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Post by Steve Yenisch on May 13, 2009 22:41:41 GMT -5
Paper: www.mae.ufl.edu/nkim/ISSMO/YoojeongNohPaper.pdfPresentation: www.mae.ufl.edu/nkim/ISSMO/YoojeongNohPresentation.pptxABSTRACTFor obtaining correct reliability-based optimum design, an input model needs to be accurately estimated in identification of marginal and joint distribution types and quantification of their parameters. However, in most industrial applications, only limited data on input variables is available due to expensive experimental testing cost. The input model generated from the insufficient data will be inaccurate, which will lead to incorrect optimum design. In this paper, reliability-based design optimization (RBDO) with the confidence level is proposed to offset the inaccurate estimation of the input model due to limited data by using the upper bound of confidence interval of the standard deviation. Using the upper bound of confidence interval of the standard deviation, a confidence level on the input model can be assessed to obtain the confidence level of the output performance, i.e. a desired probability of failure, through the simulation-based design. For RBDO, the estimated input model with the associated confidence level is integrated with the most probable point (MPP)-based dimension reduction method (DRM), which improves accuracy over the first order reliability method (FORM). A mathematical example and a fatigue problem are used to illustrate how the input model with the confidence level yields a reliable optimum design by comparing it with the input model obtained using the estimated parameters. |
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Post by haftka on May 18, 2009 16:07:23 GMT -5
I am having problem with the presentation because many of the symbols do not show accurately on my computer.
For the mathematical example, when I see huge variation in the probability of failure as in Table 7, what does the designer do with this type of information?
Also, for this example, if you treated the marginal and joint distribution as Gaussian, would there be much difference in the results?
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Post by Yoojeong Noh on May 20, 2009 16:25:08 GMT -5
I am sorry to make you difficult to read it. I will send a new presentation file to you.
The reason that Table 7 has the huge variation in the probability of failure is because the input model is estimated from a small number of samples and has the high correlation. When we have a small number of samples samples, the upper bound of the confidence interval of the standard deviation will be very high, which will lead to very small probability of failure. In addition, when the correlation coefficient is high, the calculation of probability of failure is very sensitive to the estimated input model. Thus, wrong input model sometimes yields large probability of failure. Thus, to obtain reliable and accurate RBDO result, the designer needs to increase the number of samples.
As you mentioned, if I use the marginal and joint distribution as Gaussian, the result will be different, but it will not be much different from Table 7 because the Gaussian copula shape is not much different from Frank copula. Likewise, the Gaussian distribution (margin) is also not much different from lognormal distribution in this example. However, if a copula or marginal distribution with very distinct shapes are used, the output confidence level will be higher than Table 7 because the error in identification of marginal distribution or copula will be reduced.
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Post by Ben Smarslok on May 21, 2009 9:06:45 GMT -5
Interesting paper. It's nice to see the sensitivity of the distribution (or copula) fitting presented this way. The confidence level discussion is very appropriate and informative.
Building on the idea of limited data, could a scheme be developed to use the confidence level to get an idea of the epistemic error in the tests? Then this error could be reintroduced into the probability of failure calculation. Or does this inherently already exist?
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Post by luiscelorrio on May 21, 2009 13:06:15 GMT -5
Dear Yoojeong Noh
Thank you for your interesting papers. I have recently arrived to the Structural Reliability and RBDO world. You have used in the Roadarm Example 30 paired data generated from an assumed true input model. I think that it might be very difficult to obtain experimental data. I have proposed correlated loads in my paper. Do you know another applications with correlated random input vectors?. Here, you have two + two correlated random variables. In the case you have three or more correlated input random variables, can you generalize the Frank copula or other type of copula to model this n-dimensional random vector?. Maybe, Gaussian copula is the most suited choice in this case.
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Post by Ikjin Lee on May 22, 2009 9:33:22 GMT -5
Thank you for the question, but I could not understand your question. Especially, "epistemic error". If you mean the distribution types of input, we assume that there exists error for the identification of input distribution types. The proposed method is also tested for that kind of error. Interesting paper. It's nice to see the sensitivity of the distribution (or copula) fitting presented this way. The confidence level discussion is very appropriate and informative. Building on the idea of limited data, could a scheme be developed to use the confidence level to get an idea of the epistemic error in the tests? Then this error could be reintroduced into the probability of failure calculation. Or does this inherently already exist? |
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Post by Ikjin Lee on May 22, 2009 9:39:30 GMT -5
Since we use durability analysis for the structural design optimization, so far fatigue material properties are the only ones we used as correlated random variables. Darrel Socie showed that fatigue material properties are correlated in pairs using experimental data. Theoretically, we can n-dimensional copula, but in the case we have to assume that correlation coefficient for n-dimensional variable must be the same. For example, if 3 variables (x1,x2,x3) are correlated, then, we have to assume that correlation coefficient between x1 and x2 and correlation coefficient between x2 and x3 are identical, which may not be the case. Hence, we just use two correlated random variable case. However, in this case, the number of pairs is not a problem. Thank you. Dear Yoojeong Noh Thank you for your interesting papers. I have recently arrived to the Structural Reliability and RBDO world. You have used in the Roadarm Example 30 paired data generated from an assumed true input model. I think that it might be very difficult to obtain experimental data. I have proposed correlated loads in my paper. Do you know another applications with correlated random input vectors?. Here, you have two + two correlated random variables. In the case you have three or more correlated input random variables, can you generalize the Frank copula or other type of copula to model this n-dimensional random vector?. Maybe, Gaussian copula is the most suited choice in this case. |
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Post by Yu Liu on May 29, 2009 23:04:41 GMT -5
Dear Yoojeong,
Your interesting work attracts me. I have a question for you. In Section 5.2 you only give the formulation of the confidence Interval of standard divivation for the case of Gaussian random variable, but in the following example the mariginal distribution includes various kinds. How to get the upper bound? I did not find out some relevent statement in your paper. Is there formulation for the standard diviation confidence? Or you use simulation method to reach it?
