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

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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.

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References

  • Abdi H (2010) Partial least squares regression and projection on latent structure regression (PLS regression). Wiley Interdisciplinary Rev: Computational Statistics 2(1):97–106

    Article  Google Scholar 

  • 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 332:461–474

    Article  MathSciNet  MATH  Google Scholar 

  • Alwart H. Singular value decomposition (svd) and generalized singular value decomposition (gsvd). Encyclopedia Measurement Statistics 2006; 907–912

  • 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):81–92

    MATH  Google Scholar 

  • Blatman G, Sudret B (2008) Sparse polynomial chaos expansions and adaptive stochastic finite elements using a regression approach. Comptes Rendus Mécanique 336(6):518–523

    Article  MATH  Google Scholar 

  • Blatman G, Sudret B (2010a) Efficient computation of global sensitivity indices using sparse polynomial chaos expansions. Reliab Eng Syst Saf 95(11):1216–1229

    Article  Google Scholar 

  • 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):183–197

    Article  Google Scholar 

  • Blatman G, Sudret B (2011) Adaptive sparse polynomial chaos expansion based on least angle regression. J Comput Phys 230:2345–2367

    Article  MathSciNet  MATH  Google Scholar 

  • Bratley P, Fox BL (1988) ALGORITHM 659 implementing Sobol’s quasirandom sequence generator. ACM Trans Math Softw 14(1):88–100

    Article  MathSciNet  MATH  Google Scholar 

  • Chen T, Martin E (2009) Bayesian linear regression and variable selection for spectroscopic calibration. Anal Chim Acta 631(1):13–21

    Article  Google Scholar 

  • Cheng K, Lu Z (2018a) Sparse polynomial chaos expansion based on D-MORPH regression. Appl Math Comput 323:17–30

    Google Scholar 

  • Cheng K, Lu Z (2018b) Adaptive sparse polynomial chaos expansions for global sensitivity analysis based on support vector regression. Comput Struct 194:86–96

    Article  Google Scholar 

  • Cheng K, Lu Z, Zhou Y, Shi Y, Wei Y (2017) Global sensitivity analysis using support vector regression. Appl Math Model 49:587–598

    Article  MathSciNet  Google Scholar 

  • Cherkassky V, Ma Y (2004) Practical selection of SVM parameters and noise estimation for SVM regression. Neural Netw 17(1):113–126

    Article  MATH  Google Scholar 

  • Da Veiga S (2015) Global sensitivity analysis with dependence measures. J Stat Comput Simul 85(7):1283–1305

    Article  MathSciNet  Google Scholar 

  • 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):54–62

    Article  Google Scholar 

  • Efron B, Stein C (1981) The jacknife estimate of variance. Ann Stat 9(3):586–596

    Article  MATH  Google Scholar 

  • Fang KT, Li FZ, Sudjianto A (2005) Design and modeling for computer experiments

  • 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. ChemInform

  • Friedman J, Tibshirani R, Hastie T (2010) Regularization paths for generalized linear models via coordinate descent. J Stat Softw. 33(1)

  • Ghanem RG, Spanos PD (1991) Stochastic finite element method: response statistics. In: Stochastic finite elements: a spectral approach. Springer, N Y, pp 101–119

    Chapter  MATH  Google Scholar 

  • Ghiocel DM, Ghanem RG (2002) Stochastic finite-element analysis of seismic soil–structure interaction. J Eng Mech 128(1):66–77

    Article  Google Scholar 

  • Hoerl AE, Kennard RW (1970) Ridge regression: biased estimation for nonorthogonal problems. Technometrics 12:55–70

    Article  MATH  Google Scholar 

  • 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):1212–1217

    Article  MathSciNet  MATH  Google Scholar 

  • Looss B, Lemaître P (2014) A review on global sensitivity analysis methods. Operations Res/Computer Sci Interfaces 59:101–122

    Google Scholar 

  • Lyons R (2013) Distance covariance in metric spaces. Ann Probab 41(5):3284–3305

    Article  MathSciNet  MATH  Google Scholar 

  • Molinaro AM, Simon R, Pfeiffer RM (2005) Prediction error estimation: a comparison of resampling methods. Bioinformatics 21(15):3301–3307

    Article  Google Scholar 

  • Morris MD (1991) Factorial sampling plans for preliminary computational experiments. Technometrics 33(2):161–174

    Article  Google Scholar 

  • Owen AB. A robust hybrid of lasso and ridge regression. Stanford University 2006

  • Paruggia M. (2004) Sensitivity analysis in practice: a guide to assessing scientific models. J R Stat Soc Ser A (Statistics in Society).

