Metamodel-Based Sensitivity Analysis: Polynomial Chaos Expansions and Gaussian Processes

  • Loïc Le GratietEmail author
  • Stefano Marelli
  • Bruno Sudret
Reference work entry


Global sensitivity analysis is now established as a powerful approach for determining the key random input parameters that drive the uncertainty of model output predictions. Yet the classical computation of the so-called Sobol’ indices is based on Monte Carlo simulation, which is not affordable when computationally expensive models are used, as it is the case in most applications in engineering and applied sciences. In this respect metamodels such as polynomial chaos expansions (PCE) and Gaussian processes (GP) have received tremendous attention in the last few years, as they allow one to replace the original, taxing model by a surrogate which is built from an experimental design of limited size. Then the surrogate can be used to compute the sensitivity indices in negligible time. In this chapter an introduction to each technique is given, with an emphasis on their strengths and limitations in the context of global sensitivity analysis. In particular, Sobol’ (resp. total Sobol’) indices can be computed analytically from the PCE coefficients. In contrast, confidence intervals on sensitivity indices can be derived straightforwardly from the properties of GPs. The performance of the two techniques is finally compared on three well-known analytical benchmarks (Ishigami, G-Sobol’, and Morris functions) as well as on a realistic engineering application (deflection of a truss structure).


Polynomial chaos expansions Gaussian process regression Kriging Error estimation Sobol’ indices Model selection 


