Uncertainty quantification in system identification

Quantifying the uncertainty of system parameters has become an increasing important topic in modem engineering due to the increasing demand in the confidence of response prediction and availability of analytical, computational and experimental tools for reducing uncertainty through proper instrumentation. Associated with the latter system identification has been a topic under intensive research. Due to the lack of data and imperfect models, uncertainty of system parameters still remains and this has been quantified through Bayesian and frequentist perspectives. In the former, results are often described in terms of the ‘most probable value’ (MPV) of the model parameters and their posterior (i.e., given data) uncertainty in terms of the ‘posterior covariance matrix’. In a frequentist perspective the uncertainty is quantified by the ensemble covariance matrix of the best estimates obtained from repeated experimental trials. The Bayesian and frequentist results need not coincide but intuition suggests that they should be consistent in some sense. This paper shows mathematically that when there is no modeling error these two perspectives are consistent but in general they are different. The study reveals clearly the relevance of the Bayesian measure of uncertainty to quality control, and the frequentist measure as an aggregate quantity reflecting possible modeling error and/or unknown variations in experimental conditions. These two measures are complementary rather than competing.