Library Subscription: Guest
International Journal for Uncertainty Quantification

Published 6 issues per year

ISSN Print: 2152-5080

ISSN Online: 2152-5099

The Impact Factor measures the average number of citations received in a particular year by papers published in the journal during the two preceding years. 2017 Journal Citation Reports (Clarivate Analytics, 2018) IF: 1.7 To calculate the five year Impact Factor, citations are counted in 2017 to the previous five years and divided by the source items published in the previous five years. 2017 Journal Citation Reports (Clarivate Analytics, 2018) 5-Year IF: 1.9 The Immediacy Index is the average number of times an article is cited in the year it is published. The journal Immediacy Index indicates how quickly articles in a journal are cited. Immediacy Index: 0.5 The Eigenfactor score, developed by Jevin West and Carl Bergstrom at the University of Washington, is a rating of the total importance of a scientific journal. Journals are rated according to the number of incoming citations, with citations from highly ranked journals weighted to make a larger contribution to the eigenfactor than those from poorly ranked journals. Eigenfactor: 0.0007 The Journal Citation Indicator (JCI) is a single measurement of the field-normalized citation impact of journals in the Web of Science Core Collection across disciplines. The key words here are that the metric is normalized and cross-disciplinary. JCI: 0.5 SJR: 0.584 SNIP: 0.676 CiteScore™:: 3 H-Index: 25

Indexed in

SHAPLEY EFFECTS FOR SENSITIVITY ANALYSIS WITH CORRELATED INPUTS: COMPARISONS WITH SOBOL' INDICES, NUMERICAL ESTIMATION AND APPLICATIONS

Volume 9, Issue 5, 2019, pp. 493-514
DOI: 10.1615/Int.J.UncertaintyQuantification.2019028372
Get accessGet access

ABSTRACT

The global sensitivity analysis of a numerical model aims to quantify, by means of sensitivity indices estimates, the contributions of each uncertain input variable to the model output uncertainty. The so-called Sobol' indices, which are based on functional variance analysis, present a difficult interpretation in the presence of statistical dependence between inputs. The Shapley effects were recently introduced to overcome this problem as they allocate the mutual contribution (due to correlation and interaction) of a group of inputs to each individual input within the group. In this paper, using several new analytical results, we study the effects of linear correlation between some Gaussian input variables on Shapley effects, and compare these effects to classical first-order and total Sobol' indices. This illustrates the interest, in terms of sensitivity analysis setting and interpretation, of the Shapley effects in the case of dependent inputs. For the practical issue of computationally demanding computer models, we show that the substitution of the original model by a metamodel (here, kriging) makes it possible to estimate these indices with precision at a reasonable computational cost.

REFERENCES
  1. Iooss, B. and Lemaitre, P., A Review on Global Sensitivity Analysis Methods, in Uncertainty Management in Simulation-Optimization of Complex Systems: Algorithms and Applications, C. Meloni and G. Dellino, Eds., Springer, pp. 101-122, 2015.

  2. Wei, P., Lu, Z., and Song, J., Variable Importance Analysis: A Comprehensive Review, Reliab. Eng. Syst. Saf., 142:399-432, 2015.

  3. Borgonovo, E. and Plischke, E., Sensitivity Analysis: A Review of Recent Advances, Eur. J. Oper. Res, 248:869-887,2016.

  4. Iooss, B. and Saltelli, A., Introduction: Sensitivity Analysis, in Springer Handbook on Uncertainty Quantification, R. Ghanem, D. Higdon, and H. Owhadi, Eds., Springer, pp. 1103-1122,2017.

  5. Sobol, I., Sensitivity Estimates for Non-Linear Mathematical Models, Math. Model. Comput. Exp., 1:407-414, 1993.

  6. Kurowicka, D. and Cooke, R., Uncertainty Analysis with High-Dimensional Dependence Modelling, New York: Wiley, 2006.

  7. Petelet, M., Iooss, B., Asserin, O., and Loredo, A., Latin Hypercube Sampling with Inequality Constraints, Adv. Stat. Anal, 94:325-339,2010.

