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Structural and Multidisciplinary Optimization

, Volume 59, Issue 1, pp 229–247 | Cite as

Sparse polynomial chaos expansions for global sensitivity analysis with partial least squares and distance correlation

  • Yicheng Zhou
  • Zhenzhou LuEmail author
  • Kai Cheng
Research Paper
  • 150 Downloads

Abstract

Polynomial chaos expansion (PCE) has been proven to be a powerful tool for developing surrogate models in the field of uncertainty and global sensitivity analysis. The computational cost of classical PCE is unaffordable since the number of terms grows exponentially with the dimensionality of inputs. This considerably restricts the practical use of PCE. An efficient approach to address this problem is to build a sparse PCE. Since some basis polynomials in representation are highly correlated and the number of available training samples is small, the sparse PCE obtained by the original least square (LS) regression using these samples may not be accurate. Meanwhile, correlation between the non-influential basis polynomial and the important basis polynomials may disturb the correct selection of the important terms. To overcome the influence of correlation in the construction of sparse PCE, a full PCE of model response is first developed based on partial least squares technique in the paper. And an adaptive algorithm based on distance correlation is proposed to select influential basis polynomials, where the distance correlation is used to quantify effectively the impact of basis polynomials on model response. The accuracy of the surrogate model is assessed by leave-one-out cross validation. The proposed method is validated by several examples and global sensitivity analysis is performed. The results show that it maintains a balance between model accuracy and complexity.

Keywords

Sparse Polynomial chaos expansion Global sensitivity analysis Partial least squares distance correlation Partial distance correlation 

