On the Identification of ARMA Graphical Models

This article considers the problem to estimate a graphical model corresponding to an autoregressive moving-average (ARMA) Gaussian stochastic process. We propose a new maximum entropy covariance and cepstral extension problem and we show that the problem admits an approximate solution, which represents an ARMA graphical model whose topology is determined by the selected entries of the covariance lags considered in the extension problem. Then, we show how the corresponding dual problem is connected with the maximum likelihood principle. Such connection allows us to design a Bayesian model and characterize an approximate maximum a posteriori estimator of the ARMA graphical model in the case the graph topology is unknown. We test the performance of the proposed method through some numerical experiments.