  8. Kucherenko, S., Klymenko, O., and Shah, N., Sobol' Indices for Problems Defined in Non-Rectangular Domains, Reliab. Eng. Syst. Saf., 167:218-231, 2017.

  9. Lopez-Benito, A. and Bolado-Lavin, R., A Case Study on Global Sensitivity Analysis with Dependent Inputs: The Natural Gas Transmission Model, Reliab. Eng. Syst. Saf., 165:11-21, 2017.

  10. Saltelli, A. and Tarantola, S., On the Relative Importance of Input Factors in Mathematical Models: Safety Assessment for Nuclear Waste Disposal, J. Am. Stat. Assoc, 97:702-709, 2002.

  11. Da Veiga, S., Wahl, F., and Gamboa, F., Local Polynomial Estimation for Sensitivity Analysis on Models with Correlated Inputs, Technometrics, 51(4):452-463, 2009.

  12. Shapley, L., A Value for N-Persons Game, in Contributions to the Theory of Games II, Annals ofMathematic Studies, H. Kuhn and A. Tucker, Eds., Princeton, NJ: Princeton University Press, 1953.

  13. Owen, A., Sobol' Indices and Shapley Value, SIAM/ASA J. Uncertainty Quantif., 2:245-251, 2014.

  14. Song, E., Nelson, B., and Staum, J., Shapley Effects for Global Sensitivity Analysis: Theory and Computation, SIAM/ASA J. Uncertainty Quantif., 4:1060-1083, 2016.

  15. Owen, A. and Prieur, C., On Shapley Value for Measuring Importance of Dependent Inputs, SIAM/ASA J. Uncertainty Quantif, 5:986-1002,2017.

  16. Prieur, C. and Tarantola, S., Variance-Based Sensitivity Analysis: Theory and Estimation Algorithms, in Springer Handbook on Uncertainty Quantification, R. Ghanem, D. Higdon, and H. Owhadi, Eds., Springer, pp. 1217-1239, 2017.

  17. Fang, K.T., Li, R., and Sudjianto, A., Design and Modeling for Computer Experiments, Boca Raton, FL: Chapman & Hall/CRC, 2006.

  18. Iooss, B., Van Dorpe, F., and Devictor, N., Response Surfaces and Sensitivity Analyses for an Environmental Model of Dose Calculations, Reliab. Eng. Syst. Saf., 91:1241-1251, 2006.

  19. Le Gratiet, L., Marelli, S., and Sudret, B., Metamodel-Based Sensitivity Analysis: Polynomial Chaos Expansions and Gaussian Processes, in Springer Handbook on Uncertainty Quantification, R. Ghanem, D. Higdon, and H. Owhadi, Eds., Springer, pp. 1289-1325,2017.

  20. Homma, T. and Saltelli, A., Importance Measures in Global Sensitivity Analysis of Non-Linear Models, Reliab. Eng. Syst. Saf., 52:1-17, 1996.

  21. Jacques, J., Lavergne, C., and Devictor, N., Sensitivity Analysis in Presence of Model Uncertainty and Correlated Inputs, Reliab. Eng. Syst. Saf., 91:1126-1134, 2006.

  22. Xu, C. and Gertner, G., Uncertainty and Sensitivity Analysis for Models with Correlated Parameters, Reliab. Eng. Syst. Saf., 93:1563-1573,2008.

  23. Mara, T. and Tarantola, S., Variance-Based Sensitivity Indices for Models with Dependent Inputs, Reliab. Eng. Syst. Saf., 107:115-121,2012.

  24. Li, G., Rabitz, H., Hu, J., Chen, Z., and Ju, Y., Regularized Random-Sampling High-Dimensional Model Representation, J. Math. Chem., 43(3):6022-6032, 2008.

