Risk Sensitive Identification of ARMA Processes

Abstract

In this paper we consider the problem of recursive identification of ARMA processes. This recursive procedure is parameterized by a weight-matrix acting on the stochastic gradient. The optimal weight-matrix will be defined using a risk-sensitive identification criterion. First the cost function will be expressed using LEQG-theory. Then, applying stochastic realization theory and the bounded real lemma we derive alternative expressions for the cost function. We prove among others, that the LQG functional of a properly augmented system gives the LEQG cost function of the original system. Furthermore, we point out that this cost function can be interpreted as mutual information between two stochastic processes. The optimal weight-matrix will be computed first as the optimum of a multi-dimensional constrained minimization, then a direct approach for solving the optimization problem will be presented. Finally, we briefly indicate that the results above can be extended to multivariate stochastic systems.