Variance-Based Sensitivity Analysis: Theory and Estimation Algorithms

  • Clémentine PrieurEmail author
  • Stefano Tarantola
Reference work entry


This section aims at presenting an overview of variance-based approaches for global sensitivity analysis. Starting from functional ANOVA, Sobol’ indices are first defined and then estimation algorithms are provided. The performance of these algorithms is theorically and practically discussed. The review includes recent results on the topic.


FANOVA Sobol’ sensitivity indices Global sensitivity analysis Monte Carlo sampling Quasi-Monte Carlo sampling Sampling design Replication Latin hypercube sampling Orthogonal arrays (OA) Spectral methods Fourier amplitude sensitivity test Random balance design Effective algorithm for sensitivity indices Polynomial chaos expansion effective dimension Sensitivity indices with given data 


  1. 1.
    Blatman, G., Sudret, B.: Adaptive sparse polynomial chaos expansion based on least angle regression. J. Comput. Phys. 230(6), 2345–2367 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Caflisch, R., Morokoff, W., Owen, A.: Valuation of mortgage backed securities using Brownian bridges to reduce effective dimension. J. Comput. Fin. 1(1), 27–46 (1997)CrossRefGoogle Scholar
  3. 3.
    Champion, M., Chastaing, G., Gadat, S., Prieur, C.: L2-Boosting for sensitivity analysis with dependent inputs. Stat. Sin. 25(4), (2015). doi:10.5705/ss.2013.310
  4. 4.
    Chastaing, G., Gamboa, F., Prieur, C.: Generalized Hoeffding-Sobol decomposition for dependent variables. Application to sensitivity analysis. Electron. J. Stat. 6, 2420–2448 (2012)zbMATHGoogle Scholar
  5. 5.
    Chastaing, G., Gamboa, F., Prieur, C.: Generalized Sobol sensitivity indices for dependent variables: numerical methods. J. Stat. Comput. Simul. (ahead-of-print), 1–28 (2014)Google Scholar
  6. 6.
    Cukier, H., Fortuin, C.M., Shuler, K., Petschek, A.G., Schaibly, J.H.: Study of the sensitivity of coupled reaction systems to uncertainties in rate coefficients: theory. J. Chem. Phys. 59, 3873–3878 (1973)CrossRefGoogle Scholar
  7. 7.
    Cukier, H., Levine, R.I., Shuler, K.: Nonlinear sensitivity analysis of multiparameter model systems. J. Comput. Phys. 26, 1–42 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Cukier, H., Schaibly, J.H., Shuler, K.: Study of the sensitivity of coupled reaction systems to uncertainties in rate coefficients: analysis of the approximations. J. Chem. Phys. 63, 1140–1149 (1975)CrossRefGoogle Scholar
  9. 9.
    Efron, B., Stein, C.: The jackknife estimate of variance. Ann. Stat. 9(3), 586–596 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Fort, J.-C., Klein, T., Lagnoux, A., Laurent, B.: Estimation of the Sobol indices in a linear functional multidimensional model. J. Stat. Plan. Inference 143(9), 1590–1605 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Fruth, J., Roustant, O., Kuhnt, S.: Total interaction index: a variance-based sensitivity index for second-order interaction screening. J. Stat. Plann. Inference 147, 212–223 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Fruth, J., Roustant, O., Kuhnt, S.: Sequential designs for sensitivity analysis of functional inputs in computer experiments. Reliab. Eng. Syst. Saf. 134, 260–267 (2015)CrossRefGoogle Scholar
  13. 13.
    Gamboa, F., Janon, A., Klein, T., Lagnoux, A.: Sensitivity analysis for multidimensional and functional outputs. Electron. J. Stat. 8(1), 575–603 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Garcia-Cabrejo, O., Valocchi, A.: Global sensitivity analysis for multivariate output using polynomial chaos expansion. Reliab. Eng. Syst. Saf. 126, 25–36 (2014)CrossRefGoogle Scholar
  15. 15.
    Homma, T., Saltelli, A.: Importance measures in global sensitivity analysis of nonlinear models. Reliab. Eng. Syst. Saf. 52(1), 1–17 (1996)CrossRefGoogle Scholar
  16. 16.
