Reliability assessment using probabilistic support vector machines. Anirban Basudhar and Samy Missoum*

Transcription

1 156 Int. J. Reliability and Saety, Vol. 7, o. 2, 2013 Reliability assessment using probabilistic support vector machines Anirban Basudhar and Samy Missoum* Aerospace and Mechanical Engineering Department, The University o Arizona, Tucson, AZ 85721, USA abasudhar@lstc.com smissoum@ .arizona.edu *Corresponding author Abstract: This paper presents a methodology to calculate probabilities o ailure using robabilistic Support Vector Machines (SVMs). Support Vector Machines (SVMs) have recently gained attention or reliability assessment because o several inherent advantages. Speciically, SVMs allow one to construct explicitly the boundary o a ailure domain. In addition, they provide a technical solution or problems with discontinuities, binary responses, and multiple ailure modes. However, the basic SVM boundary might be inaccurate; thereore leading to erroneous probability o ailure estimates. This paper proposes to account or the inaccuracies o the SVM boundary in the calculation o the Monte Carlo-based probability o ailure. This is achieved using a SVM which provides the probability o misclassiication o Monte Carlo samples. The probability o ailure estimate is based on a new sigmoidbased SVM model along with the identiication o a region where the probability o misclassiication is large. The SVM-based probabilities o ailure are, by construction, always more conservative than the deterministic SVM-based probability estimates. Keywords: probabilistic support vector machines; probability o misclassiication; probability o ailure; discontinuous and binary behaviour; multiple ailure modes. Reerence to this paper should be made as ollows: Basudhar, A. and Missoum, S. (2013) Reliability assessment using probabilistic support vector machines, Int. J. Reliability and Saety, Vol. 7, o. 2, pp Biographical notes: Anirban Basudhar is an Optimisation Sotware Developer at Livermore Sotware Technology Corporation (LSTC), Livermore, CA, USA. He received his hd in Mechanical Engineering rom the University o Arizona. Samy Missoum is an Associate roessor in the Aerospace and Mechanical Engineering Department at the University o Arizona. He received his hd in Mechanical Engineering rom the ational Institute o Applied Sciences, Toulouse, France. His research interests include design optimisation and design under uncertainty. Copyright 2013 Inderscience Enterprises Ltd.

2 Reliability assessment using probabilistic SVM Introduction The need or reliable, increasingly complex systems has led to a growing emphasis on eicient methods or design under uncertainty. As a result, reliability assessment techniques have been steadily improving. However, it is well understood that the calculation o a ailure probability is hampered by hurdles such as large computational or experimental costs, discontinuous and binary system behaviours, and multiple ailure modes (Missoum et al., 2007, Layman et al., 2010; Arenbeck et al., 2007). Among reliability assessment methods, moment-based methods have been widely used. These include irst- and second-order reliability methods (Haldar and Mahadevan, 2000), advanced mean value method (Youn et al., 2005), and two-oint Adaptive onlinear Approximation (TAA) (Wang and Grandhi, 1995). These methods can, however, produce signiicant errors i the limit-state unctions are highly non-linear (Bichon et al., 2007). In the last two decades, metamodelling and response surace techniques have gained prominence or the calculation o ailure probabilities (Mourelatos et al., 2006; Simpson et al., 2008). In this approach, an approximation o the system responses is constructed based on a design o experiments. Adaptive sampling schemes to reine the approximations have also been developed (Bichon et al., 2007; Wang et al., 2005; Huang et al., 2006). While these methods can eiciently handle non-linear limit-state unctions, they are limited by several other diiculties. For instance, the presence o discontinuous (Missoumet al., 2007; Basudhar et al., 2008; Basudhar and Missoum, 2009a; Basudhar and Missoum, 2009b) and binary (pass or ail) (Layman et al., 2010) responses poses diiculties or these approximation methods. In addition, the presence o multiple ailure modes (Basudhar and Missoum, 2009a; Basudhar and Missoum, 2009b; Arenbeck et al., 2007) also presents a challenge. Recently, an alternative sampling-based method was developed by the authors to overcome these diiculties. The approach, reerred to as the Explicit Design Space Decomposition (EDSD) (Basudhar et al., 2008), classiies the responses as sae or ailed. Because there is no response approximation, the EDSD method can be used irrespective o the presence o discontinuous or binary behaviour (Basudhar et al., 2008; Layman et al., 2010). A machine learning technique reerred to as Support Vector Machines (SVMs) (Vapnik, 1998; Gunn, 1998) is used to construct the explicit boundary that separates the sae and ailed samples. EDSD is also useul or multiple ailure modes because a single SVM can be used to represent the boundary o the ailure domain (Arenbeck et al., 2007). Adaptive sampling schemes have been developed to construct accurate SVM approximations using moderate number o unction evaluations (Basudhar and Missoum, 2008; Basudhar and Missoum, 2010). Once the approximate boundary o the ailure region has been built, ailure probabilities can be eiciently assessed using Monte Carlo Simulations (MCS) (Basudhar et al., 2008). Although the recently developed methods have provided a welcome lexibility in reliability assessment, local or global inaccuracies in the SVM-based approximation o the limit-state unction might lead to substantial errors in the ailure probability. For this reason, this paper proposes to quantiy the accuracy o the SVM classiication model and propagate this inormation to the calculation o the probability o ailure. More speciically, the probability o ailure is calculated by weighing the traditional indicator unction o the Monte Carlo samples by a probability o misclassiication. This

