no. 3
Numéro spécial : fiabilité
Approche décisionnelle bayésienne pour estimer une courbe de fragilité
Journal de la société française de statistique, Tome 155 (2014) no. 3, pp. 78-103.

L’étude présentée dans cet article se focalise sur les aspects décisionnels de l’estimation statistique d’une courbe de fragilité sismique. Cet objet utilisé en ingénierie du risque renseigne la probabilité de défaillance d’une structure conditionnellement à un certain niveau de sollicitation. Après avoir détaillé la modélisation statistique utilisée pour sa construction, précisé les motivations industrielles et dressé un panel des méthodes existantes dans la littérature, nous mettons l’accent sur le cadre rigoureux que permet l’analyse décisionnelle bayésienne pour estimer une courbe de fragilité en tenant compte des conséquences socio-économiques du problème. L’estimation statistique est réalisée à partir de données simulant le comportement d’une maquette de bâtiment à échelle réduite. Plusieurs estimateurs sont comparés au regard de la fonction de coût utilisée.

This study deals with decisional analysis for estimating a seismic fragility curve, a tool used by risk engineers to provide the probability of a structure to suffer a given damage level conditionally to a given seismic intensity. After having described the statistical model and emphasized both the industrial motivations and the methods usually used to assess fragility curves, we focus on Bayesian decision analysis to estimate it accounting for social-economic consequences. Datasets are collected from numerical simulations and some estimators of the fragility curve are compared with respect to the chosen loss function.

Mot clés : analyse de risque, courbes de fragilité, théorie bayésienne, analyse décisionnelle, fonction de coût, expériences numériques
Keywords: risk analysis, fragility curves, Bayesian theory, decision analysis, loss function, computer experiments 
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     journal = {Journal de la soci\'et\'e fran\c{c}aise de statistique},
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Damblin, Guillaume; Keller, Merlin; Pasanisi, Alberto; Barbillon, Pierre; Parent, Eric. Approche décisionnelle bayésienne pour estimer une courbe de fragilité. Journal de la société française de statistique, Tome 155 (2014) no. 3, pp. 78-103. http://www.numdam.org/item/JSFS_2014__155_3_78_0/

[1] Albert, J.H.; Chib, S. Bayesian Analysis of Binary and Polychotomous Response Data, Journal of the American Statistical Association, Volume 88 (1993) no. 422, pp. 669-679 | MR | Zbl

[2] Ayyub, B.M.; Foster, J.; McGill, W.L. Risk Analysis of a Protected Hurricane-Prone Region. I : Model Development, Natural Hazards Review, Volume 10 (2009) no. 2, pp. 38-53

[3] Ayyub, B.M.; Foster, J.; McGill, W.L.; Jones, H.W. Risk Analysis of a Protected Hurricane-Prone Region. II : Computations and Illustrations, Natural Hazards Review, Volume 10 (2009) no. 2, pp. 54-67

[4] Ali, S. D. S. M. ; Silvey A general class of coefficients of divergence of one distribution from another, Journal of the Royal Statistical Society, Series B, Volume 28 (1966) no. 1, pp. 131-142 | MR | Zbl

[5] Bayarri, M. J.; Berger, J. O.; Sacks, P. R.; Cafeo, J. A.; Cavendish, J.; Lin, C.-H.; Tu, J. A Framework for Validation of Computer Models, Technometrics, Volume 49 (2007), pp. 138-154 | MR

[6] Basu, S.; Chib, S. Marginal likelihood and Bayes factors for Dirichlet Process Mixture Models, Journal of the American Statistical Association, Volume 98 (2003) no. 461, pp. 224-235 | MR | Zbl

[7] Bernier, J. Décisions et comportement des décideurs face au risque hydrologique, J. Sci. Hydrol., Volume 48 (2003) no. 3, pp. 301-316

[8] Berger, J.O. Statistical Decision Theory and Bayesian Analysis, In Statistics, Springer-Verlag, New York, 1985, 617 pages | MR | Zbl

[9] Bouc, R. Forced vibration of mechanical systems with hysteresis, Proceedings of the 4th conf. on nonlinear oscillation (1967)

[10] Bernardo, J. M.; Smith, A. F. M. Bayesian Theory, Wiley, London, 1994, 586 pages | MR | Zbl

[11] Box, G. E. P.; Tiao, G. T. Bayesian Inference in Statistical Analysis, Addison-Wesley, Reading, 1973 | MR | Zbl

[12] Casella, G.; George, E. Explaining the Gibbs Sampler, Am. Stat., Volume 46 (1992), pp. 167-174 | MR

[13] Chib, S.; Greenberg, E. Understanding the Metropolis-Hastings algorithm, The American Statistician, Volume 49(4) (1995), pp. 327-335

[14] Collett, D. Modelling Survival Data in Medical Research, Chapman & Hall/CRC, Boca Raton, 2003