    Hooker, G.: Discovering additive structure in black box functions. In: Proceedings of the Tenth ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (KDD’04), Seattle, pp. 575–580. ACM (2004)Google Scholar
  17. 17.
    Iooss, B., Ribatet, M.: Global sensitivity analysis of computer models with functional inputs. Reliab. Eng. Syst. Saf. 94(7), 1194–1204 (2009)CrossRefGoogle Scholar
  18. 18.
    Janon, A., Klein, T., Lagnoux-Renaudie, A., Nodet, M., Prieur, C.: Asymptotic normality and efficiency of two Sobol index estimators. ESAIM: Probab. Stat. 18, 342–364 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Jansen, M.J.W.: Analysis of variance designs for model output. Comput. Phys. Commun. 117(1), 35–43 (1999)CrossRefzbMATHGoogle Scholar
  20. 20.
    Jansen, M.J.W., Rossing, W.A.H., Daamen, R.A.: Monte carlo estimation of uncertainty contributions from several independent multivariate sources. In: Gasman, J., van Straten, G. (eds.) Predictability and Nonlinear Modelling in Natural Sciences and Economics, pp. 334–343. Kluwer Academic, Dordrecht (1994)CrossRefGoogle Scholar
  21. 21.
    Joe, S., Kuo, F.Y.: Remark on algorithm 659: Implementing Sobols quasirandom sequence generator. ACM Trans. Math. Softw. 29(1), 49–57 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Joint Research Centre of the European Commission: Sensitivity analysis software. (2014)
  23. 23.
    Kelley, T.L.: An unbiased correlation ratio measure. Proc. Natl. Acad. Sci. U. S. A. 21(9), 554–559 (1935)CrossRefzbMATHGoogle Scholar
  24. 24.
    Kishen, K.: On latin and hyper-graeco cubes and hypercubes. Curr. Sci. 11(3), 98–99 (1942)Google Scholar
  25. 25.
    Kucherenko, S., Balazs, F., Nilay, S., Mauntz, W.: The identification of model effective dimensions using global sensitivity analysis. Reliab. Eng. Syst. Saf. 96, 440–449 (2011)CrossRefGoogle Scholar
  26. 26.
    Kucherenko, S., Tarantola, S., Annoni, P.: Estimation of global sensitivity indices for models with dependent variables. Comput. Phys. Commun. 183(4), 937–946 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Lamboni, M., Monod, H., Makowski, D.: Multivariate sensitivity analysis to measure global contribution of input factors in dynamic models. Reliab. Eng. Syst. Saf. 96(4), 450–459 (2011)CrossRefGoogle Scholar
  28. 28.
    Lemieux, C.: Monte Carlo and Quasi-Monte Carlo Sampling. Springer Series in Statistics. Springer, New York (2009)zbMATHGoogle Scholar
  29. 29.
    Li, G., Rabitz, H.: General formulation of HDMR component functions with independent and correlated variables. J. Math. Chem. 50(1), 99–130 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Lilburne, L., Tarantola, S.: Sensitivity analysis of spatial models. Int. J. Geogr. Inf. Sci. 23(2), 151–168 (2009)CrossRefGoogle Scholar
  31. 31.
    Liu, R., Owen, A.: Estimating mean dimensionality of analysis of variance decompositions. J. Am. Stat. Assoc. 101(474), 712–721 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Mara, T.A., Joseph, O.R.: Comparison of some efficient methods to evaluate the main effect of computer model factors. J. Stat. Comput. Simul. 78, 167–178 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Mara, T.A., Tarantola, S.: Variance-based sensitivity indices for models with dependent inputs. Reliab. Eng. Syst. Saf. 107, 115–121 (2012)CrossRefGoogle Scholar
  34. 34.
    McKay, M.D.: Evaluating prediction uncertainty. Technical Report NUREG/CR-6311, US Nuclear Regulatory Commission and Los Alamos National Laboratory, pp. 1–79 (1995)Google Scholar
  35. 35.
    Moore, L.M., Morris, M.D., McKay, M.D.: Using orthogonal arrays in the sensitivity analysis of computer models. Technometrics 50, 205–215 (2008)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Monod, H., Naud, C., Makowski, D.: Uncertainty and sensitivity analysis for crop models. In: Wallach, D., Makowski, D., Jones, J.W. (eds.) Working with Dynamic Crop Models: Evaluation, Analysis, Parameterization, Applications, chap. 4, pp. 55–99. Elsevier (2006)Google Scholar
