Global sensitivity analysis based on Gini’s mean difference

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Abstract

Global sensitivity analysis has been widely used to detect the relative contributions of input variables to the uncertainty of model output, and then more resources can be assigned to the important input variables to reduce the uncertainty of model output more efficiently. In this paper, a new kind of global sensitivity index based on Gini’s mean difference is proposed. The proposed sensitivity index is more robust than the variance-based first order sensitivity index for the cases with non-normal distributions. Through the decomposition of Gini’s mean difference, it shows that the proposed sensitivity index can be represented by the energy distance, which measures the difference between probability distributions. Therefore, the proposed sensitivity index also takes the probability distribution of model output into consideration. In order to estimate the proposed sensitivity index efficiently, an efficient Monte Carlo simulation method is also proposed, which avoids the nested sampling procedure. The test examples show that the proposed sensitivity index is more robust than the variance-based first order sensitivity index for the cases with non-normal distributions.

Keywords

Global sensitivity analysis Uncertainty reduction Variance Gini’s mean difference Energy distance 

Notes

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant No. 51475370, 51775439). The authors are also thankful to the anonymous reviewers for their valuable comments.

References

  1. Aven T (2016) Risk assessment and risk management: Review of recent advances on their foundation. Eur J Oper Res 253:1–13.  https://doi.org/10.1016/j.ejor.2015.12.023 MathSciNetCrossRefMATHGoogle Scholar
  2. Borgonovo E (2007) A new uncertainty importance measure. Reliab Eng Syst Saf 92:771–784.  https://doi.org/10.1016/j.ress.2006.04.015 CrossRefGoogle Scholar
  3. Borgonovo E, Peccati L (2006) Uncertainty and global sensitivity analysis in the evaluation of investment projects. Int J Prod Econ 104:62–73.  https://doi.org/10.1016/j.ijpe.2005.05.024 CrossRefGoogle Scholar
  4. Borgonovo E, Plischke E (2016) Sensitivity analysis: A review of recent advances. Eur J Oper Res 248:869–887.  https://doi.org/10.1016/j.ejor.2015.06.032 MathSciNetCrossRefMATHGoogle Scholar
  5. Campolongo F, Cariboni J, Saltelli A (2007) An effective screening design for sensitivity analysis of large models. Environ Model Softw 22:1509–1518.  https://doi.org/10.1016/j.envsoft.2006.10.004 CrossRefGoogle Scholar
  6. Cheng K, Lu Z, Wei Y, Shi Y, Zhou Y (2017) Mixed kernel function support vector regression for global sensitivity analysis. Mech Syst Signal Process 96:201–214.  https://doi.org/10.1016/j.ymssp.2017.04.014 CrossRefGoogle Scholar
  7. Chun M-H, Han S-J, Tak N-IL (2000) An uncertainty importance measure using a distance metric for the change in a cumulative distribution function. Reliab Eng Syst Saf 70:313–321.  https://doi.org/10.1016/S0951-8320(00)00068-5 CrossRefGoogle Scholar
  8. Deng S, Suresh K (2015) Multi-constrained topology optimization via the topological sensitivity. Struct Multidiscip Optim 51:987–1001.  https://doi.org/10.1007/s00158-014-1188-6 MathSciNetCrossRefGoogle Scholar
  9. Deng S, Suresh K (2016) Multi-constrained 3D topology optimization via augmented topological level-set. Comput Struct 170:1–12.  https://doi.org/10.1016/j.compstruc.2016.02.009 CrossRefGoogle Scholar
  10. Deng S, Suresh K (2017a) Stress constrained thermo-elastic topology optimization with varying temperature fields via augmented topological sensitivity based level-set. Struct Multidiscip Optim 56:1413–1427.  https://doi.org/10.1007/s00158-017-1732-2 MathSciNetCrossRefGoogle Scholar
  11. Deng S, Suresh K (2017b) Topology optimization under thermo-elastic buckling. Struct Multidiscip Optim 55:1759–1772.  https://doi.org/10.1007/s00158-016-1611-2 MathSciNetCrossRefGoogle Scholar
  12. Gerstenberger C, Vogel D (2015) On the efficiency of Gini’s mean difference. Statistical Methods & Applications 24:569–596.  https://doi.org/10.1007/s10260-015-0315-x MathSciNetCrossRefMATHGoogle Scholar
