Abstract
In RBDO, input uncertainty models such as marginal and joint cumulative distribution functions (CDFs) need to be used. However, only limited data exists in industry applications. Thus, identification of the input uncertainty model is challenging especially when input variables are correlated. Since input random variables, such as fatigue material properties, are correlated in many industrial problems, the joint CDF of correlated input variables needs to be correctly identified from given data. In this paper, a Bayesian method is proposed to identify the marginal and joint CDFs from given data where a copula, which only requires marginal CDFs and correlation parameters, is used to model the joint CDF of input variables. Using simulated data sets, performance of the Bayesian method is tested for different numbers of samples and is compared with the goodness-of-fit (GOF) test. Two examples are used to demonstrate how the Bayesian method is used to identify correct marginal CDFs and copula.
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References
Annis C (2004) Probabilistic life prediction isn’t as easy as it looks. JAI 1(2):3–14
Bravais A (1846) Analyse mathematique sur les probabilites des erreurs de situation d’un point. Memoires par divers Savan 9:255–332
Bretthorst GL (1996) An introduction to model selection using probability theory as logic. In: Heidbreder G (ed) Maximum entropy and Bayesian methods. Kluwer, Dordrecht, pp 1–42
Cirrone GAP, Donadio S, Guatelli S et al (2004) A godness-of-fit statistical toolkit. IEEE Trans Nucl Sci 51(5):2056–2063
Ditlevsen O, Madsen HO (1996) Structural reliability methods. Wiley, New York
Efstratios N, Ghiocel DM, Singhal S (2004) Engineering design reliability handbook. CRC, New York
Genest C, Favre AC (2007) Everything you always wanted to know about copula modeling but were afraid to ask. J Hydrol Eng 12(4):347–368
Genest C, Rémillard B (2005) Validity of the parametric bootstrap for goodness-of-fit testing in semiparametric models. Technical Rep. G-2005-51, Group d’Études et de Recherche en Analyse des Décision
Genest C, Rivest LP (1993) Statistical inference procedures for bivariate Archimedean copulas. J Am Stat Assoc 88:1034–1043
Haldar A, Mahadevan S (2000) Probability, reliability, and statistical methods in engineering design. Wiley, New York
Huard D, Évin G, Favre AC (2006) Bayesian copula selection. Comput Stat Data Anal 51:809–822
Kendall M (1938) A new measure of rank correlation. Biometrika 30:81–89
Kruskal WH (1958) Ordinal measures of associations. J Am Stat Assoc 53(284):814–861
Melchers RE (1999) Structural reliability analysis and prediction, 2nd edn. Wiley, New York
Nataf A (1962) Détermination des CDFs de probabilités dont les marges sont données. C R Hebd Séances Acad Sci 255:42–43
Nelsen RB (1999) An introduction to copulas. Springer, New York
Noh Y, Choi KK, Du L (2007) New transformation of dependent input variables using copula for RBDO. In: 7th world congress of structural and multidisciplinary optimization, Seoul, Korea, 21–25 May
Noh Y, Choi KK, Du L (2008) Reliability based design optimization of problems with correlated input variables using copulas. Struct Multidisc Optim 38(1):1–16
Pearson K (1896) Mathematical contributions to the theory of evolution. III. Regression, heredity and panmixia. Philos Trans Royal Soc London Ser A 187:253–318
Socie DF (2003) Seminar notes: probabilistic aspects of fatigue. http://www.fatiguecaculator.com. Cited 8 May 2008
Sterne JA, Smith GD (2000) Sifting the evidence—what’s wrong with significance tests? Br Med J 322:226–231
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Noh, Y., Choi, K.K. & Lee, I. Identification of marginal and joint CDFs using Bayesian method for RBDO. Struct Multidisc Optim 40, 35–51 (2010). https://doi.org/10.1007/s00158-009-0385-1
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DOI: https://doi.org/10.1007/s00158-009-0385-1