L2 Model Reduction and Variance Reduction

In this contribution we examine certain variance properties of model reduction. The focus is on L2 model reduction, but some general results are also presented. These general results can be used to analyze various other model reduction schemes. The models we study are finite impulse response (FIR) and output error (OE) models. We compare the variance of two estimated models. The first one is estimated directly from data and the other one is computed by reducing a high order model, by L2 model reduction. In the FIR case we show that it is never better to estimate the model directly from data, compared to estimating it via L2 model reduction of a high order FIR model. For OE models we show that the reduced model has the same variance as the directly estimated one if the reduced model class used contains the true system