Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
6th China-Japan-Korea Joint Symposium on Optimization of Structural and Mechanical Systems June 22–25, 2010, Kyoto, Japan Reliability-Based Design Optimization with Confidence Level for Problems with Correlated Input Distributions K.K. Choi, Yoojeong Noh, and Ikjin Lee Department of Mechanical & Industrial Engineering, College of Engineering The University of Iowa, Iowa City, IA 52242, USA [email protected], [email protected], [email protected] Abstract For obtaining a correct reliability-based optimum design, an input distribution model, which includes marginal and joint distributions of input random variables with their associated input parameters such as mean, standard deviation, and correlation coefficient, needs to be accurately estimated. In engineering reliability-based design optimization (RBDO) applications, most input distributions arise from loading, material properties, and manufacturing geometric variabilities. However, only limited data on these input random variables are available due to expensive experimental testing or measuring costs. The input distribution model estimated from the insufficient data could be inaccurate, which will lead to the incorrect reliability-based optimum design. Thus, RBDO with the confidence level has been addressed to offset the inaccurate estimation of the input distribution model by using an adjusted upper bound of confidence interval of the standard deviation, and an adjusted correlation coefficient. Keywords: Reliability-based design optimization, input distribution model, confidence level, bootstrap method, Adjusted Parameters 1. Introduction In many engineering applications, since input random variables such as fatigue material properties are correlated [1-4], the input statistical uncertainty needs to be modeled by identifying the input joint cumulative distribution functions (CDFs) as well as marginal CDFs. In addition, the input physical uncertainty needs to be modeled by quantifying the input parameters, such as mean, standard deviation, and correlation coefficient, of the identified marginal and joint distributions. In industrial applications, since only limited data on input variables is available due to expensive experimental testing or measuring costs, it is difficult to accurately model the input distributions, and thus, the design engineer will not have confidence on the RBDO result obtained using the input distribution model [5] to meet the target reliability. In this paper, to offset the possible inexact identification of the input joint and marginal CDFs and inexact quantification of the parameters, an input distribution model with a target confidence level is developed for RBDO by appropriately enlarging the estimated t-contour, for the given the target reliability index t, using the confidence intervals of the estimated parameters. If the input variables have marginal Gaussian distributions, the confidence intervals of mean and standard deviation can be obtained explicitly, which are accurate [6]. On the other hand, if the input variables have non-Gaussian marginal distributions, a bootstrap method [7,8] is used to predict the confidence intervals of the input parameters for input variables with non-Gaussian marginal distributions. Once these confidence intervals are obtained, instead of directly using the estimated upper or lower bounds, an adjusted upper bound of confidence interval of the standard deviation and an adjusted correlation coefficient calculated using confidence interval of the correlation coefficient are proposed. A mathematical example with Gaussian and non-Gaussian correlated variables are used to illustrate how the proposed input distribution model with target confidence level provides the RBDO design that meets the target confidence level. 2. Generation of Input Distribution Model If the input random variables are independent, the joint distribution can be obtained easily by multiplying the marginal distributions. However, if the input random variables are correlated, the joint distribution needs to be generated. Since only limited data is available in practical engineering applications, it is difficult to obtain the joint distribution directly from limited data. Thus, the copula [9-15], which is a function of marginal distributions and correlation parameters, is used to model the input joint distribution. 