DETC A GENERALIZED MAX-MIN SAMPLE FOR RELIABILITY ASSESSMENT WITH DEPENDENT VARIABLES

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1 Proceedings o the ASME International Design Engineering Technical Conerences & Computers and Inormation in Engineering Conerence IDETC/CIE August 7-,, Bualo, USA DETC- A GENERALIZED MAX-MIN SAMPLE FOR RELIABILITY ASSESSMENT WITH DEPENDENT VARIABLES Sylvain Lacaze Computational Optimal Design o Engineering Systems Aerospace and Mechanical Engineering Department University o Arizona Tucson, Arizona, 87 lacaze@ .arizona.edu Samy Missoum Computational Optimal Design o Engineering Systems Aerospace and Mechanical Engineering Department University o Arizona Tucson, Arizona, 87 smissoum@ .arizona.edu ABSTRACT This paper introduces a novel approach or reliability assessment with dependent variables. In this work, the boundary o the ailure domain, or a computational problem with expensive unction evaluations, is approximated using a Support Vector Machine and an adaptive sampling scheme. The approximation is sequentially reined using a new adaptive sampling scheme reerred to as generalized max-min. This technique eiciently targets high probability density regions o the random space. This is achieved by modiying an adaptive sampling scheme originally tailored or deterministic spaces (Explicit Space Design Decomposition). In particular, the approach can handle any joint probability density unction, even i the variables are dependent. In the latter case, the joint distribution might be obtained rom copula. In addition, uncertainty on the probability o ailure estimate are estimated using bootstrapping. A bootstrapped coeicient o variation o the probability o ailure is used as an estimate o the true error to determine convergence. The proposed method is then applied to analytical examples and a beam bending reliability assessment using copulas. Introduction The calculation o probability o ailure is oten done using sampling techniques such as Monte-Carlo simulations, or using approximations such as moment-based methods (FORM, SORM) [, ]. It is now well known that the computational cost associated with computer simulations as well as the potential complexity o the ailure domain (e.g., nonlinearity o the limit states) are major hurdles to an eicient calculation o probabilities o ailure. For this reason, surrogate-based methods, whereby a limit state is approximated by surrogates such as Kriging [] or Support Vector Machines (SVM) [], are oten used since these approximations are computationally eicient and enable the use o many Monte-Carlo samples. In order to tackle the diiculties stemming rom nonlinear limit-states, adaptive sampling techniques have been developed or Kriging (e.g., Eicient Global Reliability Assessment (EGRA) [], AK-MCS [6]) as or SVM (e.g., Explicit Design Space Decomposition(EDSD) [7], SMART [8]). Loosely presented, these approaches aim at reining the approximation o the limit states by locating and evaluating samples in regions where inormation is needed. I major strides have been achieved in the areas o surrogates and adaptive sampling or reliability assessment, existing approaches are still mostly limited in the important case o dependent variables. Indeed, it was shown in [9] that it is essential to tailor adaptive sampling schemes or reliability assessment when the variables are dependent. This stems rom the act that the approximation o the ailure domain must be accurate in places where the probability densities are the highest. This can only be obtained i the adaptive sampling scheme includes inormation on the joint distribution o the variables. We propose to extend the original EDSD sampling scheme to the case o dependent variables. The original scheme used Copyright c by ASME