  25. Kucherenko, S., Tarantola, S., and Annoni, P., Estimation of Global Sensitivity Indices for Models with Dependent Variables, Comput. Phys. Commun., 183:937-946, 2012.

  26. Mara, T., Tarantola, S., and Annoni, P., Non-Parametric Methods for Global Sensitivity Analysis of Model Output with Dependent Inputs, Env. Model. Software, 72:173-183, 2015.

  27. Li, G., Rabitz, H., Yelvington, P., Oluwole, O., Bacon, F., Kolb, C., and Schoendorf, J., Global Sensitivity Analysis for Systems with Independent and/or Correlated Inputs, j. Phys. Chem., 114:6022-6032, 2010.

  28. Sudret, B. and Caniou, Y., Analysis of Covariance (ANCOVA) Using Polynomial Chaos Expansions, in Proc. of 11th Int. Conf. Struct. Safety and Reliability (ICOSSAR 2013), G. Deodatis, Ed., New York, 2013.

  29. Zhang, K., Lu, Z., Cheng, L., and Xu, F., A New Framework of Variance based Global Sensitivity Analysis for Models with Correlated Inputs, Struct. Saf, 55:1-9, 2015.

  30. Hoeffding, W., A Class of Statistics with Asymptotically Normal Distributions, Ann. Math. Stat, 19:293-325,1948.

  31. Stone, C.J., The Use of Polynomial Splines and Their Tensor Products in Multivariate Function Estimation, Ann. Stat., 22(1):118-184,1994.

  32. Hooker, G., Generalized Functional Anova Diagnostics for High-Dimensional Functions of Dependent Variables, J. Comput. Graph. Stat., 16:709-732,2007.

  33. Chastaing, G., Gamboa, F., and Prieur, C., Generalized Hoeffding-Sobol Decomposition for Dependent Variables-Application to Sensitivity Analysis, Electron. J. Stat., 6:2420-2448, 2012.

  34. Borgonovo, E., A New Uncertainty Importance Measure, Reliab. Eng. Syst. Saf., 92:771-784, 2007.

  35. Borgonovo, E., Castaings, W., and Tarantola, S., Moment Independent Importance Measures: New Results and Analytical Test Cases, Risk Anal, 31:404-428,2011.

  36. Zhou, C., Lu, Z., Zhang, L., and Hu, J., Moment Independent Sensitivity Analysis with Correlations, Appl. Math. Model, 38:4885-4896, 2014.

  37. Rosenblatt, M., Remarks on a Multivariate Transformation, Ann. Math. Stat., 23(3):470-472, 1952.

  38. Iman, R. and Conover, W., A Distribution-Free Approach to Inducing Rank Correlation among Input Variables, Commun. Stat., 11(3):311-334, 1982.

  39. Saltelli, A., Tarantola, S., Campolongo, F., and Ratto, M., Sensitivity Analysis in Practice: A Guide to Assessing Scientific Models, New York: Wiley, 2004.

  40. Benoumechiara, N. and Elie-Dit-Cosaque, K., Shapley Effects for Sensitivity Analysis with Dependent Inputs: Bootstrap and Kriging-Based Algorithms, ESAIM: Proc. Surveys, 65:266-293, 2019.

  41. Sacks, J., Welch, W., Mitchell, T., and Wynn, H., Design and Analysis of Computer Experiments, Stat. Sci., 4:409-435,1989.

  42. Santner, T., Williams, B., andNotz, W., The Design and Analysis of Computer Experiments, Berlin: Springer, 2003.

  43. Marrel, A., Iooss, B., VanDorpe, F., and Volkova, E., An Efficient Methodology for Modeling Complex Computer Codes with Gaussian Processes, Comput. Stat. Data Anal., 52:4731-4744, 2008.

  44. Roustant, O., Ginsbourger, D., and Deville, Y., DiceKriging, DiceOptim: Two R Packages for the Analysis of Computer Experiments by Kriging-Based Metamodeling and Optimization, J. Stat. Software, 21:1-55, 2012.

