The effect of regularization on variance error

This note addresses the problem of quantifying the effect of noise induced error(so called “variance error”) in system estimates found via a regularised cost criterion. It builds on recent work by the authors in which expressions for nonregularised criterions are derived which are exact for finite model order. Those new expressions were established to be very different to previous quantifications that are widely used but based on asymptotic in model order arguments. A key purpose of this note is to expose a rapprochement between these new finite model order, and the preexisting asymptotic model order quantifications. In so doing, a further new result is established. Namely, that variance error in the frequency domain is dependent on the choice of the point about which regularization is affected.