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Derivative-based global sensitivity measures: General links with Sobol’ indices and numerical tests

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  • Lamboni, M.
  • Iooss, B.
  • Popelin, A.-L.
  • Gamboa, F.

Abstract

The estimation of variance-based importance measures (called Sobol’ indices) of the input variables of a numerical model can require a large number of model evaluations. It turns to be unacceptable for high-dimensional model involving a large number of input variables (typically more than ten). Recently, Sobol and Kucherenko have proposed the derivative-based global sensitivity measures (DGSM), defined as the integral of the squared derivatives of the model output, showing that it can help to solve the problem of dimensionality in some cases. We provide a general inequality link between DGSM and total Sobol’ indices for input variables belonging to the class of Boltzmann probability measures, thus extending the previous results of Sobol and Kucherenko for uniform and normal measures. The special case of log-concave measures is also described. This link provides a DGSM-based maximal bound for the total Sobol indices. Numerical tests show the performance of the bound and its usefulness in practice.

Suggested Citation

  • Lamboni, M. & Iooss, B. & Popelin, A.-L. & Gamboa, F., 2013. "Derivative-based global sensitivity measures: General links with Sobol’ indices and numerical tests," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 87(C), pages 45-54.
    • Handle: RePEc:eee:matcom:v:87:y:2013:i:c:p:45-54
    DOI: 10.1016/j.matcom.2013.02.002
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    1. Kucherenko, S. & Rodriguez-Fernandez, M. & Pantelides, C. & Shah, N., 2009. "Monte Carlo evaluation of derivative-based global sensitivity measures," Reliability Engineering and System Safety, Elsevier, vol. 94(7), pages 1135-1148.
    2. Jack P.C. Kleijnen, 2015. "Design and Analysis of Simulation Experiments," International Series in Operations Research and Management Science, Springer, edition 2, number 978-3-319-18087-8, December.
    3. Sobol’, I.M. & Tarantola, S. & Gatelli, D. & Kucherenko, S.S. & Mauntz, W., 2007. "Estimating the approximation error when fixing unessential factors in global sensitivity analysis," Reliability Engineering and System Safety, Elsevier, vol. 92(7), pages 957-960.
    4. Kucherenko, Sergei & Feil, Balazs & Shah, Nilay & Mauntz, Wolfgang, 2011. "The identification of model effective dimensions using global sensitivity analysis," Reliability Engineering and System Safety, Elsevier, vol. 96(4), pages 440-449.
    5. Sobol’, I.M. & Kucherenko, S., 2009. "Derivative based global sensitivity measures and their link with global sensitivity indices," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 79(10), pages 3009-3017.
    6. Sobol′ , I.M, 2001. "Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 55(1), pages 271-280.
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    7. Matieyendou Lamboni, 2020. "Uncertainty quantification: a minimum variance unbiased (joint) estimator of the non-normalized Sobol’ indices," Statistical Papers, Springer, vol. 61(5), pages 1939-1970, October.
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    9. Lamboni, Matieyendou, 2021. "Derivative-based integral equalities and inequality: A proxy-measure for sensitivity analysis," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 179(C), pages 137-161.
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    15. Matieyendou Lamboni, 2018. "Global sensitivity analysis: a generalized, unbiased and optimal estimator of total-effect variance," Statistical Papers, Springer, vol. 59(1), pages 361-386, March.
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