  13. Homma T, Saltelli A (1996) Importance measures in global sensitivity analysis of nonlinear models. Reliab Eng Syst Saf 52:1–17.  https://doi.org/10.1016/0951-8320(96)00002-6 CrossRefGoogle Scholar
  14. Hu Z, Mahadevan S (2016) Global sensitivity analysis-enhanced surrogate (GSAS) modeling for reliability analysis. Struct Multidiscip Optim 53:501–521.  https://doi.org/10.1007/s00158-015-1347-4 MathSciNetCrossRefGoogle Scholar
  15. Iman RL (1987) A Matrix-Based Approach to Uncertainty and Sensitivity Analysis for Fault Trees1. Risk Anal 7:21–33.  https://doi.org/10.1111/j.1539-6924.1987.tb00966.x CrossRefGoogle Scholar
  16. Iman RL, Johnson ME, Watson CC (2005a) Sensitivity Analysis for Computer Model Projections of Hurricane Losses. Risk Anal 25:1277–1297.  https://doi.org/10.1111/j.1539-6924.2005.00673.x CrossRefGoogle Scholar
  17. Iman RL, Johnson ME, Watson CC (2005b) Uncertainty analysis for computer model projections of hurricane losses. Risk Anal 25:1299–1312.  https://doi.org/10.1111/j.1539-6924.2005.00674.x CrossRefGoogle Scholar
  18. Jiang C, Li WX, Han X, Liu LX, Le PH (2011) Structural reliability analysis based on random distributions with interval parameters. Comput Struct 89:2292–2302.  https://doi.org/10.1016/j.compstruc.2011.08.006 CrossRefGoogle Scholar
  19. Kala Z (2016) Global sensitivity analysis in stability problems of steel frame structures. J Civ Eng Manag 22:417–424.  https://doi.org/10.3846/13923730.2015.1073618 CrossRefGoogle Scholar
  20. Kala Z, Valeš J (2017) Global sensitivity analysis of lateral-torsional buckling resistance based on finite element simulations. Eng Struct 134:37–47.  https://doi.org/10.1016/j.engstruct.2016.12.032 CrossRefGoogle Scholar
  21. Li C, Mahadevan S (2016) An efficient modularized sample-based method to estimate the first-order Sobol’ index. Reliab Eng Syst Saf 153:110–121.  https://doi.org/10.1016/j.ress.2016.04.012 CrossRefGoogle Scholar
  22. Liu Q, Homma T (2010) A new importance measure for sensitivity analysis. J Nucl Sci Technol 47:53–61.  https://doi.org/10.1080/18811248.2010.9711927 CrossRefGoogle Scholar
  23. Möller B, Beer M (2008) Engineering computation under uncertainty – Capabilities of non-traditional models. Comput Struct 86:1024–1041.  https://doi.org/10.1016/j.compstruc.2007.05.041 CrossRefGoogle Scholar
  24. Morris MD (1991) Factorial sampling plans for preliminary computational experiments. Technometrics 33:161–174.  https://doi.org/10.2307/1269043 CrossRefGoogle Scholar
  25. Nannapaneni S, Hu Z, Mahadevan S (2016) Uncertainty quantification in reliability estimation with limit state surrogates. Struct Multidiscip Optim 54:1509–1526.  https://doi.org/10.1007/s00158-016-1487-1 MathSciNetCrossRefGoogle Scholar
  26. Patil SR, Frey HC (2004) Comparison of Sensitivity Analysis Methods Based on Applications to a Food Safety Risk Assessment Model. Risk Anal 24:573–585.  https://doi.org/10.1111/j.0272-4332.2004.00460.x CrossRefGoogle Scholar
  27. Rashki M, Miri M, Azhdary Moghaddam M (2012) A new efficient simulation method to approximate the probability of failure and most probable point. Struct Saf 39:22–29.  https://doi.org/10.1016/j.strusafe.2012.06.003 CrossRefGoogle Scholar
  28. Rizzo ML, Székely GJ (2010) DISCO Analysis: A Nonparametric Extension of Analysis of Variance. The Annals of Applied Statistics 4:1034–1055MathSciNetCrossRefMATHGoogle Scholar
  29. Rizzo ML, Székely GJ (2016) Energy distance. Wiley Interdisciplinary Reviews: Computational Statistics 8:27–38.  https://doi.org/10.1002/wics.1375 MathSciNetCrossRefGoogle Scholar
  30. Saltelli A (1999) Sensitivity analysis: Could better methods be used? J Geophys Res 104:3789–3793CrossRefGoogle Scholar
  31. Saltelli A, Annoni P, Azzini I, Campolongo F, Ratto M, Tarantola S (2010) Variance based sensitivity analysis of model output. Design and estimator for the total sensitivity index. Comput Phys Commun 181:259–270.  https://doi.org/10.1016/j.cpc.2009.09.018 MathSciNetCrossRefMATHGoogle Scholar
  32. Saltelli A, Tarantola S (2002) On the Relative Importance of Input Factors in Mathematical Models: Safety Assessment for Nuclear Waste Disposal. J Am Stat Assoc 97:702–709MathSciNetCrossRefMATHGoogle Scholar
  33. Saltelli A, Tarantola S, Campolongo F, Ratto M (2004) Sensitivity analysis in practice: A guide to assessing scientific models. John Wiley, New YorkMATHGoogle Scholar
