ARMA Covariance realization from noisy data

Stochastic realization algorithms are based on the premise that the sequence to be realized is in actuality a covariance sequence. In practice such a sequence is often arrived at by estimation, and it may therefore not be an actual covariance sequence. Failure of common stochastic realization algorithms occurs, so that a more robust approach must be followed, guaranteeing that the final result is the best approximation to the given sequence while being a member of the class of specified stability ARMA covariance sequences. A non-linear optimization problem is formulated such that for the chosen covariance sequence parametrization the gradient and Hessian of the ARMA covariance parametrization require the computation of ARMA cross-covariances, which can be executed efficiently.