2.1 Identification of Input Distribution Model Copulas are multivariate distribution functions whose one-dimensional margins are uniform on the interval [0, 1]. According to Sklar’s theorem [11], a joint cumulative distribution function (CDF) FX1 X n x1 ,, xn of random variables Xi can be expressed in terms of marginal CDFs FX i xi of Xi for i=1,…,n, and copula function C as FX1 ,..., X n x1 ,..., xn C FX1 x1 ,..., FX n xn θ (1) where is the matrix of correlation parameters among X1,…,Xn. The dimension of the correlation matrix used in different copulas may or may not depend on the number of correlated random variables [5]. It has been observed that two input variables are correlated in many cases [7-10], so that only bivariate copulas are considered in this paper. In this case, is a scalar quantity. For a given limited data, the joint distribution can be identified by a one-step procedure, which directly tests all candidate joint distributions; or by a two-step procedure, which first identifies marginal distributions and then a copula [9,10]. In identifications of the input distributions, it is shown in Ref. 10 that the two-step approach is more efficient and accurate than the one-step approach. Regarding the measure of identification, the weight-based Bayesian method [9,10] and Markov chain Monte Carlo (MCMC)-based Bayesian method [16] can be used. It is shown that the weight-based method is much more efficient than the MCMC-based method [10]. Thus, in this paper, the two-step weight-based Bayesian method is used to identify the input distribution model. 2.2 Quantification of Input Distribution Model With the marginal and joint CDF types are identified, it is necessary to determine their parameters such as means and standard deviations of the marginal distributions, and the correlation coefficient. The parameters obtained from limited data are the sample mean ns xi (2) i 1 ns and the sample variance 1 ns 2 (3) xi 2 ns 1 i 1 On the other hand, all two-parameter marginal distributions, except the Gaussian distribution, which are used in many engineering applications, have their own parameters a and b, which can be expressed explicitly in terms of the mean and standard deviation ( and ) as shown in Table 3 in Ref. 9. Since the mean and standard deviation determine the location and variability of the distribution, respectively, the standard deviation can be used to enlarge the t-contour of the distribution. Accordingly, once the sample mean and standard deviation are calculated from given data, the parameters a and b of the identified input distributions can be calculated using explicit functions. Since copula functions have their own correlation parameters, it is desirable to use a common correlation coefficient. The Kendall’s tau, which is a widely used correlation coefficient [9], is used in this paper. The sample version of Kendall’s tau is expressed as cd ns (4) c d / cd 2 can be used, where c and d are the numbers of concordant and discordant pairs, respectively, and ns is the number of samples. Once the sample version of Kendall’s tau is obtained from Eq. (5), the correlation parameter can be obtained using explicit formulas g ( ) presented in Table 1 in Ref. 9. 3. Generation of Input Distribution Models with Target Confidence Level In this paper, the output confidence level of the RBDO designs is defined as the probability of RBDO designs, which are obtained using input distribution models, meet the target reliability. In this definition, the input distribution models, which are generated using limited test data sets, are treated as random events. However, direct assessment of the confidence level of the RBDO design is a very difficult undertaking. Thus, in this paper, it is proposed to generate the input distribution model that provides confidence level. If the t-contour of the proposed input distribution model, which is obtained using the adjusted standard deviation and correlation coefficient, covers the true t-contour, then the RBDO design will satisfy the target reliability since the Most Probable Point (MPP) of the RBDO design is on the t-contour. To generate an input distribution model with appropriately enlarged t-contour to cover the true t-contour with a certain confidence