Advertisement

Multivariate output global sensitivity analysis using multi-output support vector regression

  • Kai Cheng
  • Zhenzhou LuEmail author
  • Kaichao Zhang
Research Paper
  • 44 Downloads

Abstract

Models with multivariate outputs are widely used for risk assessment and decision-making in practical applications. In this paper, multi-output support vector regression (M-SVR) is employed for global sensitivity analysis (GSA) with multivariate output models. The orthogonal polynomial kernel is used to build the M-SVR meta-model, and the covariance-based sensitivity indices of multivariate output are obtained analytically by post-processing the coefficients of M-SVR model. In order to improve the performance of the orthogonal polynomial kernel M-SVR model, a kernel function iteration algorithm is introduced further. The proposed method take advantage of the information of all outputs to get robust meta-model. To validate the performance of the proposed method, two high-dimensional analytical functions and a hydrological model (HYMOD) with multiple outputs are examined, and a detailed comparison is made with the sparse polynomial chaos expansion meta-model developed in UQLab Toolbox. Results show that the proposed methods are efficient and accurate for GSA of the complex multivariate output models.

Keywords

Multivariate output sensitivity analysis Multi-output support vector regression Orthogonal polynomial kernel function Polynomial chaos expansion 

Notes

Funding information

This work was supported by the National Natural Science Foundation of China (Grant No. NSFC 51475370 and NSFC 51775439).