Notes

References

  1. Abdi H (2010) Partial least squares regression and projection on latent structure regression (PLS regression). Wiley Interdisciplinary Rev: Computational Statistics 2(1):97–106CrossRefGoogle Scholar
  2. Abraham S, Raisee M, Ghorbaniasl G, Contino F, Lacor C (2017) A robust and efficient stepwise regression method for building sparse polynomial chaos expansions. J Comput Phys 332:461–474MathSciNetzbMATHCrossRefGoogle Scholar
  3. Alwart H. Singular value decomposition (svd) and generalized singular value decomposition (gsvd). Encyclopedia Measurement Statistics 2006; 907–912Google Scholar
  4. Berveiller M, Sudret B, Lemaire M (2006) Stochastic finite element: a non-intrusive approach by regression. European J Computational Mechanics/Revue Européenne de Mécanique Numérique 15(1–3):81–92zbMATHGoogle Scholar
  5. Blatman G, Sudret B (2008) Sparse polynomial chaos expansions and adaptive stochastic finite elements using a regression approach. Comptes Rendus Mécanique 336(6):518–523zbMATHCrossRefGoogle Scholar
  6. Blatman G, Sudret B (2010a) Efficient computation of global sensitivity indices using sparse polynomial chaos expansions. Reliab Eng Syst Saf 95(11):1216–1229CrossRefGoogle Scholar
  7. Blatman G, Sudret B (2010b) An adaptive algorithm to build up sparse polynomial chaos expansions for stochastic finite element analysis. Probabilistic Engineering Mechanics 25(2):183–197CrossRefGoogle Scholar
  8. Blatman G, Sudret B (2011) Adaptive sparse polynomial chaos expansion based on least angle regression. J Comput Phys 230:2345–2367MathSciNetzbMATHCrossRefGoogle Scholar
  9. Bratley P, Fox BL (1988) ALGORITHM 659 implementing Sobol’s quasirandom sequence generator. ACM Trans Math Softw 14(1):88–100MathSciNetzbMATHCrossRefGoogle Scholar
  10. Chen T, Martin E (2009) Bayesian linear regression and variable selection for spectroscopic calibration. Anal Chim Acta 631(1):13–21CrossRefGoogle Scholar
  11. Cheng K, Lu Z (2018a) Sparse polynomial chaos expansion based on D-MORPH regression. Appl Math Comput 323:17–30Google Scholar
  12. Cheng K, Lu Z (2018b) Adaptive sparse polynomial chaos expansions for global sensitivity analysis based on support vector regression. Comput Struct 194:86–96CrossRefGoogle Scholar
  13. Cheng K, Lu Z, Zhou Y, Shi Y, Wei Y (2017) Global sensitivity analysis using support vector regression. Appl Math Model 49:587–598MathSciNetCrossRefGoogle Scholar
  14. Cherkassky V, Ma Y (2004) Practical selection of SVM parameters and noise estimation for SVM regression. Neural Netw 17(1):113–126zbMATHCrossRefGoogle Scholar
  15. Da Veiga S (2015) Global sensitivity analysis with dependence measures. J Stat Comput Simul 85(7):1283–1305MathSciNetCrossRefGoogle Scholar
  16. Daghir-Wojtkowiak E, Wiczling P, Bocian S, Kubik Ł, Kośliński P, Buszewski B et al (2015) Least absolute shrinkage and selection operator and dimensionality reduction techniques in quantitative structure retention relationship modeling of retention in hydrophilic interaction liquid chromatography. J Chromatogr A 1403(4):54–62CrossRefGoogle Scholar
  17. Efron B, Stein C (1981) The jacknife estimate of variance. Ann Stat 9(3):586–596zbMATHCrossRefGoogle Scholar
  18. Fang KT, Li FZ, Sudjianto A (2005) Design and modeling for computer experimentsGoogle Scholar
  19. Farkas O, Héberger K (2015) Comparison of ridge regression, partial least-squares, pairwise correlation, forward- and best subset selection methods for prediction of retention indices for aliphatic alcohols. ChemInformGoogle Scholar
  20. Friedman J, Tibshirani R, Hastie T (2010) Regularization paths for generalized linear models via coordinate descent. J Stat Softw. 33(1)Google Scholar
  21. Ghanem RG, Spanos PD (1991) Stochastic finite element method: response statistics. In: Stochastic finite elements: a spectral approach. Springer, N Y, pp 101–119zbMATHCrossRefGoogle Scholar
  22. Ghiocel DM, Ghanem RG (2002) Stochastic finite-element analysis of seismic soil–structure interaction. J Eng Mech 128(1):66–77CrossRefGoogle Scholar
  23. Hoerl AE, Kennard RW (1970) Ridge regression: biased estimation for nonorthogonal problems. Technometrics 12:55–70zbMATHCrossRefGoogle Scholar
  24. Kucherenko S (2010) A new derivative based importance criterion for groups of variables and its link with the global sensitivity indices. Comput Phys Commun 181(7):1212–1217MathSciNetzbMATHCrossRefGoogle Scholar
  25. Looss B, Lemaître P (2014) A review on global sensitivity analysis methods. Operations Res/Computer Sci Interfaces 59:101–122Google Scholar
  26. Lyons R (2013) Distance covariance in metric spaces. Ann Probab 41(5):3284–3305MathSciNetzbMATHCrossRefGoogle Scholar
  27. Molinaro AM, Simon R, Pfeiffer RM (2005) Prediction error estimation: a comparison of resampling methods. Bioinformatics 21(15):3301–3307CrossRefGoogle Scholar