3 158 A. Basudhar and S. Missoum probability o misclassiication is only calculated within a region o the sae region based on the spatial distribution o the training samples. Thereore, the modiied, MCS-based, measure o probability o ailure is constructed so that it is always more conservative compared to the probability o ailure based on a deterministic SVM, thus mitigating the consequences o an inaccurate SVM approximation. The probability o misclassiication is obtained using a robabilistic Support Vector Machine (SVM). This paper proposes necessary enhancements to existing SVM models (Vapnik, 1998; Wahba, 1999; latt, 1999). The proposed new SVM model, reerred to as distance-based SVM (DSVM), accounts or the spatial distribution o data points. The proposed method is tested on two analytical examples. The main objective o the examples is to compare the probability o ailure estimates to the actual probability but also to the case where the SVM is deterministic. This study is carried out or various designs o experiment sizes. The organisation o the paper is as ollows. A review o the basic SVM-based reliability assessment methodology is presented in Section 2. Section 3 presents the proposed SVM-based reliability assessment method. The ormulation o the ailure probability accounting or the probability o misclassiication by the SVM is explained. Section 4 provides an overview o the current SVM models and their limitations. In Section 5, the modiied DSVM model proposed to overcome these limitations is presented. An error measure to compare the SVM models is provided in Section 6. Finally, analytical test examples are presented in Section 7 to show the eicacy o the proposed method, ollowed by a discussion in Section 8. 2 Reliability assessment using support vector machines This section provides a review o the previously developed method or reliability assessment using SVMs (Vapnik, 1998; Gunn, 1998). SVM is a machine learning technique that is widely used or the classiication o data. In the context o reliability assessment, any given coniguration or sample in the space is either sae or ailed. Thereore, the deinition o the ailure domain can easily be treated as a binary classiication problem (Basudhar et al., 2008; Hurtado, 2004). An SVM is used to deine the explicit boundary o the ailure region that separates the sae and ailed samples: s( x) = b y K( x, x ) = 0 (1) i i i i=1 Here, x i is the i-th training sample, i is the corresponding Lagrange multiplier, is the number o training samples, K is a kernel unction, y i is the class label corresponding to x i that can take values +1 and 1, and b is the bias. The optimal SVM boundary is constructed by maximising the margin between s( x )= 1 and s( x )= 1 in a eature space (Vapnik, 1998), where the boundary is linear. I a so-called hard classiier is used, none o the training samples can lie within the margin. Once the SVM is constructed, the class o any point x is predicted based on the sign o s(x). I s( x )> 0 (resp. s( x )< 0), it is classiied as +1 (resp. 1).