Thanks
Best.
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Post by Yoojeong Noh on Jun 1, 2009 16:30:58 GMT -5
Thank you for the question. Like the copulas, the marginal distributions also have their own parameters and those parameters can be explicitly expressed as functions of mean and standard deviation in most cases. Once the upper bound of the confidence interval of the standard deviation and the mean are calculated from samples, the upper bound of the confidence interval of those parameters can be obtained using the explicit functions, which are shown in the following paper. Yoojeong Noh, K.K. Choi, Ikjin Lee, Identification of Marginal and Joint CDFs Using Bayesian Method for RBDO, SMO Journal, 2009, to appear. Even though the parameters are different according to distribution types, the upper bound of confidence interval of the standard deviation always yields a large PDF contour for a target reliability index. Thank you. Yoojeong Noh Dear Yoojeong, Your interesting work attracts me. I have a question for you. In Section 5.2 you only give the formulation of the confidence Interval of standard divivation for the case of Gaussian random variable, but in the following example the mariginal distribution includes various kinds. How to get the upper bound? I did not find out some relevent statement in your paper. Is there formulation for the standard diviation confidence? Or you use simulation method to reach it? Thanks Best. |
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Post by Yu Liu on Jun 1, 2009 19:39:05 GMT -5
Dear Yoojeong, Thank you for your reply. I think maybe I did not describe my question clear in last message. My question is the Eq.(21) is only for Guassian random variable, if it is not Guassian, say exponential, Weibull etc., how to get the result like Eq.(22),(23)or(24) which are the upper and lower bounds of the standard deviation. Thank you! Kind Thank you for the question. Like the copulas, the marginal distributions also have their own parameters and those parameters can be explicitly expressed as functions of mean and standard deviation in most cases. Once the upper bound of the confidence interval of the standard deviation and the mean are calculated from samples, the upper bound of the confidence interval of those parameters can be obtained using the explicit functions, which are shown in the following paper. Yoojeong Noh, K.K. Choi, Ikjin Lee, Identification of Marginal and Joint CDFs Using Bayesian Method for RBDO, SMO Journal, 2009, to appear. Even though the parameters are different according to distribution types, the upper bound of confidence interval of the standard deviation always yields a large PDF contour for a target reliability index. Thank you. Yoojeong Noh Dear Yoojeong, Your interesting work attracts me. I have a question for you. In Section 5.2 you only give the formulation of the confidence Interval of standard divivation for the case of Gaussian random variable, but in the following example the mariginal distribution includes various kinds. How to get the upper bound? I did not find out some relevent statement in your paper. Is there formulation for the standard diviation confidence? Or you use simulation method to reach it? Thanks Best. |
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Post by Yoojeong Noh on Jun 5, 2009 14:40:13 GMT -5
Dear Yu Liu, I am sorry to give the wrong answer. In my paper, the confidence interval of the standard deviation is calculated using a function “normfit” in MATLAB, which uses Eqs. (22)-(24) even though the data follow nonnormal distributions. In that case, the normality assumption will be violated in that case as you mentioned. However, through simulation studies, we have found that even if the input variables follows nonnormal distributions such as weibull and lognormal, the confidence interval includes the true standard deviation with reasonable target confidence levels. I will try to find refereces dealing with confidence interval of the stnadard deviation for nonnormal distributions. Thank you for your question. Dear Yoojeong, Thank you for your reply. I think maybe I did not describe my question clear in last message. My question is the Eq.(21) is only for Guassian random variable, if it is not Guassian, say exponential, Weibull etc., how to get the result like Eq.(22),(23)or(24) which are the upper and lower bounds of the standard deviation. Thank you! Kind Thank you for the question. Like the copulas, the marginal distributions also have their own parameters and those parameters can be explicitly expressed as functions of mean and standard deviation in most cases. Once the upper bound of the confidence interval of the standard deviation and the mean are calculated from samples, the upper bound of the confidence interval of those parameters can be obtained using the explicit functions, which are shown in the following paper. Yoojeong Noh, K.K. Choi, Ikjin Lee, Identification of Marginal and Joint CDFs Using Bayesian Method for RBDO, SMO Journal, 2009, to appear. Even though the parameters are different according to distribution types, the upper bound of confidence interval of the standard deviation always yields a large PDF contour for a target reliability index. Thank you. Yoojeong Noh |
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Post by mqfsl on Aug 18, 2009 20:20:01 GMT -5
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