{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T19:42:01Z","timestamp":1787341321353,"version":"build-2736575974"},"reference-count":28,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","funder":[{"DOI":"10.13039\/501100006489","name":"Commissariat \u00e0 l'\u00c9nergie Atomique et aux \u00c9nergies Alternatives","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100006489","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM\/ASA J. Uncertainty Quantification"],"published-print":{"date-parts":[[2021,1]]},"abstract":"<jats:p>In this paper, we address the estimation of sensitivity indices called \u201cShapley effects.\u201d These sensitivity indices enable one to handle dependent input variables. The Shapley effects are generally difficult to estimate, but they are easily computable in the Gaussian linear framework. The aim of this work is to use the values of the Shapley effects in an approximated Gaussian linear framework as estimators of the true Shapley effects corresponding to a nonlinear model. First, we consider Gaussian input variables with small variances. We provide rates of convergence of the estimated Shapley effects to the true Shapley effects. Then, we focus on the case where the inputs are given by a non-Gaussian empirical mean. We prove that, under some mild assumptions, when the number of terms in the empirical mean increases, the difference between the true Shapley effects and the estimated Shapley effects given by the Gaussian linear approximation converges to 0. Our theoretical results are supported by numerical studies, showing that the Gaussian linear approximation is accurate and enables one to decrease the computational time significantly.<\/jats:p>","DOI":"10.1137\/20m1342884","type":"journal-article","created":{"date-parts":[[2021,8,26]],"date-time":"2021-08-26T10:25:33Z","timestamp":1629973533000},"page":"1132-1151","source":"Crossref","is-referenced-by-count":0,"title":["Gaussian Linear Approximation for the Estimation of the Shapley Effects"],"prefix":"10.1137","volume":"9","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1333-2149","authenticated-orcid":true,"given":"Baptiste","family":"Broto","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5336-5714","authenticated-orcid":true,"given":"Fran\u00e7ois","family":"Bachoc","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Marine","family":"Depecker","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jean-Marc","family":"Martinez","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2021,8,26]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.13182\/NSE12-55"},{"key":"atypb2","first-page":"266","author":"Benoumechiara N.","year":"2019","journal-title":"France"},{"key":"atypb3","doi-asserted-by":"crossref","unstructured":"R. N. Bhattacharya and R. R. Rao,\n                      Normal Approximation and Asymptotic Expansions\n                      , Classics Appl. Math. 64, SIAM, Philadelphia, 1986,https:\/\/doi.org\/10.1137\/1.9780898719895.","DOI":"10.1137\/1.9780898719895"},{"key":"atypb4","unstructured":"B. Broto, F. Bachoc, L. Clouvel, and J.M. Martinez,\n                      Block-diagonal Covariance Estimation and Application to the Shapley Effects in Sensitivity Analysis\n                      , 2020,https:\/\/hal.archives-ouvertes.fr\/hal-02196583v2."},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1137\/18M1234631"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1016\/j.matcom.2019.02.008"},{"key":"atypb7","doi-asserted-by":"crossref","unstructured":"D. G. Cacuci,\n                      Sensitivity and Uncertainty Analysis, Volume 1: Theory\n                      , Chapman & Hall\/CRC, Boca Raton, FL, 2003.","DOI":"10.1201\/9780203498798"},{"key":"atypb8","unstructured":"G. Chastaing,\n                      Indices de Sobol g\u00e9n\u00e9ralis\u00e9s pour variables d\u00e9pendantes\n                      , Ph.D. Thesis, Universit\u00e9 de Grenoble, 2013,https:\/\/tel.archives-ouvertes.fr\/tel-00930229\/document(accessed 2017-03-27)."},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1214\/12-EJS749"},{"key":"atypb10","unstructured":"L. Clouvel,\n                      Uncertainty Quantification of the Fast Flux Calculation for a PWR Vessel\n                      , Ph.D. thesis, Universit\u00e9 Paris-Saclay, 2019."},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1080\/02331888.2015.1105803"},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1016\/j.csda.2010.06.008"},{"key":"atypb13","unstructured":"L. Hasco\u00ebt and V. Pascual,\n                      Tapenade 2.1 User's Guide\n                      , 2004,https:\/\/hal.inria.fr\/inria-00069880\/document."},{"key":"atypb14","unstructured":"B. Iooss, J. Alexandre, and G. Pujol,\n                      Sensitivity: Global Sensitivity Analysis of Model Outputs\n                      , 2020,https:\/\/CRAN.R-project.org\/package=sensitivity(accessed 2020-02-14)."},{"key":"atypb15","doi-asserted-by":"publisher","DOI":"10.1615\/Int.J.UncertaintyQuantification.2019028372"},{"key":"atypb16","unstructured":"A. Koning, C. Dean, U. Fischer, and R. Mills,\n                      Validation of the JEFF-3.1 Nuclear Data Library: JEFF Report $23$\n                      , 2013,https:\/\/www.oecd-nea.org\/upload\/docs\/application\/pdf\/2019-12\/nea7079-jeff23_2019-12-20_14-54-21_729.pdf."},{"key":"atypb17","doi-asserted-by":"publisher","DOI":"10.1080\/18811248.2011.9711675"},{"key":"atypb18","doi-asserted-by":"publisher","DOI":"10.13182\/NSE06-A2589"},{"key":"atypb19","doi-asserted-by":"publisher","DOI":"10.1016\/j.ress.2011.08.008"},{"key":"atypb20","doi-asserted-by":"publisher","DOI":"10.1016\/j.envsoft.2015.07.010"},{"key":"atypb21","unstructured":"V. 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