2 a max-min sample whose purpose was to populate sparse regions while updating an SVM approximation o the ailure domain [7, ]. This was done by locating the sample on the approximated domain boundary. However, this sampling scheme did not include any inormation on the distributions o random variables. This work generalizes the max-min sample by including the joint probability density unctions. This generalized max-min has proven to be beneicial in the case o independent variables o any distribution []. In this paper, we investigate its use in the case o correlated variables. It is shown that only the joint probability density unction needs to be known. An examples is shown or an analytical joint probabilistic density unction and an example shows the derivations or use o copulas []. This paper is structured as ollows. Section presents the generalization o the max-min sample or independent variables along with its key eatures. Section urther extends the ormulation to dependent variables, either with known joint PDF or using copulas. Section briely describes an additional eature using bootstrapping to assess the error on the estimated probability o ailure due to SVM approximation. Section presents results or an analytical example and a simple mechanical example. Generalized max-min This section introduces the key elements o the proposed approach: a new distance-based adaptive sampling scheme able to account or the joint distribution o the random variables. For the sake o simplicity, it is irst introduced or independent variables. It is subsequently extended to the case o dependent variables.. Classic max-min The generalized max-min is an extension o a previous adaptive sampling scheme reerred to as Explicit Design Space Decomposition (EDSD) [7,]. In its most basic orm, the sample is based on the identiication o a point in the space that maximizes the distance to its closest neighbor while lying on the current approximation o the limit state: x mm = argmax x min j=,...,n s x x ( j) () s.t. s (k) (x) = l i x i u i i =,...,n where N s is the number o existing samples, l i and u i are the lower and upper bounds or the i th random variable. s (k) is the SVM approximation at iteration k (used in EDSD as approximation o the limit state). Figure (a) shows the distribution o such adaptive samples along a given limit state. (a) Classic max-min (b) Generalized max-min FIGURE. Distribution o classic and generalized max-min samples along a given limit state with independent normal random variables. Because EDSD was developed or deterministic spaces, it does not take into account the probabilistic deinition o the problem. More speciically, the distribution o samples shown on Figure (a) would remain the same regardless o the stochastic model. This leads to scalability problems and waste o unction calls during the adaptive sampling process. This is due to the act that a sample that maximizes the distance to its neighbors in low probabilistic content regions does not bring much inormation or estimating the probability o ailure. In this work, we propose a generalization o this sample that takes advantage o the probabilistic deinition o the problem.. Generalization to account or arbitrary joint PDF In order to tackle the aorementioned hurdles, the maxmin problem is reormulated as: x gmm = argmax x n x X (x) min x ( j) () j=,...,n s s.t. s(x) = This ormulation weights the Euclidean distance with probabilistic inormation. Hence, the density o adaptive samples is higher in regions that are statistically important. Note that there are no more side constraints in this deinition. This comes rom the act that the inormation about the domain is embedded within the joint PDF X. Figure (b) shows the distribution o such adaptive samples along a given limit state. The key argument behind this ormulation is that adaptive samples drawn rom: max x n x X (x) min x ( j) () j=,...,n s will actually ollow the distribution X. A proo o concept o the ability o the proposed ormulation to actually ollow the joint Copyright c by ASME

3 where F Xi is the i th marginal CDF. The joint PDF unction is deined as: X (x,...,x n ) = dn F X (x,...,x n ) () dx...dx n n = Xi (x i ) dn C(ν,...,ν n ) (6) dν...dν n i= ν j =F X j (x j ) (a) Classic max-min (b) Generalized max-min FIGURE. Distribution o classic and generalized max-min samples along a given limit state with correlated normal random variables. distribution X is proposed using numerical experiments o up to dimensions in the case o a joint normal PDF. This was done both graphically (Figure ) and numerically by means o a Kolmogorov-Smirnov test at the % signiicance level. This approach has shown promising results or reliability assessment using independent random variables []. The case o dependent variables As introduced in the previous section, the generalized maxmin works with any arbitrary joint PDF, without any assumptions o independence. Hence, as long as dependent joint PDFs can be provided, the proposed approach can be used.. Known joint PDF I an analytical joint PDF is known or the given problem, the ormulation o the generalized max-min can be used in a straightorward manner. An example o correlated normally distributed variables is provided in Figure. Unortunately, only ew analytical dependent joint PDFs exist. In most real world scenarios, it is unlikely that one would be able to it a joint distribution accurately. However, rather recently, copulas have become an attractive tool to model dependency and obtain an approximation o a joint distribution.. Use o copulas Copulas [] have received a large amount o attention in the past years in ields such as economics [], biostatistics [], hydrology [], as well as engineering design [6]. The theoretical oundation comes rom the Sklar s theorem [7]. It essentially states that or any joint CDF F X, there exists a unique copula C such that: F X (x,...,x n ) = C(F X (x ),...,F Xn (x n )) () Maybe the most widely used amilies o copulas are reerred to as elliptical or Archimedean. As long as one is able to compute the copula, the PDF can be obtained numerically through successive inite dierences. In the case o elliptical copulas, the derivation o the joint PDF can be carried out analytically. For example, consider a Gaussian copula deined as: C(ν,...,ν n ) = Φ R (Φ (ν ),...,Φ (ν n )) (7) where R is a correlation matrix. By carrying out the derivation, we obtain: X (x,...,x n ) = n i= [ ] Xi (x i ) ( φ (Φ φ (ν i )) R Φ (ν i ) ) (8) where ν i = F Xi (x i ). This process is illustrated on Figure. Estimation o the probability o ailure Recall that the proposed approach is an iterative process, or which at each iteration k, a new SVM is constructed and then reined by the addition o a generalized max-min sample. At iteration k, the probability o ailure is estimated using Monte Carlo simulations such that: P (k) [ [ ]] = E I s (k) (x) 6 Monte Carlo samples are used so that the error is minimal. However, the error on the estimated probability o ailure due to the SVM approximation o the ailure domain should not be ignored. In order to account or it, bootstrapping [8] is used. This is achieved by bootstrapping the training samples and obtaining a new SVM or each bootstrap. More speciically, at iteration k, let x (k) = [x (),...,x (Ns) ] be the training samples set. A bootstrap training set x (k, j) is obtained, a new SVM s (k, j) (x) is constructed on this new training set and a new probability o ailure P (k, j) = E [ I [ s (k, j) (x) (9) ]] is estimated. This process is repeated n bs times (i.e., j = [,...,n bs ]). Based on the bootstrap sample o the probability o ailure P (k, ) = [P (k,),...,p (k,n bs) ], Copyright c by ASME