References

  1. Abraham S et al (2017) A robust and efficient stepwise regression method for building sparse polynomial chaos expansions. J Comput Phys 332:461–474MathSciNetCrossRefGoogle Scholar
  2. Blatman G, Sudret B (2011) Adaptive sparse polynomial chaos expansion based on least angle regression. J Comput Phys 230:2345–2367MathSciNetCrossRefGoogle Scholar
  3. Borgonovo E (2007) A new uncertainty importance measure. Reliab Eng Syst Saf 92(6):771–784CrossRefGoogle Scholar
  4. Bourinet J-M (2016) Rare-event probability estimation with adaptive support vector regression surrogates. Reliab Eng Syst Saf 150:210–221CrossRefGoogle Scholar
  5. Cheng K, Lu ZZ et al (2017a) Global sensitivity analysis using support vector regression. Appl Math Model 49:587–598MathSciNetCrossRefGoogle Scholar
  6. Cheng K, Lu ZZ et al (2017b) Mixed kernel function support vector regression for global sensitivity analysis. Mech Syst Signal Process 96:201–214CrossRefGoogle Scholar
  7. Chowdhury R, Rao BN, Prasad AM (2010) Stochastic sensitivity analysis using HDMR and score function. Sadhana 34:967–986CrossRefGoogle Scholar
  8. Gamboa F et al (2013) Sensitivity indices for multivariate outputs. C R Math 351:307–310MathSciNetCrossRefGoogle Scholar
  9. Garcia-Cabrejo O, Valocchi A (2014) Global sensitivity analysis for multivariate output using polynomial chaos expansion. Reliab Eng Syst Saf 126:25–36CrossRefGoogle Scholar
  10. Gratiet LL, Marelli S, Sudret B, Metamodel-based sensitivity analysis: polynomial chaos expansions and Gaussian processes. In Handbook on Uncertainty Quantification, Ghanem, R., Higdon, D. &Owhadi, H. (Eds)Google Scholar
  11. Hao WR, Lu ZZ, Li LY (2013) A new interpretation and validation of variance based importance measure analysis for model with correlated inputs. Comput Phys Commun 184:1401–1413MathSciNetCrossRefGoogle Scholar
  12. Iman RL, Johnson ME, Schroeder TA (2002) Assessing hurricane effects. Part 1. Sensitivity analysis. Reliab Eng Syst Saf 78:131–145CrossRefGoogle Scholar
  13. Kollat JB, Reed PM, Wagener T (2012) When are multiobjective calibration trade-offs in hydrologic models meaningful? Water Resour Res 48(3):3520CrossRefGoogle Scholar
  14. Konakli K, Sudret B (2010) Efficient computation of global sensitivity indices using sparse polynomial chaos expansions. Reliab Eng Syst Saf 95:1216–1229CrossRefGoogle Scholar
  15. Lambert RS, Lemke F et al (2016) Global sensitivity analysis using sparse high dimensional model representations generated by the group method of data handing. Math Comput Simul 128:42–54CrossRefGoogle Scholar
  16. Li LY et al (2012) Moment-independent importance measure of basic variable and its state dependent parameter solution. Struct Saf 38:40–47CrossRefGoogle Scholar
  17. Lozzo MD, Marrel A (2017) Sensitivity analysis with dependence and variance-based measures for spatio-temporal numerical simulators. Stoch Env Res Risk A 31:1–17CrossRefGoogle Scholar
  18. Mao WT et al (2014a) A fast and robust model selection algorithm for multi-input multi-output support vector machine. Neurocomputing 130:10–19CrossRefGoogle Scholar
  19. Mao WT et al (2014b) Leave-one-out cross-validation-based model selection for multi-input multi-output support vector machine. Neural Comput & Applic 24:441–451CrossRefGoogle Scholar
  20. Marelli S, Sudret B (2014) UQlab: a framework for uncertainty quantification in MATLAB. In: Proceedings of the 2nd international conference on vulnerability, risk analysis and management (ICVRA2014). Liverpool, United KingdomGoogle Scholar
  21. Marrel A, Perot N, Mottet C (2015) Development of a surrogate model and sensitivity analysis for spatio-temporal numerical simulators. Stoch Env Res Risk A 29:959–974CrossRefGoogle Scholar
  22. Morris MD (1991) Factorial sampling plans for preliminary computational experiments. Technometrics 33(2):161–174CrossRefGoogle Scholar
  23. Pianosi F, Sarrazin F, Wagener T (2015) A Matlab toolbox for global sensitivity analysis. Environ Model Softw 70:80–85CrossRefGoogle Scholar
  24. Rahman S (2011) Global sensitivity analysis by polynomial dimensional decomposition. Reliab Eng Syst Saf 96:825–837CrossRefGoogle Scholar
  25. Saltelli A (2002) Making best use of model evaluations to compute sensitivity indices. Comput Phys Commun 145:280–297CrossRefGoogle Scholar
  26. Sánchez-fernández M et al (2004) SVM multiregression for nonlinear channel estimation in multi-input multi-output systems. IEEE Trans Signal Process 52(8):2298–2307MathSciNetCrossRefGoogle Scholar
  27. Sobol IM (1993) Sensitivity estimates for nonlinear mathematical models. Math Model Comput Exp 1(4):407–414MathSciNetzbMATHGoogle Scholar
  28. Sobol IM, Kucherenko S (2009) Derivative based global sensitivity measures and their link with global sensitivity indices. Math Comput Simul 79(10):3009–3017MathSciNetCrossRefGoogle Scholar
  29. Sobol IM, 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):1212–1217MathSciNetCrossRefGoogle Scholar
  30. Sorooshian S et al (1983) Evaluation if maximum likelihood parameter estimation techniques for conceptual rainfall-runoff models: influence of calibration data variability and length on model credibility. Water Resour Res 19(1):251–259CrossRefGoogle Scholar
  31. Sudret B (2008) Global sensitivity analysis using polynomial chaos expansion. Reliab Eng Syst Saf 93:964–979CrossRefGoogle Scholar
  32. Wagener T et al (2001) A framework for development and application of hydrological models. Hydrol Earth Syst Sci Discuss 5(1):13–26CrossRefGoogle Scholar
  33. Wang ZQ, Wang DH et al (2016) Global sensitivity analysis using a Gaussian radial basis function metamodel. Reliab Eng Syst Saf 154:171–179Google Scholar
  34. Xiao SN, Lu ZZ, Xu LY (2017) Multivariate sensitivity analysis based on the direction of eigenspace through principal component analysis. Reliab Eng Syst Saf 165:1–10CrossRefGoogle Scholar
  35. Xiao SN, Lu ZZ, Wang P (2018) Multivariate global sensitivity analysis for dynamic models based on energy distance. Struct Multidiscip Optim 57:279–291MathSciNetCrossRefGoogle Scholar
  36. Zeng X, Dong W (2012) Sensitivity analysis of the probability distribution of groundwater level series based on information entropy. Stoch Env Res Risk A 26:345–356CrossRefGoogle Scholar
  37. Zhang KC, Lu ZZ, Cheng L, Xu F (2015) A new framework of variance based global sensitivity analysis for models with correlated inputs. Struct Saf 55:1–9CrossRefGoogle Scholar
  38. Zhong Z, Carr TR (2016) Application of mixed kernels function (MKF) based support vector regression model (SVR) for CO2-reservoir oil minimum miscibility pressure prediction. Fuel 184:590–603CrossRefGoogle Scholar
  39. Ziehn T, Tomlin AS (2009) GUI-HDMR-A software tool for global sensitivity analysis of complex models. Environ Model Softw 24:775–785CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

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

Personalised recommendations