Hyperbolic decomposition of Dirichlet distance for ARMA models

We investigate the hyperbolic decomposition of the Dirichlet norm and distance between autoregressive moving average (ARMA) models. With the Kähler information geometry of linear systems in Hardy spaces and weighted Hardy spaces, we demonstrate that the Dirichlet norm and distance of ARMA models, corresponding to the mutual information between the past and future, are decomposed into functions of the hyperbolic distances between the poles and zeros of the ARMA models. Moreover, the distance is also expressed with separate terms from AR parts, MA parts, and AR-MA cross terms. Furthermore, the hyperbolic decomposition is helpful for the model order reduction of ARMA models.