  1. 1.
    Abramowitz, M., Stegun, I.: Handbook of Mathematical Functions. Dover Publications, New York (1970)zbMATHGoogle Scholar
  2. 2.
    Andrianov, G., Burriel, S., Cambier, S., Dutfoy, A., Dutka-Malen, I., de Rocquigny, E., Sudret, B., Benjamin, P., Lebrun, R., Mangeant, F., Pendola, M.: Open TURNS, an open source initiative to Treat Uncertainties, Risks’N Statistics in a structured industrial approach. In: Proceedings of the ESREL’2007 Safety and Reliability Conference, Stavenger (2007)Google Scholar
  3. 3.
    Bachoc, F.: Cross validation and maximum likelihood estimations of hyper-parameters of Gaussian processes with model misspecification. Comput. Stat. Data Anal. 66, 55–69 (2013)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bates, R.A., Buck, R., Riccomagno, E., Wynn, H.: Experimental design and observation for large systems. J. R. Stat. Soc. Ser. B 58(1), 77–94 (1996)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Bect, J., Ginsbourger, D., Li, L., Picheny, V., Vazquez, E.: Sequential design of computer experiments for the estimation of a probability of failure. Stat. Comput. 22, 773–793 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    van Beers, W., Kleijnen, J.: Customized sequential designs for random simulation experiments: Kriging metamodelling and bootstrapping. Eur. J. Oper. Res. 186, 1099–1113 (2008)CrossRefzbMATHGoogle Scholar
  7. 7.
    Berveiller, M., Sudret, B., Lemaire, M.: Presentation of two methods for computing the response coefficients in stochastic finite element analysis. In: Proceedings of the 9th ASCE Specialty Conference on Probabilistic Mechanics and Structural Reliability, Albuquerque (2004)Google Scholar
  8. 8.
    Berveiller, M., Sudret, B., Lemaire, M.: Stochastic finite elements: a non intrusive approach by regression. Eur. J. Comput. Mech. 15(1–3), 81–92 (2006)zbMATHGoogle Scholar
  9. 9.
    Bieri, M., Schwab, C.: Sparse high order FEM for elliptic sPDEs. Comput. Methods Appl. Mech. Eng. 198, 1149–1170 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Blatman, G.: Adaptive sparse polynomial chaos expansions for uncertainty propagation and sensitivity analysis. PhD thesis, Université Blaise Pascal, Clermont-Ferrand (2009)Google Scholar
  11. 11.
    Blatman, G., Sudret, B.: Sparse polynomial chaos expansions and adaptive stochastic finite elements using a regression approach. C. R. Mécanique 336(6), 518–523 (2008)CrossRefzbMATHGoogle Scholar
  12. 12.
    Blatman, G., Sudret, B.: An adaptive algorithm to build up sparse polynomial chaos expansions for stochastic finite element analysis. Prob. Eng. Mech. 25, 183–197 (2010)CrossRefGoogle Scholar
  13. 13.
    Blatman, G., Sudret, B.: Efficient computation of global sensitivity indices using sparse polynomial chaos expansions. Reliab. Eng. Syst. Saf. 95, 1216–1229 (2010)CrossRefGoogle Scholar
  14. 14.
    Blatman, G., Sudret, B.: Adaptive sparse polynomial chaos expansion based on Least Angle Regression. J. Comput. Phys. 230, 2345–2367 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Brown, S., Beck, J., Mahgerefteh, H., Fraga, E.: Global sensitivity analysis of the impact of impurities on CO2 pipeline failure. Reliab. Eng. Syst. Saf. 115, 43–54 (2013)CrossRefGoogle Scholar
  16. 16.
    Buzzard, G.: Global sensitivity analysis using sparse grid interpolation and polynomial chaos. Reliab. Eng. Syst. Saf. 107, 82–89 (2012)CrossRefGoogle Scholar
  17. 17.
    Buzzard, G., Xiu, D.: Variance-based global sensitivity analysis via sparse-grid interpolation and cubature. Commun. Comput. Phys. 9(3), 542–567 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Chastaing, G., Le Gratiet, L.: Anova decomposition of conditional Gaussian processes for sensitivity analysis with dependent inputs. J. Stat. Comput. Simul. 85(11), 2164–2186 (2015)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Chilès, J., Delfiner, P.: Geostatistics: Modeling Spatial Uncertainty. Wiley Series in Probability and Statistics (Applied Probability and Statistics Section). Wiley, New York (1999)CrossRefzbMATHGoogle Scholar
  20. 20.
    Crestaux, T., Le Maître, O., Martinez, J.M.: Polynomial chaos expansion for sensitivity analysis. Reliab. Eng. Syst. Saf. 94(7), 1161–1172 (2009)CrossRefGoogle Scholar
  21. 21.
    De Lozzo, M., Marrel, A.: Estimation of the derivative-based global sensitivity measures using a Gaussian process metamodel (2015, submitted)Google Scholar
  22. 22.
    Ditlevsen, O., Madsen, H.: Structural reliability methods. Wiley, Chichester (1996)Google Scholar
  23. 23.
    Doostan, A., Owhadi, H.: A non-adapted sparse approximation of PDEs with stochastic inputs. J. Comput. Phys. 230(8), 3015–3034 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Dubreuil, S., Berveiller, M., Petitjean, F., Salaün, M.: Construction of bootstrap confidence intervals on sensitivity indices computed by polynomial chaos expansion. Reliab. Eng. Syst. Saf. 121, 263–275 (2014)CrossRefGoogle Scholar
  25. 25.