  28. Morris MD (1991) Factorial sampling plans for preliminary computational experiments. Technometrics 33(2):161–174CrossRefGoogle Scholar
  29. Owen AB. A robust hybrid of lasso and ridge regression. Stanford University 2006Google Scholar
  30. Paruggia M. (2004) Sensitivity analysis in practice: a guide to assessing scientific models. J R Stat Soc Ser A (Statistics in Society).Google Scholar
  31. Pettersson MP, Iaccarino G, Nordstrom J (2015) Polynomial chaos methods for hyperbolic partial differential equations. Springer Math Eng 10:978–973zbMATHGoogle Scholar
  32. Polat E, Gunay S (2015) The comparison of partial least squares regression, principal component regression and ridge regression with multiple linear regression for predicting pm10 concentration level based on meteorological parameters. J Data Sci 13(4):663–692Google Scholar
  33. Raisee M, Kumar D, Lacor C (2015) A non-intrusive model reduction approach for polynomial chaos expansion using proper orthogonal decomposition. Int J Numer Methods Eng 103(4):293–312MathSciNetzbMATHCrossRefGoogle Scholar
  34. Rajabi MM, Ataie-Ashtiani B, Simmons CT (2015) Polynomial chaos expansions for uncertainty propagation and moment independent sensitivity analysis of seawater intrusion simulations. J Hydrol 520:101–122CrossRefGoogle Scholar
  35. Razavi S, Gupta HV (2015) What do we mean by sensitivity analysis? The need for comprehensive characterization of “global” sensitivity in earth and environmental systems models. Water Resour Res 51(5):3070–3092CrossRefGoogle Scholar
  36. Saltelli A, Chan K, Scott EM (2000) Sensitivity analysis. J. Wiley & SonsGoogle Scholar
  37. Shao Q, Younes A, Fahs M, Mara TA (2017) Bayesian sparse polynomial chaos expansion for global sensitivity analysis. Comput Methods Appl Mech Eng 318:474–496MathSciNetCrossRefGoogle Scholar
  38. Sobol IM (1993) Sensitivity estimates for nonlinear mathematical models. Mathematical Modelling Computational Experiments 1(4):407–414MathSciNetzbMATHGoogle Scholar
  39. Sobol IM (2001) Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates. Math Comput Simul 55(1–3):271–280MathSciNetzbMATHCrossRefGoogle Scholar
  40. Sobol IM (2003) Theorems and examples on high dimensional model representation. Reliab Eng Syst Saf 79(2):187–193MathSciNetCrossRefGoogle Scholar
  41. Soize C, Ghanem R (2004) Physical systems with random uncertainties: chaos representations with arbitrary probability measure. SIAM J Sci Comput 26(2):395–410MathSciNetzbMATHCrossRefGoogle Scholar
  42. Song S, Wang L (2017) Modified GMDH-NN algorithm and its application for global sensitivity analysis. J Comput Phys 348:534–548MathSciNetzbMATHCrossRefGoogle Scholar
  43. Stone M. (1974) Cross-validatory choice and assessment of statistical predictions. J R Stat Soc Ser B (Methodological) 111–47Google Scholar
  44. Sudret B (2008) Global sensitivity analysis using polynomial chaos expansions. Reliab Eng Syst Saf 93(7):964–979CrossRefGoogle Scholar
  45. Szekely GJ, Rizzo ML (2014) Partial distance correlation with methods for dissimilarities. Ann Stat 42(6):2382–2412MathSciNetzbMATHCrossRefGoogle Scholar
  46. Székely GJ, Rizzo ML, Bakirov NK. Measuring and testing dependence by correlation of distances. Ann Stat 2007; 2769–94Google Scholar
  47. Szepietowska K et al (2017) Sensitivity analysis based on non-intrusive regression-based polynomial chaos expansion for surgical mesh modelling. Struct Multidiscip Optim 2:1–19MathSciNetGoogle Scholar
  48. Tibshirani R (1996) Regression shrinkage and selection via the lasso. J R Stat Soc Ser B 58(1):267–288MathSciNetzbMATHGoogle Scholar
  49. Wang HW, Wu ZB, Meng J (2006) Partial least-squares regression-linear and nonlinear methods. National Defense Industry Press, BeijingGoogle Scholar
  50. Wang P, Lu ZZ, Xiao SN (2017) A generalized separation for the variance contributions of input variables and their distribution parameters. Appl Math Model 47:381–399MathSciNetCrossRefGoogle Scholar
  51. Wold H. (1985) Partial least squares. Encyclopedia Statistical SciGoogle Scholar
  52. Xiao S, Lu Z, Wang P (2018a) Multivariate global sensitivity analysis for dynamic models based on wavelet analysis. Reliab Eng Syst Saf 170Google Scholar
  53. Xiao S, Lu Z, Wang P (2018b) Multivariate global sensitivity analysis for dynamic models based on energy distance. Struct Multidiscip Optim 57:279–291MathSciNetCrossRefGoogle Scholar
  54. Zhao W, Wang W (2013) Application of partial least squares regression in response surface for analysis of structural reliability. Engineering Mechanics 30(2):272–277Google Scholar

Copyright information

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

Authors and Affiliations

  1. 1.School of AeronauticNorthwestern Polytechnical UniversityXi’anPeople’s Republic of China

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