A Theoretical Framework for Calibration in Computer Models: Parametrization, Estimation and Convergence Properties
Rui Tuo, Changbao Wu · SIAM/ASA Journal on Uncertainty Quantification · 2016
Calibration parameters in deterministic computer experiments are those attributes that cannot be measured or are not available in physical experiments. Kennedy and O'Hagan [M.C. Kennedy and A. O'Hagan, J. R. Stat. Soc. Ser. B Stat. Methodol., 63 (2001), pp. 425--464] suggested an approach to estimating them by using data from physical experiments and computer simulations. A theoretical framework is given which allows us to study the issues of parameter identifiability and estimation. We define the $L_2$-consistency for calibration as a justification for calibration methods. It is shown that a simplified version of the original Kennedy--O'Hagan (KO) method leads to asymptotically $L_2$-inconsistent calibration. This $L_2$-inconsistency can be remedied by modifying the original estimation procedure. A novel calibration method, called $L_2$ calibration, is proposed, proven to be $L_2$-consistent, and enjoys optimal convergence rate. A numerical example and some mathematical analysis are used to illustrate the source of the $L_2$-inconsistency problem.