Efficient Evaluation of Sobol’ Indices Utilizing Samples from an Auxiliary Probability Density Function
Publication: Journal of Engineering Mechanics
Volume 142, Issue 5
Abstract
A sample-based evaluation of Sobol’ sensitivity indices is discussed in this paper relying on Kernel Density Estimation (KDE) for achieving computational efficiency. The foundation of the approach is the definition of an auxiliary probability density function (PDF) for the vector of model parameters (i.e., random variables representing system input). The sensitivity index for each input can be then expressed through the marginal density related to this auxiliary joint PDF. The efficient estimation of the indices for all model parameters is ultimately facilitated by simulating first a single sample-set from the joint PDF and then utilizing this information to approximate all marginal distributions of interest through KDE. The same sample set is used for all approximations whereas this set is further exploited to improve the accuracy of the estimation of the integrals defining the sensitivity indices. An extension to facilitate calculation of higher order indices and total sensitivity indices is also examined. Once the sample set is generated, the computational burden of the approach is small and practically independent of the dimension of the model parameters. This makes the proposed approach particularly attractive for applications for which such samples are readily available (or can be efficiently obtained), as it can provide sensitivity information with small overall computational cost.
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References
Archer, G., Saltelli, A., and Sobol, I. (1997). “Sensitivity measures, ANOVA-like techniques and the use of bootstrap.” J. Stat. Comput. Simul., 58(2), 99–120.
Arwade, S. R., Moradi, M., and Louhghalam, A. (2010). “Variance decomposition and global sensitivity for structural systems.” Eng. Struct., 32(1), 1–10.
Au, S. K., and Beck, J. L. (2003). “Subset simulation and its applications to seismic risk based on dynamic analysis.” J. Eng. Mech., 901–917.
Boore, D. M. (2003). “Simulation of ground motion using the stochastic method.” Pure Appl. Geophys., 160(3), 635–676.
Chen, W., Jin, R. C., and Sudjianto, A. (2005). “Analytical variance-based global sensitivity analysis in simulation-based design under uncertainty.” J. Mech. Des., 127(5), 875–886.
Cukier, R. I., Levine, H. B., and Shuler, K. E. (1978). “Nonlinear sensitivity analysis of multiparameter model systems.” J. Comput. Phys., 26(1), 1–42.
Doucet, A., Godsill, S., and Andrieu, C. (2000). “On sequential Monte Carlo sampling methods for Bayesian filtering.” Stat. Comput., 10(3), 197–208.
Gatelli, D., Kucherenko, S., Ratto, M., and Tarantola, S. (2009). “Calculating first-order sensitivity measures: A benchmark of some recent methodologies.” Reliab. Eng. Syst. Saf., 94(7), 1212–1219.
Hall, F. F., Heaton, T. H., Halling, M. W., and Wald, D. J. (1995). “Near-source ground motion and its effects on flexible buildings.” Earthquake Spectra, 11(4), 569–605.
Halldórsson, B., Mavroeidis, G., and Papageorgiou, A. (2011). “Near-fault and far-field strong ground-motion simulation for earthquake engineering applications using the specific barrier model.” J. Struct. Eng., 433–444.
Homma, T., and Saltelli, A. (1996). “Importance measures in global sensitivity analysis of nonlinear models.” Reliab. Eng. Syst. Saf., 52(1), 1–17.
Ishigami, T., and Homma, T. (1990). “An importance quantification technique in uncertainty analysis for computer models.” Proc., 1st Int. Symp. on Uncertainty Modeling and Analysis, IEEE, New York, 398–403.
Jia, G., Gidaris, I., Taflanidis, A. A., and Mavroeidis, G. P. (2014). “Reliability-based assessment/design of floor isolation systems.” Eng. Struct., 78, 41–56.
Jia, G., and Taflanidis, A. A. (2013). “Non-parametric stochastic subset optimization for optimal-reliability design problems.” Comput. Struct., 126, 86–99.
Jia, G., and Taflanidis, A. A. (2014). “Sample-based evaluation of global probabilistic sensitivity measures.” Comput. Struct., 144, 103–118.
Jia, G., Taflanidis, A. A., and Beck, J. L. (2015). “A new adaptive rejection sampling method using kernel density approximations and its application to Subset Simulation.” J. Risk Uncertainty Eng. Syst. Part A: Civ. Eng., D4015001.
Karunamuni, R. J., and Zhang, S. (2008). “Some improvements on a boundary corrected kernel density estimator.” Stat. Probab. Lett., 78(5), 499–507.
Mavroeidis, G. P., and Papageorgiou, A. S. (2003). “A mathematical representation of near-fault ground motions.” Bull. Seismol. Soc. Am., 93(3), 1099–1131.
Medina, J. C., and Taflanidis, A. (2014). “Adaptive importance sampling for optimization under uncertainty problems.” Comput. Methods Appl. Mech. Eng., 279, 133–162.
Michailidis, P. D., and Margaritis, K. G. (2013). “Accelerating kernel density estimation on the GPU using the CUDA framework.” Appl. Math. Sci., 7(30), 1447–1476.
Oakley, J. E., and O’Hagan, A. (2004). “Probabilistic sensitivity analysis of complex models: A Bayesian approach.” J. R. Stat. Soc. Series B (Stat. Methodol.), 66(3), 751–769.
Ratto, M., Pagano, A., and Young, P. (2007). “State dependent parameter metamodelling and sensitivity analysis.” Comput. Phys. Commun., 177(11), 863–876.
Robert, C. P., and Casella, G. (2004). Monte Carlo statistical methods, Springer, New York.
Saltelli, A., et al. (2008). Global sensitivity analysis: The primer, Wiley, West Sussex, England.
Saltelli, A., and Sobol’, I. M. (1995). “About the use of rank transformation in sensitivity analysis of model output.” Reliab. Eng. Syst. Saf., 50(3), 225–239.
Scott, D. W. (1992). Multivariate density estimation: Theory, practise and visualization, Wiley, New York.
Silverman, B. W. (1998). Density estimation for statistics and data analysis, Chapman and Hall/CRC, Boca Raton, FL.
Sobol’, I. M. (1990). “On sensitivity estimation for nonlinear mathematical models.” Matematicheskoe Modelirovanie, 2(1), 112–118.
Sobol’, I. M. (2001). “Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates.” Math. Comput. Simul., 55(1–3), 271–280.
Sudret, B. (2008). “Global sensitivity analysis using polynomial chaos expansions.” Reliab. Eng. Syst. Saf., 93(7), 964–979.
Tarantola, S., Gatelli, D., and Mara, T. A. (2006). “Random balance designs for the estimation of first order global sensitivity indices.” Reliab. Eng. Syst. Saf., 91(6), 717–727.
Zhang, X., and Pandey, M. D. (2014). “An effective approximation for variance-based global sensitivity analysis.” Reliab. Eng. Syst. Saf., 121, 164–174.
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Published In
Journal of Engineering Mechanics
Volume 142 • Issue 5 • May 2016
Copyright
© 2016 American Society of Civil Engineers.
History
Received: Jul 1, 2015
Accepted: Nov 20, 2015
Published online: Jan 14, 2016
Published in print: May 1, 2016
Discussion open until: Jun 14, 2016
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