4 Reliability assessment using probabilistic SVM 159 The main steps o the reliability assessment procedure using an SVM are as ollows. Sampling o the space using a Design o Experiments (DOE) (Montgomery, 2005; Beachkoski and Grandhi, 2002; Romero et al., 2006). Classiication o the samples into two classes sae and ailed. The convention is to denote the sae samples as +1 and the ailed ones as 1. Construction o the SVM limit-state boundary using equation (1). Calculation o the probability o ailure using MCS (Melchers, 1999) based on the SVM boundary. The region s( x ) 0 is considered as the ailure region. The probability o ailure is = = ( x ), (2) 1 MCS Ig i MCS MCS i=1 where is the number o MCS samples classiied as ailure by the SVM classiier [equation (1)] and MCS is the total number o MCS samples. I g (x) is an indicator unction given as: I g x 1 s( x) 0 = 0 s ( x )> 0 An example o ailure probability calculation using SVM is shown in Figure 1. (3) Figure 1 Calculation o the ailure probability using an SVM (let). Misclassiication o the MCS samples by the SVM (right) (see online version or colours) The calculation o probability o ailure using the SVM-based method has several advantages. It is applicable to a wider range o problems compared to the current methods. These include problems with discontinuous and binary responses (Basudhar et al., 2008; Layman et al., 2010), and multiple ailure modes (Arenbeck et al., 2007). Treating the reliability assessment process as a classiication problem allows one to easily handle these diiculties.

5 160 A. Basudhar and S. Missoum The construction o the SVM limit-state unction is based on a DOE. Thereore, there is, in general, an error associated with the approximation o the boundary. This can result in an inaccurate probability o ailure (Figure 1). The ollowing section introduces the use o SVMs to estimate the probability o ailure. 3 Reliability assessment using robabilistic Support Vector Machines (SVMs) This section presents a method or calculating the probability o ailure that accounts or the error associated with the SVM approximation. To do so, the probability o ailure, evaluated using MCS, is calculated based on the probability o misclassiication o the Monte Carlo samples in addition to the probability density unctions o the variables. The probability o misclassiication is calculated using a SVM (latt, 1999; Vapnik, 1998). A deterministic SVM (Section 2) only provides a binary classiication. However, a SVM provides the probability that a particular coniguration (or sample) will belong to a speciic class (+1 or 1). This probability, which is the conditional probability o belonging to the +1 (resp. 1) class is denoted as ( 1 x ) (resp. ( 1 x )). It is noticed in equation (3) that the indicator unction I g (x) can be interpreted as the probability o being in the ailure class based on a deterministic SVM boundary. Thereore, the probability o being in the ailure class 1 or any Monte Carlo sample is either 0 or 1. It is equal to 0 or a sample lying in the sae or +1 class and equal to 1 or a sample lying in the ailure or 1 class. The use o SVM allows one to replace the binary indicator unction by ( 1 x i ), thus leading to: SVM 1 MCS = 1 MCS i=1 x i (4) A relatively conservative measure o the probability o ailure is obtained i the probability o misclassiication is considered only or the Monte Carlo samples belonging to the sae class, that is, or s( x ) > 0: SVM 1 MCS = 1 i, x MCS i=1 1 s( x) 0 1 x= 1 x s( x) > 0 In equation (5), the probability o misclassiication is considered only or the samples lying in the sae domain based on the SVM s( x ) > 0. Thereore, it would naturally provide a probability o ailure that is greater than the one using the deterministic SVM [equation (2)]. However, the ailure probability estimate using equation (5) may be over- 1 x or the entire sae conservative. Thereore, instead o considering a non-zero domain, it is reasonable to consider it only in the regions with high probability o misclassiication. Such regions can be identiied as the ones lacking data in the vicinity o the SVM boundary. Thereore, the region misc or considering the probability o misclassiication is a sub-region o the s( x ) > 0 regions. It is deined based on the (5)

6 Reliability assessment using probabilistic SVM 161 distances to the closest +1 and 1 samples and the SVM margin, which does not contain any samples. Outside this region, the classiication provided by the deterministic SVM is trusted (i.e. ( 1 x ) is either 1 or 0). The region misc or considering the probability o misclassiication by the SVM is misc = sd s( x) <1 ( d( x) d( x ) 0), (6) where sd is the sae domain based on the deterministic SVM, and d + (x) and d (x) are the distances o x to the closest +1 and 1 training samples. misc consists o two kinds o regions in the +1 class. One is the SVM margin in the sae class and another is the region with d ( x) d ( x ) (Figure 2). The probability o ailure is given as: SVM 1 MCS = 1, x i MCS i=1 1 x 1 x= 0 xsd 1 x xmisc misc (7) Figure 2 Deinition o the region misc or considering the probability o misclassiication. misc is the union o the two shaded regions in the let and the right igures (see online version or colours) The calculation o the ailure probability using equation (7) requires the calculation o ( 1 x ) in the region misc. The details o the SVM models or calculating ( 1 x ) or ( 1 x ) are presented in the ollowing sections. 4 Traditional sigmoid-based SVM model robabilistic support vector machines are used to map the SVM values to the probability o belonging to a speciic class. One o the commonly used SVM models proposed by latt (1999) is based on the itting o a sigmoid unction. Some other models have also been proposed by Wahba (1999) and Vapnik (1998). The sigmoid SVM model is presented in the ollowing section.