level, information of the confidence intervals of the estimated parameters of marginal distributions are used. 3.1 Dependence of t-contour on Marginal Input Distribution Parameters To make the estimated t-contour cover the true t-contour, it is necessary to understand how the input parameters affect the estimated t -contour shapes. First, consider the independent input random variables, where the joint distribution can be obtained easily by multiplying the marginal distributions. If we can assume the sample mean is accurate, then a larger standard deviation can be used to enlarge the estimated t-contour. On the other hand, a larger or smaller sample mean is related to the position of the estimated t-contour. Thus, enlargement of the estimated t-contour also needs to consider inaccuracy of the sample mean obtained from the data. For this purpose, we need first obtain the confidence intervals of the sample mean and sample standard deviations, which will be discussed in Sections 3.2. Second, if the random variables are correlated, then the problem is more complicate since we have to deal with the correlation coefficient. Like the mean, neither larger nor smaller sample correlation coefficient makes the estimated t-contour fully covers the true t-contour. Therefore, instead of using the estimated mean and correlation coefficient, adjusted parameters using the confidence intervals of the mean and correlation coefficient are proposed to ensure the estimated t-contour with the adjusted parameters covers the true t-contour, which will lead to a desirable confidence level of the input model. 3.2 Confidence Intervals of Input Parameters for Gaussian and Non-Gaussian Marginal Distributions Since only limited data are available, the sample mean in Eq. (2) and sample standard deviation from Eq. (3) of the marginal distributions are used to estimate the population mean and population standard deviation . A range of values in which they may be located is called the confidence interval. 3.2.1 Confidence Interval of Sample Mean If the underlying random variable is Gaussian, the two sided confidence interval that the population mean belongs to this interval with the confidence level of 100(1)% is [5,6] (5) Pr t / 2, ns 1 t / 2, ns 1 1 ns ns where ns is the number of samples and t / 2,ns 1 are the values of Student’s t-distribution with (ns1) degree of freedom evaluated at probabilities of (1/2) and /2, respectively. Note that, as the number of samples increases, the lower and upper bounds of the confidence interval of the sample mean approach the population mean. If input variables follow Gaussian distribution, Eq. (5) is accurate. However, if not, the estimated confidence interval may not be accurate. To circumvent this, a bootstrap method, which does not require the normality assumption on the input variable can be used. In this study, since the confidence interval of the mean is rather accurately calculated even for the non-Gaussian distribution [17,18], the bootstrap method is used only for confidence interval of the standard deviation of non-Gaussian distribution. 3.2.2 Confidence Interval of Sample Standard Deviation Using a similar procedure of calculating the confidence interval of the mean, if the underlying random variable is Gaussian, the two sided confidence interval that the population variance belongs to this interval with the confidence level of 100(1) is [5,6] ns 1 2 ns 1 2 (6) 2 Pr 1 c / 2, ns 1 c1 / 2, ns 1 where c / 2,ns 1 and c1 / 2, ns 1 are the values of the chi-square distribution with (ns1) degrees of freedom evaluated at probabilities of /2 and (1/2), respectively. Thus, the lower and upper bounds of the 100×(1−α)% confidence interval for the sample standard deviation are respectively calculated as 1L ns 1 2 c1 / 2, ns 1 and 1U ns 1 2 c / 2, ns 1 (7) If input variables follow Gaussian distribution, Eq. (7) is accurate. However, if not, the estimated confidence interval is not accurate. Thus, the bootstrap method, which does not use the Gaussian distribution of input variables to calculate the confidence interval of the mean and standard deviation, needs to be used. The bootstrap method calculates the confidence interval of estimated standard deviation by constructing a distribution of the standard deviation using the frequency distribution of * obtained from