A new $$L_2$$ L 2 calibration procedure of computer models based on the smoothing spline ANOVA

Computer models cannot always fit the physical systems well in practice. Most of these models contain unknown parameters with high uncertainty. Calibration is an approach to identify these unknown parameters in computer models. Inspired by Tuo and Wu (Ann Stat 43(6):2331–2352, 2015), we propose a new calibration procedure based on the smoothing spline ANOVA, which can be regarded as an extension of the methods of Tuo and Wu (Ann Stat 43(6):2331–2352, 2015). The proposed procedure mainly comprises two steps: computing the improved $$L_2$$ L 2 estimator of the calibration parameters, and estimating the discrepancy function. We derive the rate of convergence of the proposed estimator of the calibration parameters and investigate its asymptotic properties. The proposed method combines the advantages of $$L_2$$ L 2 calibration and ordinary least square (OLS) calibration, and exhibits better performances than Tuo and Wu (Ann Stat 43(6):2331–2352, 2015) and Wong et al. (J R Stat Soc Ser B 79(2):635–645, 2017). We conduct some numerical simulations and apply the proposed method to two real examples, which demonstrate the advantages of the proposed procedure.