  45. Le Gratiet, L., Cannamela, C., and Iooss, B., A Bayesian Approach for Global Sensitivity Analysis of (Multifidelity) Computer Codes, SIAM/ASA J. Uncertainty Quantif., 2:336-363,2014.

  46. Rupin, F., Blatman, G., Lacaze, S., Fouquet, T., and Chassignole, B., Probabilistic Approaches to Compute Uncertainty Intervals and Sensitivity Factors of Ultrasonic Simulations of a Weld Inspection, Ultrasonics, 54:1037-1046, 2014.

  47. Moysan, J., Apfel, A., Corneloup, C., and Chassignole, B., Modelling the Grain Orientation of Austenitic Stainless Steel Multipass Welds to Improve Ultrasonic Assessment of Structural Integrity, Int. j. Pressure Vessels Piping, 80:77-85, 2003.

  48. Janon, A., Nodet, M., and Prieur, C., Uncertainties Assessment in Global Sensitivity Indices Estimation from Metamodels, Int. J. Uncertainty Quantif, 4:21-36, 2014.

  49. Janon, A., Klein, T., Lagnoux, A., Nodet, M., and Prieur, C., Asymptotic Normality and Efficiency of Two Sobol Index Estimators, ESAIM: Probab. Stat, 18:342-364, 2014.

  50. De Lozzo, M. and Marrel, A., Estimation of the Derivative-Based Global Sensitivity Measures Using a Gaussian Process Metamodel, SIAM/ASA J. Uncertainty Quantif., 4:708-738, 2016.

CITED BY
  1. Antoniadis Anestis, Lambert-Lacroix Sophie, Poggi Jean-Michel, Random forests for global sensitivity analysis: A selective review, Reliability Engineering & System Safety, 206, 2021. Crossref

  2. Rabitti Giovanni, Borgonovo Emanuele, Is mortality or interest rate the most important risk in annuity models? A comparison of sensitivity analysis methods, Insurance: Mathematics and Economics, 95, 2020. Crossref

  3. Broto Baptiste, Bachoc François, Depecker Marine, Martinez Jean-Marc, Gaussian Linear Approximation for the Estimation of the Shapley Effects, SIAM/ASA Journal on Uncertainty Quantification, 9, 3, 2021. Crossref

  4. Melillo Nicola, Darwich Adam S., A latent variable approach to account for correlated inputs in global sensitivity analysis, Journal of Pharmacokinetics and Pharmacodynamics, 48, 5, 2021. Crossref

  5. Taverniers Søren, Hall Eric J., Katsoulakis Markos A., Tartakovsky Daniel M., Mutual information for explainable deep learning of multiscale systems, Journal of Computational Physics, 444, 2021. Crossref

  6. Plischke Elmar, Rabitti Giovanni, Borgonovo Emanuele, Computing Shapley Effects for Sensitivity Analysis, SIAM/ASA Journal on Uncertainty Quantification, 9, 4, 2021. Crossref

  7. Bethencourt Loic, Dabachine Walid, Dejouy Vincent, Lalmiche Zakaria, Neuberger Karl, Ibnouhsein Issam, Chereau Sandrine, Mathelin Carole, Savy Nicolas, Pierre Philippe Saint, Bousquet Nicolas, Guiding Measurement Protocols of Connected Medical Devices Using Digital Twins: A Statistical Methodology Applied to Detecting and Monitoring Lymphedema, IEEE Access, 9, 2021. Crossref

  8. Bénard Clément, Da Veiga Sébastien, Scornet Erwan, Mean decrease accuracy for random forests: inconsistency, and a practical solution via the Sobol-MDA, Biometrika, 2022. Crossref

  9. Schröder Laura, Dimitrov Nikolay Krasimirov, Aasted Sørensen John, Uncertainty propagation and sensitivity analysis of an artificial neural network used as wind turbine load surrogate model, Journal of Physics: Conference Series, 1618, 4, 2020. Crossref

