IDEAS home Printed from https://ideas.repec.org/a/spr/stpapr/v61y2020i5d10.1007_s00362-018-1010-4.html
   My bibliography  Save this article

Uncertainty quantification: a minimum variance unbiased (joint) estimator of the non-normalized Sobol’ indices

Author

Listed:
  • Matieyendou Lamboni

    (University of Guyane
    228-UMR Espace-Dev, University of Guyane, University of Réunion, University of Montpellier, IRD)

Abstract

Often, uncertainty quantification is followed by the computation of sensitivity indices of input factors. Variance-based sensitivity analysis and multivariate sensitivity analysis (MSA) aim to apportion the variability of the model output(s) into input factors and their interactions. Sobol’ indices (first-order and total indices), which quantify the effects of input factor(s), serve as a practical tool to assess interactions among input factors, the order of interactions, and the magnitude of interactions. In this paper, we investigate a novel way of estimating both the first-order and total indices based on U-statistics, including the statistical properties of the new estimator. First, we provide a minimum variance unbiased estimator of the non-normalized Sobol’ indices as well as its optimal rate of convergence and its asymptotic distribution. Second, we derive a joint estimator of Sobol’ indices, its consistency and its asymptotic distribution, and third, we demonstrate the applicability of these results by means of numerical tests. The new estimator allows for improving the estimation of Sobol’ indices for some degrees of the kernel.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:stpapr:v:61:y:2020:i:5:d:10.1007_s00362-018-1010-4
    DOI: 10.1007/s00362-018-1010-4
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s00362-018-1010-4
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s00362-018-1010-4?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    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. Storlie, Curtis B. & Swiler, Laura P. & Helton, Jon C. & Sallaberry, Cedric J., 2009. "Implementation and evaluation of nonparametric regression procedures for sensitivity analysis of computationally demanding models," Reliability Engineering and System Safety, Elsevier, vol. 94(11), pages 1735-1763.
    3. E. Borgonovo & S. Tarantola & E. Plischke & M. D. Morris, 2014. "Transformations and invariance in the sensitivity analysis of computer experiments," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 76(5), pages 925-947, November.
    4. Bolado-Lavin, R. & Castaings, W. & Tarantola, S., 2009. "Contribution to the sample mean plot for graphical and numerical sensitivity analysis," Reliability Engineering and System Safety, Elsevier, vol. 94(6), pages 1041-1049.
    5. Sudret, Bruno, 2008. "Global sensitivity analysis using polynomial chaos expansions," Reliability Engineering and System Safety, Elsevier, vol. 93(7), pages 964-979.
    6. 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.
    7. Mara, Thierry A. & Tarantola, Stefano, 2012. "Variance-based sensitivity indices for models with dependent inputs," Reliability Engineering and System Safety, Elsevier, vol. 107(C), pages 115-121.
    8. Lamboni, Matieyendou & Monod, Hervé & Makowski, David, 2011. "Multivariate sensitivity analysis to measure global contribution of input factors in dynamic models," Reliability Engineering and System Safety, Elsevier, vol. 96(4), pages 450-459.
    9. Jeremy E. Oakley & Anthony O'Hagan, 2004. "Probabilistic sensitivity analysis of complex models: a Bayesian approach," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 66(3), pages 751-769, August.
    10. Buzzard, Gregery T., 2012. "Global sensitivity analysis using sparse grid interpolation and polynomial chaos," Reliability Engineering and System Safety, Elsevier, vol. 107(C), pages 82-89.
    11. Marco Ratto & Andrea Pagano, 2010. "Using recursive algorithms for the efficient identification of smoothing spline ANOVA models," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 94(4), pages 367-388, December.
    12. Kucherenko, S. & Delpuech, B. & Iooss, B. & Tarantola, S., 2015. "Application of the control variate technique to estimation of total sensitivity indices," Reliability Engineering and System Safety, Elsevier, vol. 134(C), pages 251-259.
    13. 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.
    14. Plischke, Elmar & Borgonovo, Emanuele & Smith, Curtis L., 2013. "Global sensitivity measures from given data," European Journal of Operational Research, Elsevier, vol. 226(3), pages 536-550.