randomly generated bootstrap samples based on the given data. For the bootstrap method, the first step is to construct an empirical distribution Fns x or a parametric distribution F x a, b from given samples x x1 , x2 ,, xns . In the second step, bootstrap samples are generated from the constructed empirical distribution or parametric distribution. If a random sample of size ns with replacement is drawn from the empirical distribution Fns x , then this is called a non-parametric approach. If the resample is drawn from the parametric distribution F x a, b , this is called a parametric approach. In this paper, the parametric distribution type is used for the two-step weight-based Bayesian method [10]. The third step is to calculate from the bootstrap samples, yielding bs* . In the fourth step, the second and third steps are repeated B times (e.g., B=1000). Finally, the fifth step is to construct a probability distribution from 1* , 2* , , B* . This distribution is the bootstrap sampling distribution of , G * * , which is used to calculate the confidence interval of . To obtain the bootstrap sampling distribution of , there are five different methods: the normal approximation, percentile, bias corrected (BC), percentile-t, or bias corrected accelerated (BCa) methods. The normal approximation method assumes that the distribution of is a Gaussian distribution [19]. The percentile method calculates the confidence interval for the parameter based on the bootstrap sampling distribution G * * approximating the population distribution G [20]. When the number of samples is small, G * * might be a biased estimator of G , i.e., * is a biased estimator of . In that case, the percentile method can be inaccurate. The bias corrected (BC) method corrects the bias term by introducing an adjusted parameter [19]. The BC method corrects the bias term, but it still requires the parametric assumption that there exist monotonic transformations of * and . The bias corrected and accelerated (BCa) method generalizes the BC method. The BC method only corrects the bias, whereas the BCa method corrects both the bias and the skewness. The percentile-t method uses the distribution of a standardized estimator to calculate the confidence interval. The percentile-t interval is expected to be accurate to the extent that standardizing depends less on the boot sampling estimator, * , than the percentile method. These five methods are tested using a lognormal marginal distribution as the true model. The confidence level of the standard deviation is assessed by calculating the probability that the upper bound of the confidence interval of the standard deviation is larger than the true standard deviation over M data sets with ns samples generated from the true model. It is the most desirable if the estimated confidence level is close to the target confidence level. However, even if the estimated confidence level is close to the target confidence level, an unnecessarily large upper bound for standard deviation is not desirable because it will yield unnecessarily conservative RBDO design. Therefore, the most desirable confidence interval should just include the true standard deviation for the target confidence level. In summary, when the input marginal distribution is not Gaussian, the parametric percentile method has the most desirable performance. Even though the bootstrap methods do not achieve the target confidence level for a small number of samples, as the number of samples increases, the obtained confidence levels tend to converge to the target confidence level while the method using Gaussian distribution of input variable does not. The bootstrap method can be applied to any types of distribution, and the test results for various types of distributions are presented in Ref. 18. 