral estimates are analyzed using some recent results on the valid parameter space of Gaussian Markov random fields (GMRFs). For several nontrivial examples, explicit conditions for extendability and, using those conditions, sketches of the space of extendible covariance sequences are obtained. For the general case, a cutting-plane algorithm is proposed as an alternative to the two existing numerical procedures for ascertaining extendibility, namely, the linear programming procedure and the expanding-hull algorithm. The duality between the valid parameter space and the space of extendible covariances, and the relationship between those two spaces and the space of admissible covariances for finite-size data sequences, are examined. The connection between extendability and maximum-likelihood (ML) estimation is established, and some properties extendibility of covariances specified over increasing window sizes are presented