Global sensitivity analysis: a Bayesian learning based polynomial chaos approach. (English) Zbl 1440.62093

Summary: A novel sparse polynomial chaos expansion (PCE) is proposed in this paper for global sensitivity analysis (GSA). The proposed model combines variational Bayesian (VB) inference and automatic relevance determination (ARD) with the PCE model. The VB inference is utilized to compute the PCE coefficients. The PCE coefficients are obtained through a simple optimization procedure in the VB framework. On the other hand, the curse of dimensionality issue of PCE model is tackled using the ARD which reduces the number of polynomial bases significantly. The applicability of the proposed approach is illustrated by performing GSA on five numerical examples. The results show that the proposed approach outperforms a similar state-of-art surrogate model in obtaining an accurate sensitivity index using limited number of model evaluations. For all the examples, the PCE models are highly sparse, which require very few polynomial bases to assess an accurate sensitivity index.


62F15 Bayesian inference
86A05 Hydrology, hydrography, oceanography
Full Text: DOI HAL


[1] Borgonovo, E.; Plischke, E., Sensitivity analysis: a review of recent advances, Eur. J. Oper. Res., 248, 3, 869-887 (2016) · Zbl 1346.90771
[2] Sobol, I. M., Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates, Math. Comput. Simul., 55, 1-3, 271-280 (2001) · Zbl 1005.65004
[3] Helton, J. C., Uncertainty and sensitivity analysis techniques for use in performance assessment for radioactive waste disposal, Reliab. Eng. Syst. Saf., 42, 2-3, 327-367 (1993)
[4] Iman, R. L.; Hora, S. C., A robust measure of uncertainty importance for use in fault tree system analysis, Risk Anal., 10, 3, 401-406 (1990)
[5] Sobol, I. M., Sensitivity analysis for nonlinear mathematical models, Math. Model. Comput. Exp., 1, 4, 407-414 (1993) · Zbl 1039.65505
[6] Borgonovo, E., A new uncertainty importance measure, Reliab. Eng. Syst. Saf., 92, 6, 771-784 (2007)
[7] Chakraborty, S.; Chowdhury, R., A hybrid approach for global sensitivity analysis, Reliab. Eng. Syst. Saf., 158, 50-57 (2017)
[8] Dell’Oca, A.; Riva, M.; Guadagnini, A., Moment-based metrics for global sensitivity analysis of hydrological systems, Hydrol. Earth Syst. Sci., 21, 12, 6219-6234 (2017)
[9] Pearson, K., On the General Theory of Skew Correlation and Non-linear Regression (1905), Dulau and Co.
[10] Efron, B.; Stein, C., The jackknife estimate of variance, Ann. Stat., 9, 3, 586-596 (1981) · Zbl 0481.62035
[11] 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) · Zbl 1219.93116
[12] Sobol, I. M., Quasi-Monte Carlo methods, Prog. Nucl. Energy, 24, 1-3, 55-61 (1990)
[13] Sudret, B., Global sensitivity analysis using polynomial chaos expansions, Reliab. Eng. Syst. Saf., 93, 7, 964-979 (2008)
[14] Blatman, G.; Sudret, B., Efficient computation of global sensitivity indices using sparse polynomial chaos expansions, Reliab. Eng. Syst. Saf., 95, 11, 1216-1229 (2010)
[15] Shao, Q.; Younes, A.; Fahs, M.; Mara, T. A., Bayesian sparse polynomial chaos expansion for global sensitivity analysis, Comput. Methods Appl. Mech. Eng., 318, 474-496 (2017) · Zbl 1439.62088
[16] Park, J.; Sandberg, I. W., Universal approximation using radial-basis-function networks, Neural Comput., 3, 2, 246-257 (1991)
[17] Wu, Z.; Wang, W.; Wang, D.; Zhao, K.; Zhang, W., Global sensitivity analysis using orthogonal augmented radial basis function, Reliab. Eng. Syst. Saf., 185, 291-302 (2019)
[18] Ge, Q.; Ciuffo, B.; Menendez, M., Combining screening and metamodel-based methods: an efficient sequential approach for the sensitivity analysis of model outputs, Reliab. Eng. Syst. Saf., 134, 334-344 (2015)
[19] Zuniga, M. M.; Kucherenko, S.; Shah, N., Metamodelling with independent and dependent inputs, Comput. Phys. Commun., 184, 6, 1570-1580 (2013) · Zbl 1303.65002