[15] Clough, RW.; Penzien, J. Dynamics of structures, New York : McGraw-Hill, Inc, 1977

[16] Dempster, A.P.; Laird, N.M.; Rubin, D.B Maximum Likelihood from Incomplete Data via the EM Algorithm, Journal of the Royal Statistical Society, Series B, Volume 39 (1977) no. 1, pp. 1-38 | MR | Zbl

[17] Doucet, A.; de Freitas, N.; Gordon, N. Sequential Monte Carlo Methods in Practice, Springer Verlag, New York, 2001 | Zbl

[18] Ellingwood, R.; Kinali, K. Quantifying and communicating uncertainty in seismic risk assessment, Structural Safety, Volume 31 (2009) no. 2, pp. 179-187 | DOI

[19] EPRI Methodology for Developing Seismic fragilities (Final Report TR-103959, 1994)

[20] Ferguson, T. A Bayesian analysis of some nonparametric problems, The American Statistician, Volume 1 (1973), pp. 209-230 | MR | Zbl

[21] Fabbrocino, G.; Iervolino, I.; Orlando, F.; Salzano, E. Quantitative risk analysis of oil storage facilities in seismic areas, Journal of Hazardous Materials (2005), pp. 61-69

[22] Gelman, A.; Carlin, J.B.; Stern, H.S.; Rubin, D.B. Bayesian Data Analysis. Second Edition, Col. Texts in Statistical Science., Chapman & Hall, New-York USA, 2004 | MR

[23] Geman, S.; Geman, D. Stochastic relaxation, Gibbs distributions and the Bayesian restorationof image, IEEE Trans. Pattern Anal. Mach. Intell., Volume 6 (1984), pp. 721-741 | Zbl

[24] Ghosh, J.K.; Ramamoorthi, R.V. Bayesian nonparametrics, Springer, 2002 | MR | Zbl

[25] Gelfand, A.E.; Smith, A.F.M. Sampling Based Approaches to Calculating Marginal Densities, Journal of the American Statistical Association, Volume 85 (1990), pp. 398-409 | MR | Zbl

[26] HAZUS Hazus multy hazard (FEMA, 2003)

[27] Herbin, A.H.; Barbato, M. Fragility curves for building envelope components subject to windborne debris impact, Journal of Wind Engineering and Industrial Aerodynamics, Volume 107,108 (2012) no. 0, pp. 285-298 | DOI

[28] Hoeting, J.A.; Madigan, D.; Raftery, A.E.; Volinsky, C.T. Bayesian Model Averaging : A Tutorial, Statistical Science, Volume 14 (1999) no. 4, pp. 382-417 | MR | Zbl

[29] Hoshi, S.; Maruyama, Y.; Yamazaki, F. Reevaluation Method of fragility Curves of Wooden House Based on Collected Damage Information, Procedia Engineering, Volume 14 (2011), pp. 227-232

[30] Huang, Y.; Whittaker, S.; Luco, N. A probabilistic seismic risk assessment procedure for nuclear power plants : (I) Methodology, Nuclear Engineering and Design, Volume 241 (2011) no. 9, pp. 3996-4003 | DOI

[31] Huang, Y.; Whittaker, S.; Luco, N. A probabilistic seismic risk assessment procedure for nuclear power plants : (II) Application, Nuclear Engineering and Design, Volume 241 (2011) no. 9, pp. 3985-3995 | DOI

[32] Probabilistic safety assessment for seismic events (1993)

[33] Jordaan, I Decisions Under Uncertainty : Probabilistic Analysis For Engineering Decisions, Cambridge University Press, 2005 (275 pages)

[34] Kalalo, E.; Brenot, D. Rôles et limites des EPS, Contrôle, Volume 155 (2003), pp. 39-42

[35] Kaplan, E. L.; Meier, P. Nonparametric Estimation from Incomplete Observations, Journal of the American Statistical Association, Volume 53 (1958), pp. 457-481 | MR | Zbl

[36] Kennedy, M.; O’Hagan, A. Bayesian Calibration of Computer Models, Journal of the Royal Statistical Society, Series B, Methodological, Volume 63 (2001), pp. 425-464 | MR | Zbl

[37] Keller, M.; Pasanisi, A.; Parent, E. Réflexions sur l’analyse d’incertitudes dans un contexte industriel : information disponible et enjeux décisionnels, Journal de la Société Française de Statistique, Volume 152 (2011) no. 4, pp. 60-77 | Numdam | MR | Zbl

[38] Kass, R.E.; Raftery, A.E. Bayes factors, Journal of the American Statistical Association, Volume 90 (1995) no. 430, pp. 773-795 | MR | Zbl

[39] Lanore, J. Perspectives sur le développement des EPS de niveau 1,2 et 3, Contrôle, Volume 155 (2003), pp. 53-57