7 162 A. Basudhar and S. Missoum 4.1 Sigmoid model or robabilistic Support Vector Machines (SVMs) The sigmoid SVM model proposed by latt is presented. In this model, the conditional probability ( 1 x ) is represented as a unction o two parameters A and B: 1 x (8) e ( 1 )= 1 As ( x) B The parameters A (A < 0) and B o the sigmoid unction are ound using maximum likelihood. The conditional probability o correctly classiying the sample x i is 1 x i i it belongs to the +1 class and is equal to 1 1 x i i it belongs to the 1 class. The inormation available to train the SVM is the class o the training samples. According to equation (8), the likelihood that all the training samples are classiied correctly is a unction o A and B: p n i i LAB (, )= 1 x 1 1 x, (9) i=1 i=1 where p is the number o +1 sample and n is the number o 1 sample. A good SVM model should provide a high likelihood o correctly classiying all the samples. Thereore, the parameters A and B are ound by maximising L(A, B). In the work o latt (1999), the log o the likelihood unction was used as: min AB, tlog i ( 1 xi) 1 tilog1 ( 1 x i), (10) i where is the number o training samples and t i is given as: yi 1 ti =, 2 where y i is the class labels. Thus, t = 1 or the +1 samples and t = 0 or the samples belonging to the 1 class. 4.2 Limitations o the basic sigmoid model One o the limitations o the model mentioned in Section 4.1 is that it depends only on the SVM values and not on the spatial distribution o the samples. As a result, i the classes o the evaluated samples are considered deterministic, it does not satisy one o the conditions that requires ( 1 x ) to be either 0 or 1 at these samples. Instead, it provides a non-zero probability o misclassiication (e.g. ( 1 x )> 0 or s( x )< 0) even or the samples that have already been evaluated. An example o the probability o misclassiication misc using latt s sigmoid model is shown in Figure 3. It is seen that the probability o misclassiication is high even or regions that are ar rom the boundary and consist o already evaluated samples. (11)

8 Reliability assessment using probabilistic SVM 163 Figure 3 Map o the probabilities o misclassiication using latt s sigmoid model (see online version or colours) 5 Improved Distance-based robabilistic Support Vector Machines (DSVMs) using a modiied sigmoid model The biggest limitation o the basic sigmoid SVM stems rom neglecting the spatial distribution o the evaluated samples. To overcome this issue, a modiied sigmoid model is presented in this section. The proposed model depends not only on the SVM values, but also on the distances to the evaluated samples used to train the SVM. Because the proposed model depends on the spatial distribution o the samples, it is also reerred to as Distance-based robabilistic Support Vector Machine (DSVM). It is assumed that in this model the class o any evaluated sample is deterministic. The modiied sigmoid model is deined as: x = A<, B < 0, (12) d d As( x) B( ) min( smax, smin ) d d 1 e where d and d + are the distances to the closest 1 and +1 samples, respectively. s min and s max are the minimum and maximum SVM values among the training samples. is a small quantity (set equal to in this work) added in order to avoid numerical issues at the evaluated training sample. The proposed model satisies the ollowing boundary conditions: 1 x 1 i s( x ) or d 0 1 x 0 i s( x ) or d 0 1 x 0.5 i s( x ) 0 and d d