[20] Tang, K.; Congedo, P. M.; Abgrall, R., Adaptive surrogate modeling by ANOVA and sparse polynomial dimensional decomposition for global sensitivity analysis in fluid simulation, J. Comput. Phys., 314, 557-589 (2016) · Zbl 1349.62329
[21] Abraham, S.; Raisee, M.; Ghorbaniasl, G.; Contino, F.; Lacor, C., A robust and efficient stepwise regression method for building sparse polynomial chaos expansions, J. Comput. Phys., 332, 461-474 (2017) · Zbl 1384.62216
[22] Blatman, G.; Sudret, B., Sparse polynomial chaos expansions and adaptive stochastic finite elements using a regression approach, C. R., Méc., 336, 6, 518-523 (2008) · Zbl 1138.74046
[23] Blatman, G.; Sudret, B., Adaptive sparse polynomial chaos expansion based on least angle regression, J. Comput. Phys., 230, 6, 2345-2367 (2011) · Zbl 1210.65019
[24] Cheng, K.; Lu, Z., Adaptive sparse polynomial chaos expansions for global sensitivity analysis based on support vector regression, Comput. Struct., 194, 86-96 (2018)
[25] Peng, J.; Hampton, J.; Doostan, A., A weighted ℓ1-minimization approach for sparse polynomial chaos expansions, 267, 92-111 (2014) · Zbl 1349.65198
[26] Peng, J.; Hampton, J.; Doostan, A., On polynomial chaos expansion via gradient-enhanced ℓ1-minimization, J. Comput. Phys., 310, 440-458 (2016) · Zbl 1349.65538
[27] Zhou, Y.; Lu, Z.; Cheng, K.; Shi, Y., An expanded sparse Bayesian learning method for polynomial chaos expansion, Mech. Syst. Signal Process., 128, 153-171 (2019)
[28] Zhou, Y.; Lu, Z.; Cheng, K., Sparse polynomial chaos expansions for global sensitivity analysis with partial least squares and distance correlation, Struct. Multidiscip. Optim., 59, 1, 229-247 (2019)
[29] Attias, H., A variational Bayesian framework for graphical models, (12th International Conference on Neural Information Processing Systems (1999), MIT Press), 209-215
[30] Bishop, C. M., Pattern Recognition and Machine Learning (2006), Springer · Zbl 1107.68072
[31] Burden, F. R.; Ford, M. G.; Whitley, D. C.; Winkler, D. A., Use of automatic relevance determination in QSAR studies using Bayesian neural networks, J. Chem. Inf. Model., 40, 6, 1423-1430 (2000)
[32] Ghahramani, Z.; Beal, M. J., Propagation algorithms for variational Bayesian learning, (Advances in Neural Information Processing Systems (2001)), 507-513
[33] Homma, T.; Saltelli, A., Importance measures in global sensitivity analysis of nonlinear models, Reliab. Eng. Syst. Saf., 52, 1, 1-17 (1996)
[34] Jansen, M. J., Analysis of variance designs for model output, Comput. Phys. Commun., 117, 1-2, 35-43 (1999) · Zbl 1015.68218
[35] Sobol, I. M., Global sensitivity indices for the investigation of nonlinear mathematical models, Mat. Model., 19, 11, 23-24 (2007) · Zbl 1140.60323
[36] Xiu, D.; Karniadakis, G. E., The Wiener-Askey polynomial chaos for stochastic differential equation, SIAM J. Sci. Comput., 24, 2, 619-644 (2002) · Zbl 1014.65004
[37] Koekoek, R.; Swarttouw, R. F., The Askey-Scheme of Hypergeometric Orthogonal Polynomials and Its q-Analogue (1996)
[38] Bhattacharyya, B., A critical appraisal of design of experiments for uncertainty quantification, Arch. Comput. Methods Eng., 25, 3, 727-751 (2018) · Zbl 1397.62271
[39] Lei, H.; Li, J.; Gao, P.; Stinis, P.; Baker, N. A., A data-driven framework for sparsity-enhanced surrogates with arbitrary mutually dependent randomness, Comput. Methods Appl. Mech. Eng., 350, 199-227 (2019) · Zbl 1441.60004
[40] Rahman, S., Wiener-Hermite polynomial expansion for multivariate Gaussian probability measures, J. Math. Anal. Appl., 454, 1, 303-334 (2017) · Zbl 1373.60071
[41] Zhao, W.; Bu, L., Global sensitivity analysis with a hierarchical sparse metamodeling method, Mech. Syst. Signal Process., 115, 769-781 (2019)
