Symbolic computation of Fisher information matrices for parametrized state-space systems

The asymptotic Fisher information matrix (FIM) has several applications in linear systems theory and statistical parameter estimation. It occurs in relation to the Cramer-Rao lower bound for the covariance of unbiased estimators. It is explicitly used in the method of scoring and it determines the asymptotic convergence properties of various system identification methods. It defines the Fisher metric on manifolds of systems and it can be used to analyze questions on identifiability of parametrized model classes. For many of these applications, exact symbolic computation of the FIM can be of great use. In this paper two different methods are described for the symbolic computation of the asymptotic FIM. The first method applies to parametrized MIMO state-space systems driven by stationary Gaussian white noise and proceeds via the solution of discrete-time Lyapunov and Sylvester equations, for which a method based on Faddeev sequences is used. This approach also leads to new short proofs of certain well-known results on the structure of the FIM for SISO ARMA systems. The second method applies to parametrized SISO state-space systems and uses an extended Faddeev algorithm. For both algorithms the concept of a Faddeev reachability matrix and the solution of discrete-time Sylvester equations in controller companion form are central issues. The methods are illustrated by two worked examples. The first concerns three different parametrizations of the class of stable AR(n) systems. The second concerns a model for an industrial mixing process, in which the value of exact computation to answer identifiability questions becomes particularly clear.