4 FX i, Φ FXi, Φ FX i, Φ (a) D Xi Xi Xi (b) D (c) D FIGURE. Graphical proo o concept o the ability o the generalized ormulation to ollow a given joint distribution. Empirical marginal CDFs FXi compared to standard normal CDF Φ (red thick dotted line) or,, and dimensions based on generalized max-min samples. Y = T (X) X Z = T (Y) (Yi = FX i (Xi ))! " Zi = Φ (Yi ).8 Z Y X X (X) = Q.6.8 Y X Xi (X i ) φ[φ (F Xi (X i ))] 6 Y (Y) = φr [T (T (X))] Q Z - 6 dφ dyi -Y φr (T (Y)) Z (Z) = φr (Z) i.8 Z Y X FIGURE..6.8 Y X Z Graphical illustration o the derivation o the joint PDF or a Gaussian copula. an empirical conidence interval can be derived. In addition, the (k,?) median o the repetition P is used as the estimated proba- Results The proposed approach is applied to an analytical example and a cantilever beam. For the sake o comparison, the results using the generalized max-min are compared to results using the classic max-min sample. In addition, to motivate the idea o (k) bility or iteration k. Finally, the coeicient o variation cvbs o (k,?) P is used as an estimate o the error due to SVM. In this work nbs = repetitions and a 9% conidence interval are used. Copyright c by ASME

5 Limit State PDF X P X FIGURE. Probability density unction and limit state or the demonstrative example. using the coeicient o variation cv (k) bs o the bootstrapped P (k, ) as an estimate o the error, it is compared to the true absolute relative error deined as: P P (k) ε (k) = () P. Demonstrative example: Two dimensional correlated Gaussian variables The irst example used in this paper eatures a complex D limit state deined as: 6 sgn(x ) x x sgn(x ) () where sgn is the sign or signum unction. For this example, the random variables X and X ollow a Gaussian correlated joint PDF with ρ =.7. Figure provides an overview o the stochastic space along with the true limit state. Figure 6 shows the convergence o the estimated probability o ailure or the two dierent schemes. Figure 7 shows the evolution o the true error ε along with the bootstrapped coeicient o variation cv bs when generalized max-min samples are used or the reinement scheme. In addition, Table reports values o 6 Iteration Classic max-min 9% CI Generalized max-min 9% CI P FIGURE 6. Convergence o the estimated probability o ailure using two adaptive sampling schemes, classic or generalized max-min sample or the demonstrative example. the estimated probability o ailure at iterations, 6, and. As a reerence, the conidence interval o the Monte Carlo simulation itsel on the true limit state is provided. It can clearly be seen that using the generalized max-min drastically improves the convergence o the algorithm in addition to the coeicient o variation. When the generalized max-min sample is used, a coeicient o variation below % is irst achieved at iteration 8 and below % at iteration. The actual errors made on the estimated probability o ailure at these iterations are respectively.% and.9%.. Cantilever Beam: Three dimensional example using copula. PL EI () The marginal distributions o b, h and P are shown in Table. The copula or this problem is a Gaussian with ρ = ρ = ρ =.7. A scatter plot o the projected dependent joint PDF is provided on Figure 9. Figure shows the convergence o the estimated probability o ailure or the two dierent schemes. Figure shows the Copyright c by ASME