  34. Saltelli A et al (2008) Global sensitivity analysis, The primer. John Wiley & Sons, New YorkMATHGoogle Scholar
  35. Shi Y, Lu Z, Cheng K, Zhou Y (2017) Temporal and spatial multi-parameter dynamic reliability and global reliability sensitivity analysis based on the extreme value moments. Struct Multidiscip Optim 56:117–129.  https://doi.org/10.1007/s00158-017-1651-2 MathSciNetCrossRefGoogle Scholar
  36. Sobol IM (1976) Uniformly distributed sequences with additional uniformity properties. USSR Comput Math Math Phys 16:236–242CrossRefMATHGoogle Scholar
  37. Sobol IM (1993) Sensitivity analysis for non-linear mathematical models. Mathematical Modeling & Computational Experiment 1:407–414MathSciNetMATHGoogle Scholar
  38. Sobol IM (2001) Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates. Math Comput Simul 55:271–280.  https://doi.org/10.1016/S0378-4754(00)00270-6 MathSciNetCrossRefMATHGoogle Scholar
  39. Sobol IM, Asotsky D, Kreinin A, Kucherenko S (2011) Construction and comparison of high-dimensional Sobol’ generators. Wilmott 2011:64–79.  https://doi.org/10.1002/wilm.10056 CrossRefGoogle Scholar
  40. Székely GJ, Rizzo ML (2004) Testing for Equal Distributions in High Dimension. InterStat 5:1–6Google Scholar
  41. Székely GJ, Rizzo ML (2005) A new test for multivariate normality. J Multivar Anal 93:58–80.  https://doi.org/10.1016/j.jmva.2003.12.002 MathSciNetCrossRefMATHGoogle Scholar
  42. Székely GJ, Rizzo ML (2013) Energy statistics: A class of statistics based on distances. Journal of Statistical Planning and Inference 143:1249–1272.  https://doi.org/10.1016/j.jspi.2013.03.018 MathSciNetCrossRefMATHGoogle Scholar
  43. Székely GJ, Rizzo ML, Bakirov NK (2007) Measuring and Testing Dependence by Correlation of Distances. Ann Stat 35:2769–2794MathSciNetCrossRefMATHGoogle Scholar
  44. Tian W (2013) A review of sensitivity analysis methods in building energy analysis. Renew Sust Energ Rev 20:411–419.  https://doi.org/10.1016/j.rser.2012.12.014 CrossRefGoogle Scholar
  45. Xiao S, Lu Z (2016) Structural reliability analysis using combined space partition technique and unscented transformation. J Struct Eng 142:04016089.  https://doi.org/10.1061/(ASCE)ST.1943-541X.0001553 CrossRefGoogle Scholar
  46. Xiao S, Lu Z, Wang P (2018a) Multivariate global sensitivity analysis for dynamic models based on energy distance. Struct Multidiscip Optim 57:279–291.  https://doi.org/10.1007/s00158-017-1753-x MathSciNetCrossRefGoogle Scholar
  47. Xiao S, Lu Z, Wang P (2018b) Multivariate global sensitivity analysis for dynamic models based on wavelet analysis. Reliab Eng Syst Saf 170:20–30.  https://doi.org/10.1016/j.ress.2017.10.007 CrossRefGoogle Scholar
  48. Xiao S, Lu Z, Xu L (2016) A new effective screening design for structural sensitivity analysis of failure probability with the epistemic uncertainty. Reliab Eng Syst Saf 156:1–14.  https://doi.org/10.1016/j.ress.2016.07.014 CrossRefGoogle Scholar
  49. Xiao S, Lu Z, Xu L (2017) Global sensitivity analysis based on random variables with interval parameters by metamodel-based optimisation. International Journal of Systems Science: Operations & Logistics:1–14.  https://doi.org/10.1080/23302674.2017.1296600
  50. Xiong F, Greene S, Chen W, Xiong Y, Yang S (2010) A new sparse grid based method for uncertainty propagation. Struct Multidiscip Optim 41:335–349.  https://doi.org/10.1007/s00158-009-0441-x MathSciNetCrossRefMATHGoogle Scholar
  51. Xu X, Lu Z, Luo X (2017) A kernel estimate method for characteristic function-based uncertainty importance measure. Appl Math Model 42:58–70.  https://doi.org/10.1016/j.apm.2016.09.028 MathSciNetCrossRefGoogle Scholar
  52. Yitzhaki S (2003) Gini's mean difference: a superior measure of variability for non-normal distributions. METRON 61:285–316MathSciNetGoogle Scholar
  53. Zhang K, Lu Z, Wu D, Zhang Y (2017) Analytical variance based global sensitivity analysis for models with correlated variables. Appl Math Model 45:748–767.  https://doi.org/10.1016/j.apm.2016.12.036 MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of AeronauticsNorthwestern Polytechnical UniversityXi’anChina

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