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Robust Calibration of Computer Models Based on Huber Loss

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Abstract

Recently, uncertainty quantification is getting more and more attention, especially for computer model calibration. However, most of the existing papers assume the errors follow a Gaussian or sub-Gaussian distribution, which would not be satisfied in practice. To overcome the limitation of the traditional calibration procedures, the authors develop a robust calibration procedure based on Huber loss, which can deal with responses with outliers and heavy-tail errors efficiently. The authors propose two different estimators of the calibration parameters based on ordinary least estimator and L2 calibration respectively, and investigate the nonasymptotic and asymptotic properties of the proposed estimators under certain conditions. Some numerical simulations and a real example are conducted, which verifies good performance of the proposed calibration procedure.

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References

  1. Tuo R and Wu C F J, Efficient calibration for imperfect computer models, Annals of Statistics, 2015, 43(6): 2331–2352.

    Article  MathSciNet  MATH  Google Scholar 

  2. Wong K W, Storlie C B, and Lee C M, A frequentist approach to computer model calibration, Journal of Royal Statistical Society Series B, 2017, 79(2): 635–648.

    Article  MathSciNet  MATH  Google Scholar 

  3. Tuo R, Adjustments to computer models via projected kernel calibration, SIAM/ASA Journal on Uncertainty Quantification, 2019, 7(2): 553–578.

    Article  MathSciNet  MATH  Google Scholar 

  4. Xie F Z and Xu Y X, Bayesian projected calibration of computer models, Journal of American Statistical Association, 2021, 116(536): 1965–1982.

    Article  MathSciNet  MATH  Google Scholar 

  5. Wang Y and Tuo R, Semi-parametric adjustment to computer models, Statistics, 2020, 54(6): 1255–1275.

    Article  MathSciNet  MATH  Google Scholar 

  6. Kennedy M C and O’Hagan A, Bayesian calibration of computer models, Journal of Royal Statistical Society Series B, 2001, 63: 425–464.

    Article  MathSciNet  MATH  Google Scholar 

  7. Tuo R, Wang Y, and Wu C F J, On the improved rates of convergence for matern-type kernel ridge regression with application to calibration of computer models, SIAM/ASA Journal on Uncertainty Quantification, 2020, 8(4): 1522–1547.

    Article  MathSciNet  MATH  Google Scholar 

  8. Tuo R, He S Y, Pourhabib A, et al., A reproducing kernel hilbert space approach to functional calibration of computer models, Journal of American Statistical Association, 2021, DOI: https://doi.org/10.1080/01621459.2021.1956938.

  9. Zhou W X, Bose K, Fan J Q, et al., A new perspective on robust M-estimations: Finite sample theory and applications to dependence-adjusted multiple testing, Annals of Statistics, 2018, 46(5): 1904–1931.

    Article  MathSciNet  MATH  Google Scholar 

  10. Huber P J, Robust estimation of a location parameter, Annals of Mathematical Statistics, 1964, 35: 73–101.

    Article  MathSciNet  MATH  Google Scholar 

  11. Santner T J, Williams B J, and Notz W I, The Design and Analysis of Computer Experiments, Springer, New York, 2003.

    Book  MATH  Google Scholar 

  12. Gu C, Smoothing spline anova models: R package gss, Journal of Statictical Software, 2014, 58: 1–25.

    Google Scholar 

  13. van de Geer S, Empirical Processes in M-Estimation, Cambridge University Press, Cambridge, 2020.

    MATH  Google Scholar 

  14. Sun Q, Zhou W X, and Fan J Q, Adaptive huber regression, Journal of the American Statistical Association, 2020, 115(529): 254–265.

    Article  MathSciNet  MATH  Google Scholar 

  15. Spokoiny V, Parametric estimation, finite sample theory, Annals of Statistics, 2012, 40: 2877–2909.

    Article  MathSciNet  MATH  Google Scholar 

  16. Vershynin R, Introduction to the Non-Asymptotic Analysis of Random Matrices, Compressed Sensing 210–268, Cambridge University Press, Cambridge, 2012.

    Google Scholar 

  17. Spokoiny V, Bernstein-von Mises theorem for growing parameter dimension, 2013, Available at arXiv: 1302.3430.

  18. Mammen E and Van de Geer S, Penalized quasi-likelihood estimation in partial linear models, Annals of Statistics, 1997, 25: 1014–1035.

    Article  MathSciNet  MATH  Google Scholar 

  19. Serfling R J, Approximation Theorems of Mathematical Statistics, John Wiley and Sons, New York, 1980.

    Book  MATH  Google Scholar 

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Authors and Affiliations

  1. School of Mathematical Sciences, Peking University, Beijing, 100871, China

    Yang Sun & Xiangzhong Fang

Authors
  1. Yang Sun
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  2. Xiangzhong Fang
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Correspondence to Yang Sun.

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The authors declare no conflict of interest.

Additional information

This research was supported by the Science Challenge Project under Grant No. TZ2018001.

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Cite this article

Sun, Y., Fang, X. Robust Calibration of Computer Models Based on Huber Loss. J Syst Sci Complex 36, 1717–1737 (2023). https://doi.org/10.1007/s11424-023-1456-x

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  • DOI: https://doi.org/10.1007/s11424-023-1456-x

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