  • Pettersson MP, Iaccarino G, Nordstrom J (2015) Polynomial chaos methods for hyperbolic partial differential equations. Springer Math Eng 10:978–973

    MATH  Google Scholar 

  • 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):663–692

    Google Scholar 

  • 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):293–312

    Article  MathSciNet  MATH  Google Scholar 

  • 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 520:101–122

    Article  Google Scholar 

  • 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):3070–3092

    Article  Google Scholar 

  • Saltelli A, Chan K, Scott EM (2000) Sensitivity analysis. J. Wiley & Sons

  • Shao Q, Younes A, Fahs M, Mara TA (2017) Bayesian sparse polynomial chaos expansion for global sensitivity analysis. Comput Methods Appl Mech Eng 318:474–496

    Article  MathSciNet  Google Scholar 

  • Sobol IM (1993) Sensitivity estimates for nonlinear mathematical models. Mathematical Modelling Computational Experiments 1(4):407–414

    MathSciNet  MATH  Google Scholar 

  • Sobol IM (2001) Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates. Math Comput Simul 55(1–3):271–280

    Article  MathSciNet  MATH  Google Scholar 

  • Sobol IM (2003) Theorems and examples on high dimensional model representation. Reliab Eng Syst Saf 79(2):187–193

    Article  MathSciNet  Google Scholar 

  • Soize C, Ghanem R (2004) Physical systems with random uncertainties: chaos representations with arbitrary probability measure. SIAM J Sci Comput 26(2):395–410

    Article  MathSciNet  MATH  Google Scholar 

  • Song S, Wang L (2017) Modified GMDH-NN algorithm and its application for global sensitivity analysis. J Comput Phys 348:534–548

    Article  MathSciNet  MATH  Google Scholar 

  • Stone M. (1974) Cross-validatory choice and assessment of statistical predictions. J R Stat Soc Ser B (Methodological) 111–47

  • Sudret B (2008) Global sensitivity analysis using polynomial chaos expansions. Reliab Eng Syst Saf 93(7):964–979

    Article  Google Scholar 

  • Szekely GJ, Rizzo ML (2014) Partial distance correlation with methods for dissimilarities. Ann Stat 42(6):2382–2412

    Article  MathSciNet  MATH  Google Scholar 

  • Székely GJ, Rizzo ML, Bakirov NK. Measuring and testing dependence by correlation of distances. Ann Stat 2007; 2769–94

  • Szepietowska K et al (2017) Sensitivity analysis based on non-intrusive regression-based polynomial chaos expansion for surgical mesh modelling. Struct Multidiscip Optim 2:1–19

    MathSciNet  Google Scholar 

  • Tibshirani R (1996) Regression shrinkage and selection via the lasso. J R Stat Soc Ser B 58(1):267–288

    MathSciNet  MATH  Google Scholar 

  • Wang HW, Wu ZB, Meng J (2006) Partial least-squares regression-linear and nonlinear methods. National Defense Industry Press, Beijing

    Google Scholar 

  • Wang P, Lu ZZ, Xiao SN (2017) A generalized separation for the variance contributions of input variables and their distribution parameters. Appl Math Model 47:381–399

    Article  MathSciNet  Google Scholar 

  • Wold H. (1985) Partial least squares. Encyclopedia Statistical Sci

  • Xiao S, Lu Z, Wang P (2018a) Multivariate global sensitivity analysis for dynamic models based on wavelet analysis. Reliab Eng Syst Saf 170

  • Xiao S, Lu Z, Wang P (2018b) Multivariate global sensitivity analysis for dynamic models based on energy distance. Struct Multidiscip Optim 57:279–291

    Article  MathSciNet  Google Scholar 

  • Zhao W, Wang W (2013) Application of partial least squares regression in response surface for analysis of structural reliability. Engineering Mechanics 30(2):272–277

    Google Scholar 

Download references

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Authors and Affiliations

  1. School of Aeronautic, Northwestern Polytechnical University, Xi’an, Shaanxi, 710072, People’s Republic of China

    Yicheng Zhou, Zhenzhou Lu & Kai Cheng

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  1. Yicheng Zhou
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  2. Zhenzhou Lu
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  3. Kai Cheng
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Correspondence to Zhenzhou Lu.

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Zhou, Y., Lu, Z. & Cheng, K. Sparse polynomial chaos expansions for global sensitivity analysis with partial least squares and distance correlation. Struct Multidisc Optim 59, 229–247 (2019). https://doi.org/10.1007/s00158-018-2062-8

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  • DOI: https://doi.org/10.1007/s00158-018-2062-8

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