9 164 A. Basudhar and S. Missoum The upper bound on A ensures that 1 distances away rom the boundary. That is, the values o 1 x does not have a strong dependence on the x are close to 0 or 1 ar rom the boundary irrespective o the inluence o the distances. More speciically, or 1 x > 0.95 at the point o maximum SVM value B = 0, the upper bound ensures s max and 1 x < 0.05 at the point o minimum SVM value s min. The proo o the ormer is given below by setting B = 0: 1 1 e As max > ln A < = (13) s s s max max max The strict inequality bound on B (B < 0) ensures that x 1 at the +1 sample and x at the 1 sample. The proo or the two cases is given below. Setting = 0 or the proo, we have For the +1 samples, d 0 : x = lim 1 d As( ) d 0 As( ) B( ) 1e 1e (14) x x 1 e d For the 1 sample, d 0 : x = lim 0 d As( ) d 0 As( ) B( ) 1e 1e (15) x x d 1 e The training process or the DSVM is as ollows: The values o d + or a +1 sample and that o d or a 1 sample are zero. However, during the training process, these are assigned as the distances to the closest +1 and 1 samples other than the sample under consideration. The values o s(x), d and d + at training samples are used to calculate the likelihood unction, which is then maximised to ind A and B. The optimisation is solved using a genetic algorithm (GA). Because the proposed DSVM model accounts or both the SVM values and the spatial distribution o the evaluated samples, it overcomes the limitations o the basic SVM model mentioned in Section 4.2. A conceptual graphical comparison o the proposed model with latt s sigmoid model is provided in Figure 4. A map o the probabilities o misclassiication is shown in Figure 5.

10 Reliability assessment using probabilistic SVM 165 Figure 4 Comparison o the two SVM models (see online version or colours) Figure 5 Map o the probabilities o misclassiication using the modiied distance-based sigmoid model. A comparison with the probabilities in Figure 2 shows major dierences (see online version or colours) 6 Error quantiication o the SVM model In order to compare the proposed DSVM model with latt s SVM model, a measure to quantiy the error or the models is presented in this section. In the case where the actual limit-state unction is known, ( 1 x ) is known or any point and is equal to 0 or 1. Thereore, the error o the SVM model can be calculated at any point. A large number o test points rom a uniorm grid are used or this purpose. Because the actual class o all the test points is known, the probability o misclassiication or the i-th point is misc x i 1 1 xi s( xi) > 0 =, 1 xi s( xi) < 0 (16)

11 166 A. Basudhar and S. Missoum where x i represents the i-th test point. A good SVM model should provide a low probability o misclassiication or the test points. The error E test is deined as the mean probability o misclassiication or all the test points: E test 1 = x (17) test misc i test i=1 7 Examples This section presents analytical test examples to demonstrate the proposed ailure probability measure. The examples consist o two and three variables. For each example, the boundary o the ailure region is approximated with a polynomial kernel SVM trained with a CVT DOE. The degree o the polynomial is selected as the lowest without training misclassiication (Basudhar and Missoum, 2010). For each example, the ratio o the probability o ailure obtained with the DSVM model to the probability obtained with the deterministic SVM is provided. This ratio, reerred to as the probability ratio (R), provides a measure o the conservativeness o the DSVM model compared to the deterministic SVM. The size o the DOE is varied to study its eect on the probability o ailure. For comparing the probabilities o ailure, all the variables are assumed to have truncated Gaussian distributions with zero mean and standard deviation equal to 1.0. The lower and upper bounds o all the variables are 4.0 and 4.0, respectively. The probabilities o ailure are calculated using 10 6 MCS samples or all the examples. Example 1 is also used to compare the DSVM model with the latt s model. As explained in Section 6, a uniorm grid is used to quantiy the eicacy o the DSVM model. The ollowing notations are used in this section: latt E test : test point-based error or the latt SVM model; Modiied E test Actual SVM Modiied SVM Modiied : test point-based error or the DSVM model; : probability o ailure calculated using the actual limit-state unction; : probability o ailure calculated using the SVM limit-state unction; : SVM-based probability o ailure calculated using the DSVM model; : relative dierence between : relative dierence between R: probability ratio Modiied SVM SVM and actual ; Modiied and actual ; model to probability obtained with deterministic SVM. : ratio o ailure probability obtained with DSVM