    15. 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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Matieyendou Lamboni, 2024. "Kernel-based Measures of Association Between Inputs and Outputs Using ANOVA," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 86(2), pages 790-826, August.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. 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.
    2. Wei, Pengfei & Lu, Zhenzhou & Song, Jingwen, 2015. "Variable importance analysis: A comprehensive review," Reliability Engineering and System Safety, Elsevier, vol. 142(C), pages 399-432.
    3. Borgonovo, Emanuele & Plischke, Elmar, 2016. "Sensitivity analysis: A review of recent advances," European Journal of Operational Research, Elsevier, vol. 248(3), pages 869-887.
    4. Lamboni, Matieyendou, 2020. "Derivative-based generalized sensitivity indices and Sobol’ indices," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 170(C), pages 236-256.
    5. Konakli, Katerina & Sudret, Bruno, 2016. "Global sensitivity analysis using low-rank tensor approximations," Reliability Engineering and System Safety, Elsevier, vol. 156(C), pages 64-83.
    6. Cheng, Lei & Lu, Zhenzhou & Zhang, Leigang, 2015. "Application of Rejection Sampling based methodology to variance based parametric sensitivity analysis," Reliability Engineering and System Safety, Elsevier, vol. 142(C), pages 9-18.
    7. 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.
    8. Wu, Zeping & Wang, Donghui & Okolo N, Patrick & Hu, Fan & Zhang, Weihua, 2016. "Global sensitivity analysis using a Gaussian Radial Basis Function metamodel," Reliability Engineering and System Safety, Elsevier, vol. 154(C), pages 171-179.
    9. Becker, William, 2020. "Metafunctions for benchmarking in sensitivity analysis," Reliability Engineering and System Safety, Elsevier, vol. 204(C).
    10. Wu, Zeping & Wang, Wenjie & Wang, Donghui & Zhao, Kun & Zhang, Weihua, 2019. "Global sensitivity analysis using orthogonal augmented radial basis function," Reliability Engineering and System Safety, Elsevier, vol. 185(C), pages 291-302.
    11. Ge, Qiao & Menendez, Monica, 2017. "Extending Morris method for qualitative global sensitivity analysis of models with dependent inputs," Reliability Engineering and System Safety, Elsevier, vol. 162(C), pages 28-39.
    12. Deman, G. & Konakli, K. & Sudret, B. & Kerrou, J. & Perrochet, P. & Benabderrahmane, H., 2016. "Using sparse polynomial chaos expansions for the global sensitivity analysis of groundwater lifetime expectancy in a multi-layered hydrogeological model," Reliability Engineering and System Safety, Elsevier, vol. 147(C), pages 156-169.
    13. Sinan Xiao & Zhenzhou Lu & Pan Wang, 2018. "Multivariate Global Sensitivity Analysis Based on Distance Components Decomposition," Risk Analysis, John Wiley & Sons, vol. 38(12), pages 2703-2721, December.
    14. Ge, Qiao & Ciuffo, Biagio & Menendez, Monica, 2015. "Combining screening and metamodel-based methods: An efficient sequential approach for the sensitivity analysis of model outputs," Reliability Engineering and System Safety, Elsevier, vol. 134(C), pages 334-344.
    15. Liu, Fuchao & Wei, Pengfei & Tang, Chenghu & Wang, Pan & Yue, Zhufeng, 2019. "Global sensitivity analysis for multivariate outputs based on multiple response Gaussian process model," Reliability Engineering and System Safety, Elsevier, vol. 189(C), pages 287-298.
    16. Mirko Ginocchi & Ferdinanda Ponci & Antonello Monti, 2021. "Sensitivity Analysis and Power Systems: Can We Bridge the Gap? A Review and a Guide to Getting Started," Energies, MDPI, vol. 14(24), pages 1-59, December.
    17. Kucherenko, Sergei & Song, Shufang & Wang, Lu, 2019. "Quantile based global sensitivity measures," Reliability Engineering and System Safety, Elsevier, vol. 185(C), pages 35-48.
    18. S. Cucurachi & E. Borgonovo & R. Heijungs, 2016. "A Protocol for the Global Sensitivity Analysis of Impact Assessment Models in Life Cycle Assessment," Risk Analysis, John Wiley & Sons, vol. 36(2), pages 357-377, February.
    19. Lamboni, Matieyendou, 2019. "Multivariate sensitivity analysis: Minimum variance unbiased estimators of the first-order and total-effect covariance matrices," Reliability Engineering and System Safety, Elsevier, vol. 187(C), pages 67-92.
    20. Emanuele Borgonovo & Gordon B. Hazen & Elmar Plischke, 2016. "A Common Rationale for Global Sensitivity Measures and Their Estimation," Risk Analysis, John Wiley & Sons, vol. 36(10), pages 1871-1895, October.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:stpapr:v:61:y:2020:i:5:d:10.1007_s00362-018-1010-4. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.