Parametrization invariant covariance quantification in identification of transfer functions for linear systems

This paper addresses the variance quantification problem for system identification based on the prediction error framework. The role of input and model class selection for the auto-covariance of the estimated transfer function is explained without reference to any particular parametrization. This is achieved by lifting the concept of covariance from the parameter space to the system manifold where it is represented by a positive kernel instead of a positive definite matrix. The Fisher information metric as defined in information geometry allows an interpretation as a signal-to-noise ratio weighted standard metric after embedding the system manifold in the Hardy space of square integrable analytic functions. The reproducing kernel of the tangent space with respect to this metric is shown to provide an asymptotically tight lower bound for the positive kernel representing the covariance at the system which generated the input-output data.