12 Reliability assessment using probabilistic SVM Example 1 two-dimensional problem This example consisting o two variables x1, x2 [ 4,4] has two ailure modes. The ailure region, or a parallel system, is deined as: 2 = 8x12x2 0x2 tan ( x17) The two modes and the resulting limit-state unction are shown in Figure 6. With the SVM-based approach, both modes are represented by a single boundary. (18) Figure 6 Example 1: The let igure shows the limit-state unctions or two ailure modes. The right igure shows the net ailure region (see online version or colours) In order to study the eect o the DOE size, the SVM approximations are constructed using CVT DOE samples with increments o 20. A comparison o the latt s model and the proposed DSVM model is provided in Section Additionally, the probabilities o ailure are provided in Section Comparison o the SVM models The two SVM models are compared in this section based on the measure provided in Section 6. Sixteen hundred grid points are used to calculate the errors due to the two models. One example o the distribution o misc values in the space or 40 samples is shown in Figures 3 and 5. The errors E and latt test E are shown in Figure 7. The Modiied test proposed modiied SVM model provides lower errors irrespective o the size o the DOE Comparison o the probabilities o ailure The probabilities o ailure using varying sized DOEs are provided in Figure 8. The dashed-dotted green curve represents the probability o ailure calculated using the deterministic SVMs. There is a signiicant variation in the ailure probability depending on the DOE.

13 168 A. Basudhar and S. Missoum Figure 7 Example 1: Testing error or the SVM models (see online version or colours) Figure 8 Example 1: robabilities o ailure and relative error as a unction o the number o DOE samples. The probabilities o ailure based on a deterministic SVM and the DSVM model are presented (see online version or colours) The DSVM-based probability o ailure and the relative dierences with respect to are also shown in Figure 8 with the solid black curves in let and right plots. The actual relative dierences between the deterministic SVM-based ailure probability and the actual one are shown with the dashed olive green curve in the right hand side plot in Figure 8. It is seen rom the igures that the deterministic SVM-based ailure probability is less than the actual value (red) or several cases. The DSVM-based ailure probability is always, by construction, more conservative than the deterministic SVM case as demonstrated by the probability ratios depicted in Figure 9. ote that this ratio will tend, by construction o the DSVM model, to 1 as the number o DOE samples tends to ininity. 7.2 Example 2 Three-dimensional problem A three variable example consisting o our disjoint regions in the space (Figure 10) is presented in this section. All the variables lie between 4 and 4. The ailure region is given as: =( x1 2) x2 ( x3 2) 3x1x2x3 1 0 (19)

14 Reliability assessment using probabilistic SVM 169 Figure 9 Example 1: robability ratio with respect to the deterministic SVM-based ailure probability The SVM approximation o the limit-state unction is constructed using samples at an interval o 40. Figure 10 Example 2: Actual limit-state unction (see online version or colours) Similar to Example 1, it is seen in Figure 11 that the probability o ailure is calculated using the deterministic SVM (dashed-dotted green) and SVM shows signiicant variation with respect to the size o the DOE. Also, using the deterministic SVM, the ailure probability is less than the actual value (red) in several cases. The DSVM-based ailure probabilities are always more conservative than the deterministic SVM case as demonstrated by the probability ratios depicted in Figure 12. As expected, the ratio reduces with the number o samples because the conidence in the SVM increases. By construction o the SVM, the ratio will tend to unity as the number o samples tends to ininity.

15 170 A. Basudhar and S. Missoum Figure 11 Example 2: robabilities o ailure and relative error as a unction o the number o DOE samples. The probabilities o ailure based on a deterministic SVM and the DSVM model are presented (see online version or colours) Figure 12 Example 2: robability ratios with respect to the deterministic SVM-based ailure probability 8 Discussion This section presents a discussion on the results presented in Section 7. Conservativeness o the SVM-based probability estimate: By construction, the DSVM-based ailure probability is calculated such that it is always more conservative than the one calculated with the deterministic SVM. Figures 9 and 12 show the ratio o the DSVM-based ailure probabilities to the probabilities using the deterministic SVMs. The ratio is always greater than 1. In act, the probability ratios are higher when the sparsity is greater and reduces with the size o the DOE.