[42] Bishop, C. M.; Tipping, M., Variational relevance vector machines, (Sixteenth Conference on Uncertainty in Artificial Intelligence. Sixteenth Conference on Uncertainty in Artificial Intelligence, UAI2000 (2000)), 46-53
[43] Tipping, M. E., Sparse Bayesian learning and the relevance vector machine, J. Mach. Learn. Res., 1, 211-244 (2001) · Zbl 0997.68109
[44] Griffin, J. E.; Brown, P. J., Inference with normal-gamma prior distributions in regression problems, Bayesian Anal., 5, 1, 171-188 (2010) · Zbl 1330.62128
[45] Gilks, W. R.; Richardson, S.; Spiegelhalter, D. J., Markov Chain Monte Carlo in Practice (1996), Chapman & Hall · Zbl 0832.00018
[46] Sun, S., A review of deterministic approximate inference techniques for Bayesian machine learning, Neural Comput. Appl., 23, 7-8, 2039-2050 (2013)
[47] Franck, I. M.; Koutsourelakis, P., Sparse variational Bayesian approximations for nonlinear inverse problems: applications in nonlinear elastography, Comput. Methods Appl. Mech. Eng., 299, 215-244 (2016) · Zbl 1425.65132
[48] Parisi, G., Statistical Field Theory (1988), Addison-Wesley
[49] Peierls, R., On a minimum property of the free energy, Phys. Rev., 54, 11, 918-919 (1938) · Zbl 0020.08403
[50] Beal, M. J., Variational algorithms for approximate Bayesian inference (2003), University College London, Ph.D. thesis
[51] Tan, V. Y.F.; Fevotte, C., Automatic relevance determination in nonnegative matrix factorization with the /spl beta/-divergence, IEEE Trans. Pattern Anal. Mach. Intell., 35, 7, 1592-1605 (2013)
[52] Wipf, D.; Nagarajan, S., A new view of automatic relevance determination, (Advances in Neural Information Processing Systems (2008)), 1625-1632
[53] Jacobs, W. R.; Baldacchino, T.; Dodd, T. J.; Anderson, S. R., Sparse Bayesian nonlinear system identification using variational inference, IEEE Trans. Autom. Control, 63, 12, 4172-4187 (2018) · Zbl 1423.93378
[54] Li, G.; Rabitz, H.; Yelvington, P. E.; Oluwole, O. O.; Bacon, F.; Kolb, C. E.; Schoendorf, J., Global sensitivity analysis for systems with independent and/or correlated inputs, J. Phys. Chem. A, 114, 19, 6022-6032 (2010)
[55] Sudret, B.; Caniou, Y., Analysis of covariance (ANCOVA) using polynomial chaos expansions, (11th International Conference on Structural Safety and Reliability. 11th International Conference on Structural Safety and Reliability, ICOSSAR 2013, New York, USA (2013)), 3275-3281
[56] Marelli, S.; Sudret, B., UQLab: a framework for uncertainty quantification in Matlab, (Second International Conference on Vulnerability and Risk Analysis and Management. Second International Conference on Vulnerability and Risk Analysis and Management, ICVRAM (2014), American Society of Civil Engineers: American Society of Civil Engineers Liverpool, UK), 2554-2563
[57] Galanti, S.; Jung, A. R., Low-discrepancy sequences: Monte Carlo simulation of option prices, J. Deriv., 5, 63-83 (1997)
[58] Ishigami, T.; Homma, T., An importance quantification technique in uncertainty analysis for computer models, (First International Symposium on Uncertainty Modeling and Analysis (1990)), 398-403
[59] Sobol’, I., Theorems and examples on high dimensional model representation, Reliab. Eng. Syst. Saf., 79, 2, 187-193 (2003)
[60] Kucherenko, S.; Song, S.; Wang, L., Quantile based global sensitivity measures, Reliab. Eng. Syst. Saf., 185, 35-48 (2019)
[61] Wagener, T.; Boyle, D. P.; Lees, M. J.; Wheater, H. S.; Gupta, H. V.; Sorooshian, S., A framework for development and application of hydrological models, Hydrol. Earth Syst. Sci., 5, 1, 13-26 (2001)
[62] Sorooshian, S.; Gupta, V. K.; Fulton, J. L., Evaluation of 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, 251-259 (1983)
[63] Pianosi, F.; Sarrazin, F.; Wagener, T., A Matlab toolbox for global sensitivity analysis, Environ. Model. Softw., 70, 80-85 (2015)
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