    Dubrule, O.: Cross validation of Kriging in a unique neighborhood. Math. Geol. 15, 687–699 (1983)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Durrande, N., Ginsbourger, D., Roustant, O., Laurent, C.: Anova kernels and RKHS of zero mean functions for model-based sensitivity analysis. J. Multivar. Anal. 115, 57–67 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Efron, B., Hastie, T., Johnstone, I., Tibshirani, R.: Least angle regression. Ann. Stat. 32, 407–499 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Fajraoui, N., Ramasomanana, F., Younes, A., Mara, T., Ackerer, P., Guadagnini, A.: Use of global sensitivity analysis and polynomial chaos expansion for interpretation of nonreactive transport experiments in laboratory-scale porous media. Water Resour. Res. 47(2) (2011)Google Scholar
  29. 29.
    Ghanem, R., Spanos, P.: Stochastic Finite Elements – A Spectral Approach. Springer, New York (1991). (Reedited by Dover Publications, Mineola, 2003)CrossRefzbMATHGoogle Scholar
  30. 30.
    Gramacy, R., Taddy, M.: Categorical inputs, sensitivity analysis, optimization and importance tempering with tgp version 2, an R package for treed Gaussian process models. J. Stat. Softw. 33, 1–48 (2010)CrossRefGoogle Scholar
  31. 31.
    Gramacy, R.B., Taddy, M., Wild, S.M.: Variable selection and sensitivity analysis using dynamic trees, with an application to computer performance tuning. Ann. Appl. Stat. 7(1), 51–80 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Harville, D.: Maximum likelihood approaches to variance component estimation and to related problems. J. Am. Stat. Assoc. 72(358), 320–338 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Iooss, B., Lemaître, P.: A review on global sensitivity analysis methods. In: Meloni, C., Dellino, G. (eds.) Uncertainty management in Simulation-Optimization of Complex Systems: Algorithms and Applications. Springer, New York (2015)Google Scholar
  34. 34.
    Jakeman, J., Eldred, M., Sargsyan, K.: Enhancing 1-minimization estimates of polynomial chaos expansions using basis selection. J. Comput. Phys. 289, 18–34 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Janon, A., Klein, T., Lagnoux, A., Nodet, M., Prieur, C.: Asymptotic normality and efficiency of two Sobol’ index estimators. ESAIM: Prob. Stat. 18, 342–364 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Lantuéjoul, C., Desassis, N.: Simulation of a Gaussian random vector: a propagative version of the Gibbs sampler. In: The 9th International Geostatistics Congress, Oslo, p. 1747181, (2012)
  37. 37.
    Le Gratiet, L., Cannamela, C.: Cokriging-based sequential design strategies using fast cross-validation techniques for multifidelity computer codes. Technometrics 57(3), 418–427 (2015)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Le Gratiet, L., Cannamela, C., Iooss, B.: A Bayesian approach for global sensitivity analysis of (multifidelity) computer codes. SIAM/ASA J. Uncertain. Quantif. 2(1), 336–363 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Le Gratiet L, Couplet, M., Iooss, B., Pronzato, L.: Planification d’expériences séquentielle pour l’analyse de sensibilité. In: 46èmes Journés de Statistique de la SFdS, Rennes (2014)Google Scholar
  40. 40.
    Lebrun, R., Dutfoy, A.: An innovating analysis of the Nataf transformation from the copula viewpoint. Prob. Eng. Mech. 24(3), 312–320 (2009)CrossRefGoogle Scholar
  41. 41.
    Marelli, S., Sudret, B.: UQLab: a framework for uncertainty quantification in Matlab. In: Vulnerability, Uncertainty, and Risk (Proceedings of the 2nd International Conference on Vulnerability, Risk Analysis and Management (ICVRAM2014), Liverpool), pp. 2554–2563, doi:10.1061/9780784413609.257,, (2014)
  42. 42.
    Marrel, A., Iooss, B., Van Dorpe, F., Volkova, E.: An efficient methodology for modeling complex computer codes with Gaussian processes. Comput. Stat. Data Anal. 52(10), 4731–4744 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Marrel, A., Iooss, B., Laurent, B., Roustant, O.: Calculations of Sobol indices for the Gaussian process metamodel. Reliab. Eng. Syst. Saf. 94, 742–751 (2009)CrossRefGoogle Scholar
  44. 44.
    Marrel, A., Iooss, B., Da Veiga, S., Ribatet, M.: Global sensitivity analysis of stochastic computer models with joint metamodels. Stat. Comput. 22(3), 833–847 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    McKay, M.D., Beckman, R.J., Conover, W.J.: A comparison of three methods for selecting values of input variables in the analysis of output from a computer code. Technometrics 2, 239–245 (1979)MathSciNetzbMATHGoogle Scholar
  46. 46.
    Munoz Zuniga M, Kucherenko, S., Shah, N.: Metamodelling with independent and dependent inputs. Comput. Phys. Commun. 184, 1570–1580 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Niederreiter, H.: Random Number Generation and Quasi-Monte Carlo Methods. Society for Industrial and Applied Mathematics, Philadelphia (1992)CrossRefzbMATHGoogle Scholar
  48. 48.