[40] Lallemant, D.; Kiremidjian, A. Fitting Fragility Curves to Empirical Data, GEM Technical Report (2013)

[41] Lagaros, D.; Tsompanakis, Y.; Psarropoulos, N.; Georgopoulos, C. Computationally efficient seismic fragility analysis of geostructures, Computers & Structures, Volume 87 (2009) no. 19,20, pp. 1195-1203 | DOI

[42] McCulloch, C.E. Maximum likelihood variance components estimation for binary data, Journal of the American Statistical Association, Volume 89 (1994) no. 425, pp. 330-335 | Zbl

[43] Marano, G.C; Greco, R; Morrone, E. Analytical evaluation of essential facilities fragility curves by using a stochastic approach, Engineering Structures, Volume 33 (2011), pp. 191-201

[44] McCullagh, P.; Nelder, J.A. Generalized Linear Models, Monographs on Statistics and Applied Probability, Chapman and Hall, 1989 | MR | Zbl

[45] Metropolis, N.; Rosenbluth, A.W.; Rosenbluth, M.N.; Teller, E. A.Hand Teller Equations of State Calculations by Fast Computing Machines, J. Chem. Phys., Volume 21 (1953), pp. 1087-1091 | Zbl

[46] Parmigiani, G.; Inoue, L. Decision Theory : Principles and Approaches, Wiley, 2008 | MR | Zbl

[47] Pasanisi, A.; Keller, M.; Parent, E. Estimation of a quantity of interest in uncertainty analysis : Some help from Bayesian decision theory, Reliability Engineering and System Safety, Volume 100 (2012) no. 0, pp. 93-101 | DOI

[48] Reese, S.; Bradley, B.A.; Bind, J.; Smart, G.; Power, W.; Sturman, J. Empirical building fragilities from observed damage in the 2009 South Pacific tsunami, Earth-Science Reviews, Volume 107 (2011), pp. 156-173

[49] Robert, C.P. L’analyse statistique bayésienne, Economica, 1992 | MR | Zbl

[50] Straub, D.; Der Kiureghian, A. Improved seismic fragility modeling from empirical data, Structural Safety, Volume 30 (2008) no. 4, pp. 320-336 | DOI

[51] Shinozuka, M.; Feng, M.; Lee, J.; Naganuma, T. Statistical Analysis of Fragility Curves, Journal of Engineering Mechanics, Volume 126 (2000) no. 12, pp. 1224-1231 | DOI

[52] Schultz, T.; Gouldby, P.; Simm, D.; Wibowo, L. Beyond the Factor of Safety : Developing Fragility Curves to Characterize System Reliability (2010) no. ERDC SR-10-1 (Technical report)

[53] Shoji, G.; Moriyama, T. Evaluation of the Structural Fragility of a Bridge Structure Subjected to a Tsunami Wave Load, Journal of Natural Disaster Science, Volume 29 (2007) no. 2, pp. 73-81

[54] Stewart, G.; Netherton, D. Security risks and probabilistic risk assessment of glazing subject to explosive blast loading, Reliability Engineering and System Safety, Volume 93 (2008) no. 4, pp. 627-638 | DOI

[55] Solomos, G.; Pinto, A.; Dimova, S. A Review of the seismic hazard zonation in national building codes in the context of Eurocode 8 (2008) no. EUR 23563 EN-2008 (Technical report)

[56] Susarla, J.; Van Ryzin, J. Nonparametric Bayesian Estimation of Survival Curves from Incomplete Observations, Journal of the American Statistical Association, Volume 71 (1976) no. 356, pp. 897-902 | MR | Zbl

[57] Tanner, M. H. Tools for Statistical Inference  : Observed Data and Data Augmentation Methods, Springer-Verlag, New York, 1992 | MR

[58] Ulmo, J.; Bernier, J. Eléments de Décision Statistique, PUF, 1973 | Zbl

[59] Wald, A. Statistical Decision Functions, Wiley, 1950 | MR | Zbl

[60] Wasserman, L. Bayesian Model Selection and Model Averaging, Journal of Mathematical Psychology, Volume 44 (2000), pp. 92-107 | MR | Zbl

[61] Wen, YK. Method for random vibration of hysteretic systems, J Eng Mech Div, ASCE, Volume 4 (1976), pp. 102-150

[62] Zentner, I.; Borgonovo, E.; Tarantola, S. Use of HDMR metamodel for seismic fragility analysis, Applications of Statistics and Probability in Civil Engineering, ICASP 2011 (Nishijima, K., ed.), CRC Press (2011), pp. 653-660

[63] Zentner, I. Numerical computation of fragility curves for NPP equipment, Nuclear Engineering and Design, Volume 240 (2010) no. 6, pp. 1614-1621 | DOI