3.2.3 Confidence Interval of Sample Correlation Coefficient Genest and Rivest [21] showed that, as ns goes to infinity, the sample correlation coefficient follows a Gaussian distribution as 1 dg 1 2 ~ N , 4w (8) ns d 1 where g ( ) is presented in Table 1 in Ref. 9, w 1 ns 1 ns 1 ns 2 wi w i 2w , wi Iij , w i I ji , and ns j 1 ns j 1 ns i 1 1 ns wi . If x1j<x1i or x2j<x2i (x1i or and x2i are ith sample point for X1 and X2), then Iij=1, otherwise, Iij=0. Thus, the ns i 1 confidence interval for the correlation parameter for 100(1) of the confidence level is obtained as w Pr h h 1 where h z / 2 1 ns 4w (9) dg 1 and z / 2 is the CDF value of Gaussian distribution evaluated at / 2 . d Using the lower and upper bounds of the confidence interval for the correlation parameter , the upper and lower bounds of the confidence interval for the correlation coefficient can be calculated from g . The confidence interval of correlation parameter can be accurately estimated regardless of copula function types; and thus, the bootstrap method is not required to obtain the confidence interval of the correlation parameter. 4. Generation of Input Distribution Models with Target Confidence Level To have an input model with a target confidence level, confidence intervals of the input parameters need to be used to offset the prediction error of the estimated input parameters. However, neither the upper nor lower bounds of the confidence intervals of mean and correlation yield reliable design. On the other hand, if the sample mean and sample correlation coefficient are accurate, the upper bound of the confidence interval of the standard deviation fully covers the t-contour, and thus, it can be readily used to obtain the t-contour covering the true t-contour, which will lead to a reliable optimum design. However, the prediction errors in the mean and correlation coefficient exist in RBDO problems when the available data is insufficient. Therefore, instead of using the estimated mean and correlation coefficient, adjusted parameters using the confidence intervals of the mean and correlation coefficient are proposed to ensure that the estimated t-contour with the adjusted parameters covers the true t-contour, which will lead to a desirable confidence level of the input model. Since the confidence interval of the mean cannot be used to enlarge t -contour, the change in the sample standard deviation caused by the change in the sample mean is added to the confidence interval of the standard deviation U to obtain the adjusted standard deviation A as A U U (10) , and U L since the sample mean is the middle point of the confidence interval for the Gaussian distribution. In Eq. (10), the COV ( / ) works as a scale factor such that the effect where it is assumed that of the confidence interval of the mean on the adjusted standard deviation is proportional to / . Using the estimated mean and the adjusted standard deviation, parameters a and b of the identified distribution can be estimated using explicit functions, which are expressed as mean and standard deviation in Ref. 9. Like the mean, the t-contours for the lower and upper bounds of the correlation coefficient do not yield reliable design when the sample standard deviation is not large enough to cover the true contour. To resolve this problem, the adjusted correlation coefficient is used. As stated before, neither the lower nor upper bounds of the confidence interval of correlation coefficient cover the true t-contour. However, when the lower bound of the correlation coefficient with the adjusted standard deviation is used, it yields more reliable design than when the upper bound with the adjusted standard deviation is used. Thus, when the true correlation coefficient is small, the adjusted correlation coefficient needs to be close to the estimated one. Otherwise, the adjusted correlation coefficient needs to be underestimated. Thus, the adjusted correlation coefficient is proposed as (11) A max U , L such that it can be applied to both small and large correlation coefficients. As the number of samples increases, the adjusted correlation coefficient converges to the true correlation coefficient. 5. Numerical Example In this section, a mathematical example with correlated Gaussian and non-Gaussian input variables are used to show how the proposed methods yield reliable designs to meet the target confidence level of the output. To carry out RBDO, the MPP-based DRM [22] is used to more accurately calculate the probability of failure than the FORM. The RBDO problem is formulated to minimize cost(d) d1 d 2 subject to P Gi X 0 PFTar ( 2.275%), i 1, 2,3 i d = μ X , 0 d1 , d 2 10 G1 ( X) 1 0.9010 X 1 0.4339 X 2 1.5 2 (12) 0.4339 X 1 0.9010 X 2 2 / 20 G2 ( X) 1 X 1 X 2 2.8 / 30 X 1 X 2 12 / 120 2 2 G3 ( X) 1 200 / {2.5(0.9010 X 1 0.4339 X 2 3) 2 8 0.4339 X 1 0.9010 X 2 5} First, let X1 