    Oakley, J., O’Hagan, A.: Probabilistic sensitivity analysis of complex models a Bayesian approach. J. R. Stat. Soc. Ser. B 66(part 3), 751–769 (2004)Google Scholar
  49. 49.
    Rasmussen, C., Williams, C.: Gaussian Processes for Machine Learning. MIT, Cambridge (2006)zbMATHGoogle Scholar
  50. 50.
    Robert, C.: The Bayesian Choice: From Decision-Theoretic Foundations to Computational Implementation. Springer, New York (2007)zbMATHGoogle Scholar
  51. 51.
    Sandoval, E.H., Anstett-Collin, F., Basset, M.: Sensitivity study of dynamic systems using polynomial chaos. Reliab. Eng. Syst. Saf. 104, 15–26 (2012)CrossRefGoogle Scholar
  52. 52.
    Santner, T., Williams, B., Notz, W.: The Design and Analysis of Computer Experiments. Springer, New York (2003)CrossRefzbMATHGoogle Scholar
  53. 53.
    Sargsyan, K., Safta, C., Najm, H., Debusschere, B., Ricciuto, D., Thornton, P.: Dimensionality reduction for complex models via Bayesian compressive sensing. Int. J. Uncertain. Quantif. 4(1), 63–93 (2014)MathSciNetCrossRefGoogle Scholar
  54. 54.
    Schöbi, R., Sudret, B., Wiart, J.: Polynomial-chaos-based Kriging. Int. J. Uncertain. Quantif. 5(2), 171–193 (2015)MathSciNetCrossRefGoogle Scholar
  55. 55.
    Sobol’, I.: Sensitivity estimates for non linear mathematical models. Math. Model. Comput. Exp. 1, 407–414 (1993)Google Scholar
  56. 56.
    Sobol’, I., Kucherenko, S.: Derivative based global sensitivity measures and their link with global sensitivity indices. Math. Comput. Simul. 79(10), 3009–3017 (2009)Google Scholar
  57. 57.
    Sobol’, I., Tarantola, S., Gatelli, D., Kucherenko, S., Mauntz, W.: Estimating the approximation error when fixing unessential factors in global sensitivity analysis. Reliab. Eng. Syst. Saf. 92(7), 957–960 (2007)Google Scholar
  58. 58.
    Soize, C., Ghanem, R.: Physical systems with random uncertainties: chaos representations with arbitrary probability measure. SIAM J. Sci. Comput. 26(2), 395–410 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  59. 59.
    Stein, M.: Interpolation of Spatial Data. Springer Series in Statistics. Springer, New York (1999)CrossRefzbMATHGoogle Scholar
  60. 60.
    Storlie, C., Swiler, L., Helton, J., Sallaberry, C.: Implementation and evaluation of nonparametric regression procedures for sensitivity analysis of computationally demanding models. Reliab. Eng. Syst. Saf. 94(11), 1735–1763 (2009)CrossRefGoogle Scholar
  61. 61.
    Sudret, B.: Global sensitivity analysis using polynomial chaos expansions. In: Spanos, P., Deodatis, G. (eds.) In: Proceeding of the 5th International Conference on Computational Stochastic Mechanics (CSM5), Rhodos (2006)Google Scholar
  62. 62.
    Sudret, B.: Uncertainty propagation and sensitivity analysis in mechanical models – contributions to structural reliability and stochastic spectral methods. Tech. rep., Université Blaise Pascal, Clermont-Ferrand, France, habilitation à diriger des recherches (229 pages) (2007)Google Scholar
  63. 63.
    Sudret, B.: Global sensitivity analysis using polynomial chaos expansions. Reliab. Eng. Syst. Saf. 93, 964–979 (2008)CrossRefGoogle Scholar
  64. 64.
    Sudret, B.: Polynomial chaos expansions and stochastic finite element methods. In: Phoon, K.K., Ching, J. (eds.) Risk and Reliability in Geotechnical Engineering. Taylor and Francis, Boca Raton (2015)Google Scholar
  65. 65.
    Sudret, B., Caniou, Y.: Analysis of covariance (ANCOVA) using polynomial chaos expansions. In: Deodatis, G. (ed.) Proceeding of the 11th International Conference on Structural Safety and Reliability (ICOSSAR’2013), New York (2013)Google Scholar
  66. 66.
    Sudret, B., Mai, C.V.: Computing derivative-based global sensitivity measures using polynomial chaos expansions. Reliab. Eng. Syst. Saf. 134, 241–250 (2015)CrossRefGoogle Scholar
  67. 67.
    Svenson, J., Santner, T., Dean, A., Hyejung, M.: Estimating sensitivity indices based on Gaussian process metamodels with compactly supported correlation functions. J. Stat. Plan. Inference 114, 160–172 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  68. 68.
    Welch, W.J., Buck, R.J., Sacks, J., Wynn, H.P., Mitchell, T.J., Morris, M.D.: Screening, predicting, and computer experiments. Technometrics 34(1), 15–25 (1992)CrossRefGoogle Scholar
  69. 69.
    Xiu, D., Karniadakis, G.: The Wiener-Askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput. 24(2), 619–644 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  70. 70.
    Younes, A., Mara, T., Fajraoui, N., Lehmann, F., Belfort, B., Beydoun, H.: Use of global sensitivity analysis to help assess unsaturated soil hydraulic parameters. Vadose Zone J. 12(1) (2013)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2017

Authors and Affiliations

  • Loïc Le Gratiet
    • 1
    Email author
  • Stefano Marelli
    • 2
  • Bruno Sudret
    • 2
  1. 1.EDF R&DChatouFrance
  2. 2.Chair of Risk, Safety and Uncertainty QuantificationETH ZürichZürichSwitzerland

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