  10. D'Anna M., Idier D., Castelle B., Rohmer J., Cagigal L., Mendez F.J., Effects of stochastic wave forcing on probabilistic equilibrium shoreline response across the 21st century including sea-level rise, Coastal Engineering, 175, 2022. Crossref

  11. Broto Baptiste, Bachoc François, Clouvel Laura, Martinez Jean-Marc, Block-Diagonal Covariance Estimation and Application to the Shapley Effects in Sensitivity Analysis, SIAM/ASA Journal on Uncertainty Quantification, 10, 1, 2022. Crossref

  12. El Garroussi Siham, Ricci Sophie, De Lozzo Matthias, Goutal Nicole, Lucor Didier, Tackling random fields non-linearities with unsupervised clustering of polynomial chaos expansion in latent space: application to global sensitivity analysis of river flooding, Stochastic Environmental Research and Risk Assessment, 36, 3, 2022. Crossref

  13. D'Anna M., Castelle B., Idier D., Rohmer J., Le Cozannet G., Thieblemont R., Bricheno L., Uncertainties in Shoreline Projections to 2100 at Truc Vert Beach (France): Role of Sea‐Level Rise and Equilibrium Model Assumptions, Journal of Geophysical Research: Earth Surface, 126, 8, 2021. Crossref

  14. Mercadier Cécile, Roustant Olivier, Genest Christian, Linking the Hoeffding–Sobol and Möbius formulas through a decomposition of Kuo, Sloan, Wasilkowski, and Woźniakowski, Statistics & Probability Letters, 185, 2022. Crossref

  15. Mara Thierry A., Becker William E., Polynomial chaos expansion for sensitivity analysis of model output with dependent inputs, Reliability Engineering & System Safety, 214, 2021. Crossref

  16. Lo Piano Samuele, Benini Lorenzo, A critical perspective on uncertainty appraisal and sensitivity analysis in life cycle assessment, Journal of Industrial Ecology, 26, 3, 2022. Crossref

  17. Il Idrissi Marouane, Chabridon Vincent, Iooss Bertrand, Developments and applications of Shapley effects to reliability-oriented sensitivity analysis with correlated inputs, Environmental Modelling & Software, 143, 2021. Crossref

  18. Heredia María Belén, Prieur Clémentine, Eckert Nicolas, Global sensitivity analysis with aggregated Shapley effects, application to avalanche hazard assessment, Reliability Engineering & System Safety, 222, 2022. Crossref

  19. Ballester-Ripoll Rafael, Leonelli Manuele, Computing Sobol indices in probabilistic graphical models, Reliability Engineering & System Safety, 225, 2022. Crossref

  20. Al Ali Hannah, Daneshkhah Alireza, Boutayeb Abdesslam, Malunguza Noble Jahalamajaha, Mukandavire Zindoga, Exploring dynamical properties of a Type 1 diabetes model using sensitivity approaches, Mathematics and Computers in Simulation, 201, 2022. Crossref

  21. Vecherin S., Ketcham S., Meyer A., Dunn K., Desmond J., Parker M., Short-range near-surface seismic ensemble predictions and uncertainty quantification for layered medium, Journal of Applied Geophysics, 204, 2022. Crossref

  22. Elie-Dit-Cosaque Kevin, Maume-Deschamps Veronique, Goal-Oriented Shapley Effects with Special Attention to the Quantile-Oriented Case, SIAM/ASA Journal on Uncertainty Quantification, 10, 3, 2022. Crossref

  23. Rohmer Jeremy, Idier Deborah, Thieblemont Remi, Le Cozannet Goneri, Bachoc François, Partitioning the contributions of dependent offshore forcing conditions in the probabilistic assessment of future coastal flooding, Natural Hazards and Earth System Sciences, 22, 10, 2022. Crossref

Begell Digital Portal Begell Digital Library eBooks Journals References & Proceedings Research Collections Prices and Subscription Policies Begell House Contact Us Language English 中文 Русский Português German French Spain