and X2 have Gaussian distributions, X1 and X2~(3, 0.32), which are correlated with the Frank copula and = 0.8. For the comparison study, three types of input models – one model with estimated parameters (without confidence level), another model with upper bound for standard deviation (only with confidence level for standard deviation) and sample correlation coefficient, and the other model with adjusted parameters (with confidence levels for all input variables) – are tested. The output confidence levels are assessed using 100 data sets with ns=30, 100, and 300, which are randomly generated from the true input model. The marginal distribution, the copula type, and their parameters are determined from each data set. Using the generated input models from 100 data sets, the RBDO is carried out. The target probability of failure is given as 2.275% and the target confidence level is 97.5%. Table 1 shows the minimum, mean, and maximum values of the probabilities of failure PF1 and PF2 for two active constraints evaluated at the optimum designs using the Monte Carlo simulation (MCS). The output confidence levels are estimated by calculating the probability that the obtained probability of failure is smaller than the target probability of failure. As shown in Table 1, when the input model with the estimated parameters is used for ns=30, the mean values of PF1 and PF2 (3.426% and 2.594%, respectively) are larger than the target probability of failure 2.275%. In particular, the maximum values of PF1 and PF2 (14.29% and 7.752%, respectively) are much larger than 2.275%. Thus, the confidence levels of the output performance are significantly lower than the target confidence level 97.5% as shown in Table 1. On the other hand, when the input model with the upper bound of the standard deviation and sample correlation is used for ns=30, the mean values of PF1 and PF2 (1.074% and 1.562%, respectively) are smaller than 2.275%, which yield more reliable design than those using the estimated parameters. The obtained output confidence levels are improved over those using estimated parameters. However, they have not reached the 97.55% target confidence level. When the input model with the adjusted parameters is used, the obtained output confidence levels using the adjusted parameters become much closer to the target confidence level. Even though it offsets the quantification error of input parameters, the estimated output confidence levels are still smaller than the target confidence level due to the incorrect identification of marginal distribution and copula for a small number of samples, ns=30. However, as the number of samples increases, the minimum, mean, and maximum values of PF1 and PF2 using the input model with the adjusted parameters approaches the target probability of failure from a more reliable side than those using other input models. Moreover, the input model with adjusted parameters provides better output confidence levels than other input models. Table 1. Probabilities of Failure and Output Confidence Levels for X 1 and X 2 ~ N 3, 0.32 Par. ns Min Mean 30 Max Conf. Min Mean 100 Max Conf. Min Mean 300 Max Conf. Estimated PF1 PF2 0.142 3.426 14.29 35 0.640 2.279 5.829 37 0.854 2.394 3.835 43 0.315 2.594 7.752 54 0.516 2.087 5.941 65 1.064 1.924 3.599 77 Upper Std. PF1 PF2 0.007 1.074 5.918 83 0.231 1.425 3.694 89 0.510 1.615 2.846 94 0.119 1.562 4.719 81 0.393 1.573 4.091 88 0.894 1.624 3.150 90 Adjusted PF1 PF2 0.004 0.908 7.082 90 0.192 1.282 3.445 93 0.473 1.518 2.732 95 0.026 0.444 2.595 98 0.255 0.947 2.300 99 0.719 1.262 2.285 98 *Optimum Design Using True Input Model: (2.026, 1.163) ; PF = 2.245%, PF =2.070% 1 2 To test the effect of larger standard deviation on the output confidence level, consider a true input model with X 1 2 2 ~ N (3, 0.3 ) and X 2 ~ N (3,1.5 ) . As shown in Table 2, the input model with the estimated parameters still yields large probabilities of failure and small output confidence levels. The input model with the upper bound for standard deviation and the sample correlation coefficient provides better results than the one with the estimated parameters, but the output confidence levels are not as good as the one with the adjusted parameters, especially for a small number of samples. For ns=300, the output confidence level using the adjusted parameters is larger than the target confidence level of 97.5%, which means it yields somewhat conservative optimum designs. This is because, for large standard deviation (i.e., 1.5 ), the confidence intervals of the mean and standard deviations are slowly converged rather than for small standard deviation. Even though the output confidence levels are high, the minimum, mean, and maximum values of probabilities of failure tend to approach the target probability of failure 2.275%, as the number of samples increases as shown in Table 2. 