6 TABLE. Estimated probability o ailure (i.e.,. quantile) and associated 9% CI at iteration, 6 and or both scheme using either classic or generalized max-min or the demonstrative example. Quantile Classic Generalized Monte Carlo samples cv bs (%) ε (%) Obtained using independent Monte Carlo simulation Obtained using (??) h P P b 6 8 b 6 8 h FIGURE 9. Scatter plot o the dependent joint distributions or the cantilever beam. TABLE. beam. Marginal distribution o the parameters or the cantilever Parameter Distribution b (mm) lnn (m =.9,v =.6) h (mm) Γ(α =,β =.) P (N) Weibull(a =,b = ) evolution o the true error ε along with the bootstrapped coeicient o variation cv bs when generalized max-min samples are used or the reinement scheme. In addition, Table reports values o the estimated probability o ailure at iterations, 6 and. For reerence, the conidence interval o the Monte Carlo simulation itsel on the true limit state is provided. Once again, a aster convergence is observed when the generalized max-min sample is used. When the generalized max-min sample is used, a coeicient o variation below % is irst achieved at iteration and below % at iteration 9. The actual errors made on the estimated probability o ailure at these iterations are respectively.% and.%. 6 Conclusion In this paper, a novel adaptive sampling scheme or reliability assessment with any PDF, including dependent ones, was introduced. Promising results have been shown on two test cases. In addition, a bootstrapping-based conidence interval was derived or the estimated probability o ailure, in order to account or the error due to the SVM approximation. The main ocus o uture work will be to derive a ormal convergence criterion and investigate some possible tighter bounds or the conidence interval. 6 Copyright c by ASME

7 TABLE. Estimated probability o ailure (i.e.,. quantile) and associated 9% CI at iteration, 6 and or both schemes using either classic or generalized max-min or the cantilever beam. Quantile Classic Generalized Monte Carlo samples cv bs (%) ε (%) Obtained using independent Monte Carlo simulation Obtained using (??) 9 ε cv BS 8 7 cv BS / ε (%) 6 P 6 8 Iterations FIGURE 7. Evolution o the true error ε along with the bootstrapped coeicient o variation cv bs or the generalized max-min scheme or the demonstrative example. 6 Iteration Classic max-min 9% CI Generalized max-min 9% CI P FIGURE. Convergence o the estimated probability o ailure using two adaptive sampling schemes, classic or generalized max-min sample or the cantilever beam. E P b h ACKNOWLEDGMENT Support rom the National Science Foundation (award CMMI-97) is grateully acknowledged. FIGURE 8. L Description o the cantilever beam. REFERENCES [] Melchers, R. E., and Melchers, R. E., 999. Structural reliability analysis and prediction. 7 Copyright c by ASME

8 cv BS / ε (%) Iterations ε cv BS FIGURE. Evolution o the true error ε along with the bootstrapped coeicient o variation cv bs or the generalized max-min scheme or the cantilever beam. [] Lemaire, M.,. Structural reliability, Vol. 8. John Wiley & Sons. [] Cressie, N., 99. The origins o kriging. Mathematical Geology, (), pp. 9. [] Vapnik, V.,. The nature o statistical learning theory. Springer Verlag. [] Bichon, B., Eldred, M., Swiler, L., Mahadevan, S., and Mc- Farland, J., 8. Eicient global reliability analysis or nonlinear implicit perormance unctions. AIAA journal, 6(), pp [6] Echard, B., Gayton, N., and Lemaire, M.,. Akmcs: An active learning reliability method combining kriging and monte carlo simulation. Structural Saety, (), pp.. [7] Basudhar, A., and Missoum, S.,. An improved adaptive sampling scheme or the construction o explicit boundaries. Structural and Multidisciplinary Optimization, (), pp [8] Bourinet, J.-M., Deheeger, F., and Lemaire, M.,. Assessing small ailure probabilities by combined subset simulation and support vector machines. Structural Saety, (6), pp.. [9] Jiang, P., Basudhar, A., and Missoum, S.,. Reliability assessment with correlated variables using support vector machines. In Proceedings o the nd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conerence. [] Basudhar, A., Missoum, S., and Harrison Sanchez, A., 8. Limit state unction identiication using support vector machines or discontinuous responses and disjoint ailure domains. Probabilistic Engineering Mechanics, (), pp.. [] Lacaze, S., and Missoum, S.,. A generalized maxmin sample or surrogate update. Structural and Multidisciplinary Optimization, pp.. [] Nelsen, R. B., 6. An introduction to copulas. Springer. [] Frees, E. W., and Valdez, E. A., 998. Understanding relationships using copulas. North American actuarial journal, (), pp.. [] Li, D.,. On deault correlation: a copula unction approach. Journal o Fixed income, 9(), pp.. [] Dupuis, D., 7. Using copulas in hydrology: Beneits, cautions, and issues. Journal o Hydrologic Engineering, (), pp [6] Noh, Y., Choi, K., and Du, L., 9. Reliability-based design optimization o problems with correlated input variables using a gaussian copula. Structural and multidisciplinary optimization, 8(), pp. 6. [7] Sklar, A., 99. Fonctions de répartition à n dimensions et leurs marges. Publications o the Institute o Statistics, University o Paris, 8, pp. 9. [8] Eron, B., and Tibshirani, R., 99. An introduction to the bootstrap, Vol. 7. CRC press. 8 Copyright c by ASME

Reliability Assessment with Correlated Variables using Support Vector Machines

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