16 Reliability assessment using probabilistic SVM 171 This indicates that the conidence in the SVM increases with the amount o data, as expected. In addition to considering the probability o misclassiication by the SVM, it is also useul to consider the variance o the MCS. This is done in order to check i there is any overlap between the DSVM and the deterministic ailure probability prediction. For this purpose, 99% conidence intervals o the MCS ailure probability estimate or Example 1 are depicted in Figure 13. They show that there is no overlap between the deterministic and the modiied SVM model. DSVM vs. latt SVM model: In Example 1, the results o the comparison between the two SVM models show a very clear trend. The comparison o the errors E Modiied and E test also shows lower errors. Both the errors reduce with the size o the DOE. However, the most important point is that, unlike the modiied model, the latt s model does not satisy the condition o having probabilities equal to 0 or 1 at the evaluated samples. This latter point is to be satisied i we consider deterministic computer experiments and do not account or errors associated with the model. I uncertainties on the model itsel are to be considered, the latt s model might be used. Adaptive sampling: In the studies perormed in this paper, signiicant variations o the ailure probabilities were observed with respect to the size o the CVT DOEs. Apart rom using SVMs or quantiying the probability o misclassiication, another option to reduce the errors is to use adaptive sampling or the SVMs. Several adaptive sampling schemes or the update o SVMs have been developed in previous research (Basudhar and Missoum, 2008; Basudhar and Missoum, 2010). A new sampling scheme is also possible based on the proposed modiied SVM model. Because it provides the probability o misclassiication, samples may be added in regions with high misclassiication probability. This is, however, not the ocus o this paper. The aim o this paper is to present a measure o the probability o ailure based on SVM. Even with adaptive sampling, the SVM may not always be accurate and it might be necessary to consider the probability o misclassiication using SVMs. latt test Figure 13 99% conidence interval o the MCS ailure probability estimates or Example 1 (see online version or colours)

17 172 A. Basudhar and S. Missoum 9 Conclusion A method or reliability assessment using SVMs was presented in this paper. The main idea is to include the probability o misclassiication o Monte Carlo samples in the ailure probability calculation. This probability o misclassiication is calculated within a region in the sae domain deined using the spatial distribution o the samples. For this reason, the proposed ailure probability measure is always, by construction, more conservative than the deterministic SVM which might present signiicant error. Apart rom the ailure probability measure, a modiied SVM model was also presented in this paper. The next steps o this research will study higher dimensional examples. In addition, research will be conducted to use SVM or adaptive sampling. Acknowledgements The support o the ational Science Foundation through award CMMI is grateully acknowledged. The authors are also grateul or the help provided by Mr. eng Jiang in ormatting and prooreading this manuscript. Reerences Arenbeck, H., Missoum, S., Basudhar, A. and ikravesh,.e. (2007) Simulation based optimal tolerancing or multibody systems, ASME Conerence roceedings, Vol , pp Basudhar, A. and Missoum, S. (2008) Adaptive explicit decision unctions or probabilistic design and optimization using support vector machines, Computers & Structures, Vol. 86, os. 19/20, pp Basudhar, A. and Missoum, S. (2009a, May) Local update o support vector machine decision boundaries, 50th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conerence, alm Springs, Caliornia. Basudhar, A. and Missoum, S. (2009b) A sampling-based approach or probabilistic design with random ields, Computer Methods in Applied Mechanics and Engineering, Vol. 198, os. 47/48, pp Basudhar, A. and Missoum, S. (2010) An improved adaptive sampling scheme or the construction o explicit boundaries, Structural and Multidisciplinary Optimization, Vol. 42, o. 4, pp Basudhar, A., Missoum, S. and Harrison Sanchez, A. (2008) Limit state unction identiication using support vector machines or discontinuous responses and disjoint ailure domains, robabilistic Engineering Mechanics, Vol. 23, o. 1, pp Beachkoski, B.K. and Grandhi, R. (2002, April) Improved distributed hypercube sampling, roceedings o the 43rd Conerence AIAA/ASME/ASCE/AHS/ASC on Structures, Dynamics and Materials, aper AIAA Denver, Colorado, USA. Bichon, B.J., Eldred, M.S., Swiler, L.., Mahadevan, S. and McFarland, J.M. (2007) Multimodal reliability assessment or complex engineering applications using eicient global optimization, roceedings o the 48th conerence AIAA/ASME/ASCE/AHS/ASC on Structures, Dynamics and Materials, aper AIAA , Honolulu, Hawaii. Gunn, S.R. (1998) Support Vector Machines or Classiication and Regression, Technical Report ISIS-1-98, Department o Electronics and Computer Science, University o Southampton.

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