  37. 37.
    Owen, A.B.: Orthogonal arrays for computer experiments, integration and visualization. Stat. Sin. 2, 439–452 (1992)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Owen, A.B.: Better estimation of small Sobol’ sensitivity indices. ACM Trans. Model. Comput. Simul. 23, 11–17 (2013)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Pearson, K.: Mathematical contributions to the theory of evolution. Proc. R. Soc. Lond. 71, 288–313 (1903)CrossRefzbMATHGoogle Scholar
  40. 40.
    Plischke, E.: An effective algorithm for computing global sensitivity indices (easi). Reliab. Eng. Syst. Saf. 95, 354–360 (2010)CrossRefGoogle Scholar
  41. 41.
    Plischke, E., Borgonovo, E., Smith, C.L.: Global sensitivity measures from given data. Eur. J. Oper. Res. 226, 536–550 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Qian, P.Z.G.: Nested Latin hypercube designs. Biometrika 96(4), 957–970 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Saltelli, A.: Making best use of model evaluations to compute sensitivity indices. Comput. Phys. Commun. 145, 280–297 (2002)CrossRefzbMATHGoogle Scholar
  44. 44.
    Saltelli, A., Annoni, P., Azzini, I., Campolongo, F., Ratto, M., Tarantola, S.: Variance based sensitivity analysis of model output. Design and estimator for the total sensitivity index. Comput. Phys. Commun. 181(2), 259–270 (2010)zbMATHGoogle Scholar
  45. 45.
    Saltelli, A., Tarantola, S., Campolongo, F.: Sensitivity analysis as an ingredient of modeling. Stat. Sci. 15(4), 377–395 (2000)MathSciNetCrossRefGoogle Scholar
  46. 46.
    Saltelli, A., Tarantola, S., Chan, K.: A quantitative, model-independent method for global sensitivity analysis of model output. Technometrics 41, 39–56 (1999)CrossRefGoogle Scholar
  47. 47.
    Schaibly, J.H., Shuler, K.: Study of the sensitivity of coupled reaction systems to uncertainties in rate coefficients: applications. J. Chem. Phys. 59, 3879–3888 (1973)CrossRefGoogle Scholar
  48. 48.
    Sobol’, I.M.: Sensitivity analysis for nonlinear mathematical models. Math. Model. Comput. Exp. 1, 407–414, 1993.Google Scholar
  49. 49.
    Sudret, B.: Global sensitivity analysis using polynomial chaos expansion. Reliab. Eng. Syst. Saf. 93, 964–979 (2008)CrossRefGoogle Scholar
  50. 50.
    Tarantola, S., Gatelli, D., Mara, T.A.: Random balance designs for the estimation of first-order global sensitivity indices. Reliab. Eng. Syst. Saf. 91, 717–727 (2006)CrossRefGoogle Scholar
  51. 51.
    The Comprehensive R Archive Network: DiceDesign package.
  52. 52.
    The Comprehensive R Archive Network: Sensitivity package.
  53. 53.
    Tissot, J.Y., Prieur, C.: A bias correction method for the estimation of sensitivity indices based on random balance designs. Reliab. Eng. Syst. Saf. 107, 205–213 (2012)CrossRefGoogle Scholar
  54. 54.
    Tissot, J.Y., Prieur, C.: Variance-based sensitivity analysis using harmonic analysis. Technical report (2012)
  55. 55.
    Tissot, J.Y., Prieur, C.: A randomized orthogonal array-based procedure for the estimation of first- and second-order Sobol’ indices. J. Stat. Comput. Simul. 85(7), 1358–1381 (2015)MathSciNetCrossRefGoogle Scholar
  56. 56.
    Wang, X., Fang, K.-T.: The effective dimension and quasi-Monte Carlo integration. J. Complex. 19(2), 101–124 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  57. 57.
    Xu, C., Gertner, G.: Understanding and comparisons of different sampling approaches for the Fourier amplitudes sensitivity test (fast). Comput. Stat. Data Anal. 55(1), 184–198 (2011)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2017

Authors and Affiliations

  1. 1.Laboratoire Jean Kuntzmann (LJK)University of Grenoble Alpes, INRIAGrenobleFrance
  2. 2.Statistical Indicators for Policy AssessmentJoint Research Centre of the European CommissionIspra (VA)Italy

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