Variance Error Quantifications That Are Exact for Finite-Model Order

Abstract

This paper is concerned with the frequency domain quantification of noise induced errors in dynamic system estimates. Preceding and seminal work on this problem has suggested general expressions that are approximations whose accuracy increases with observed data length and model order. In the interests of improved accuracy, this paper provides new expressions whose accuracy depends only on data length. They are therefore exact for arbitrarily small true model order. Other authors have recognized the importance of such expressions and have derived them for the case of finite-impulse response-like model structures in which denominators are fixed at true values and only numerator terms are estimated. This paper progresses beyond this situation to address the more general output-error and Box-Jenkins structures in which full dynamics models (both numerator and denominator terms) and noise models may be estimated. A key aspect of the work here is that it establishes that the variance quantification problem is equivalent to that of deriving the reproducing kernel for a subspace that depends on the model structure being employed.