2 2 Table 2. Probabilities of Failure and Output Confidence Levels for X 1 ~ N (3, 0.3 ) and X 2 ~ N (3,1.5 ) ns Par. Min Mean 30 Max Conf. Min Mean 100 Max Conf. Min Mean 300 Max Conf. Estimated PF1 PF2 0.293 2.583 10.92 58 1.035 2.261 8.804 66 1.140 2.029 3.299 78 0.045 2.895 9.732 48 0.871 2.338 5.058 56 1.238 2.185 3.879 65 Upper Std. PF1 PF2 0.023 0.992 6.961 90 0.507 1.309 6.564 95 0.791 1.514 2.607 98 0.001 1.256 4.723 94 0.296 1.117 2.880 96 0.746 1.468 2.827 98 Adjusted PF1 PF2 0.001 0.313 4.324 96 0.196 0.715 3.385 99 0.530 1.076 1.984 100 0.001 0.380 4.457 96 0.121 0.667 1.985 100 0.512 1.100 2.330 99 *Optimum Design Using True Input Model: (2.977, 4.280) ; PF =2.075%, PF = 2.213% 1 2 In summary, the input model with the adjusted parameters has the most desirable confidence level of output performance compared with input models with the estimated parameters and those with the upper bound for standard deviation and the sample correlation coefficient. Next, let X1 and X2 have lognormal and Gaussian distributions, X 1 ~ LN 3,1.52 and X 2 ~ N 3, 0.32 , which are correlated with the Frank copula and 0.7 . Table 3 shows the minimum, mean, and maximum values of the probabilities of failure PF1 and PF2 for two active constraints evaluated at the optimum designs. As shown in Table 3, when the input model with the estimated parameters is used for ns=30, the mean value of PF1 (4.313%) is larger than the target probability of failure, 2.275%. The maximum value of PF1 (20.842%) is not even close to 2.275% due to the wrong identification and quantification of the input model. Thus, the output confidence level (54%) is significantly smaller than the target confidence level of 97.5%. On the other hand, when the input model with the adjusted parameters is used, the mean values of PF1 and PF2 are smaller than 2.275%. Accordingly, the obtained output confidence levels using the adjusted parameters become much closer to the target confidence level, 97.5%. When the number of samples is small, e.g., ns=30, the estimated output confidence levels using the bootstrap method are still smaller than the target confidence level due to the incorrect identification of marginal distribution and the copula. However, as the number of samples increases, the output confidence levels using the bootstrap method are closer to the target confidence level. Table 3. Probabilities of Failure and Output Confidence Levels Estimated Adjusted (Bootstrap) ns Par. PF1 PF2 PF1 PF2 Min 0.079 0.317 0.000 0.050 Mean 4.313 2.127 0.600 0.995 30 Max 20.842 9.577 10.633 6.176 Conf. 54 73 94 93 Min 0.190 0.454 0.004 0.394 Mean 2.668 1.725 0.562 1.467 100 Max 16.71 4.833 8.277 3.727 Conf. 48 82 97 93 Min 0.700 0.684 0.067 0.865 Mean 2.393 1.695 0.707 1.675 300 Max 4.647 3.275 2.300 2.960 Conf. 49 91 99 96 *Optimum design using true input model is (3.623, 1.770); PF1 = 2.250%, PF2 =1.922% 6. Conclusions In many engineering applications, only insufficient test data are available for input variables due to expensive experimental testing or measuring costs. The input statistical model obtained from the insufficient data could be inaccurate. Thus, the RBDO with confidence level is proposed to offset the inaccurate estimation of the input model by using the adjusted standard deviation and correlation coefficient. If the input variables have Gaussian distribution, the method using the Gaussian distribution of input variables to calculate the confidence interval of standard deviation, which is exact, is used. If not, it yields an inaccurate estimation of the confidence interval of standard deviation. Thus, in this paper, the bootstrap method is proposed to be used to calculate the confidence interval of standard deviation and, thus, the adjusted standard deviation. To offset the inaccurate estimation of the input model, the adjusted standard deviation and correlation coefficient involving confidence intervals for all input parameters (mean, standard deviation, and correlation coefficient) are proposed such that they compensate the inaccurate estimation of the input parameters. Numerical results show that the input models without confidence level and those only with confidence level for standard deviation do not yield desirable confidence levels for the input model and output performance on RBDO results. On the other hand, the input models with adjusted parameters yield desirable input confidence levels, and the obtained RBDO results are considerably reliable, which leads to desirable confidence levels of the output performances. 7. Acknowledgement Research is jointly supported by the Automotive Research Center, which is sponsored by the U.S. Army TARDEC and ARO Project W911NF-09-1-0250. These supports are greatly appreciated. References 1. Socie, D. F., Seminar notes: “Probabilistic Aspects of Fatigue”, 2003, URL: http://www.fatiguecaculator.com. 2. Annis, C., “Probabilistic Life Prediction Isn't as Easy as It Looks,” Journal of ASTM International, Vol. 1, No. 2, pp. 3-14, 2004. 3. Efstratios, N., Ghiocel, D., and Singhal, S., Engineering design reliability handbook, CRC press, New York, 2004. 4. Pham, H., Springer Handbook of Engineering Statistics, Springer, London, 2006. 5. Noh, Y., Choi, K.K., and Lee, I., “Reliability-Based Design Optimization with Confidence Level under Input Uncertainty,” 35th ASME Design Automation Conference, San Diego, CA, Aug.30- Sep. 2, 2009. 6. Haldar, A., and Mahadevan, S., Probability, Reliability, and Statistical Methods in Engineering Design, John Wiley & Sons, New York, 2000. 7. Efron, B., The Jackknife, the Bootstrap, and Other Resampling Plans, SIAM, Philadelphia, 1982. 8. Efron, B., and Tibshirani, R., An Introduction to the Bootstrap, Chapman & Hall/CRC, 1993. 9. Noh, Y., Choi, K.K., and Lee, I., “Identification of Marginal and Joint CDFs Using the Bayesian Method for RBDO,” Structural and Multidisciplinary Optimization, Vol. 40, No. 1, pp. 35-51, 2010. 10. Noh, Y., Choi, K.K., and Lee, I., “Comparison Study between MCMC-based and Weight-based Bayesian Methods for Identification of Joint Distribution.” Structural and Multidisciplinary Optimization, Submitted, to appear, 2010. 11. Nelsen, R.B., An Introduction to Copulas, Springer, New York, 1999. 12. Huard, D., Évin, G., and A.-C. Favre, “Bayesian Copula Selection,” Computational Statistics and Data Analysis, COMSTA3137, Vol. 51, No. 2, pp. 809-822, 2006. 13. Genest, C., Ghoudi, K., and Rivest, L.P., “A Semiparametric Estimation Procedure of Dependence Parameters in Multivariate Families of Distribution,” Biometrika, Vol. 82, No. 3, pp. 543-552, 1995. 14. Hürliman, W., “Fitting Bivariate Cumulative Returns with Copulas,” Computational Statistics & Data Analysis, Vol. 45, No. 2, pp. 355–372, 2004. 15. Roch, O., and Alegre, A. “Testing the Bivariate Distribution of Daily Equity Returns Using Copulas. An Application to the Spanish Stockmarket,” Computational Statistics & Data Analysis, Vol. 51, No. 2, pp. 1312–1329, 2006. 16. Silva, R.S., and Lopes, H.F., “Copula, Marginal Distributions and Model Selection: a Bayesian Note,” Statistics and Computing, Vol. 18, No. 3, pp. 313-320, 2008. 17. Shao, J., and Tu, D., The Jackknife and Bootstrap, Springer, 1995. 18. Bonate, B.L., Phamacokinetic-pharmacodynamic Modeling and Simulation, Springer, 2005. 19. Mooney, C.Z., and Duval, R.D., Bootstrapping: A Nonparametric Approach to Statistical Inference, Sage Publication, 1993. 20. Liu, R.Y., and Singh, K., “Invited Discussion on Theoretical Comparison of Bootstrap Confidence Intervals”, Annals of Statistics, Vol. 16, pp.978-979, 1988. 21. Genest, C., and Favre, A-C., “Everything You Always Wanted to Know about Copula Modeling but Were Afraid to Ask.” Journal of Hydrologic Engineering, Vol. 12, No. 4, pp. 347-368, 2007. 22. Lee, I., Choi, K.K., Du, L., and Gorsich, D., “Inverse Analysis Method Using MPP-Based Dimension Reduction for Reliability-Based Design Optimization of Nonlinear and Multi-Dimensional Systems,” Special Issue of Computer Methods in Applied Mechanics and Engineering: Computational Methods in Optimization Considering Uncertainties, Vol. 198, No. 9-12, pp. 14-27, 2008.