[American Institute of Aeronautics and Astronautics 51st AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference 18th AIAA/ASME/AHS Adaptive Structures Conference

Reliability Assessment using

Probabilistic Support Vector Machines (PSVMs)

Anirban Basudhar� and Samy Missoumy�

Aerospace and Mechanical Engineering Department, The University of Arizona, Tucson, AZ, 85721, USA

This article presents a new probability of failure measure based on the notion of proba-bilistic support vector machines (PSVMs). A PSVM allows one to quantify the probabilityof having an error in the approximation of the failure boundary using a support vectormachine (SVM). SVM can de�ne explicitly the boundaries of disjoint and non-convex fail-ure domains. The approximation of the failure boundary can be re�ned using an adaptivesampling scheme with a limited number of samples. However, the calculation of the prob-ability of failure might still be inaccurate despite the adaptive sampling. In order to re�nethe probability estimate, the \quality" of the approximated boundary is quanti�ed throughthe probability of misclassi�cation of a sample by the SVM. A new measure of probabilityis then calculated using Monte-Carlo simulations that include the probability of misclassi-�cation. The proposed measure of probability of failure is such that it is always larger (i.e.,more conservative) than the one obtained using a deterministic SVM. Several analyticalexamples are presented, including a case with two failure modes.

I. Introduction

Accurate estimation of the probability of failure is essential for any reliability assessment process, which typ-ically involves a number of computational simulations or physical experiments. However, very often the costassociated with such experiments or simulations restricts the number of design con�gurations that can betested. As a result, the accuracy of the prediction of the failure probability is limited in general. Inaccuracyof the failure probability can cause unexpected failure of the system.

Several methods can be found in the literature for reliability assessment, such as moment-based methods,1

response approximation methods2,3 and response classi�cation methods.4{6 This work uses a responseclassi�cation method developed by the authors, also referred to as the \explicit design space decomposition(EDSD)".5 The basic idea in EDSD is to explicitly approximate the failure boundary in the space of designand random variables, which facilitates the calculation of the probability of failure. The choice of EDSD forreliability assessment is justi�ed by a number of advantages:

� Handling of highly nonlinear limit state functions: The presence of a highly nonlinear limit statefunction restricts the application of the traditional moment-based methods such as �rst and secondorder methods (FORM and SORM).1 More advanced methods such as the two-point non-linear adap-tive approximations (TANA)7 can signi�cantly improve the probability of failure estimate, but thesemethods can also produce large errors when applied to complex multimodal functions as shown in.8 Itis possible to implicitly approximate highly nonlinear failure boundaries using response approximationmethods such as metamodeling.3 The accuracy of the probability of failure estimate can be improvedusing adaptive sampling techniques.8,9 In the context of EDSD, a class of machine learning techniquesreferred to as support vector machines (SVMs) have been found to be very exible in de�ning highlynonlinear failure boundaries.5 Adaptive sampling techniques have also been developed to improve theaccuracy of the probability of failure with limited sample evaluations.10,11

�Graduate Student, Aerospace and Mechanical Engineering Department, The University of Arizona, Tucson, AZ, 85721,USA, and AIAA Student Member.yAssistant Professor, Aerospace and Mechanical Engineering Department, The University of Arizona, Tucson, AZ, 85721,

USA, and AIAA Member.

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American Institute of Aeronautics and Astronautics

51st AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference<BR> 18th12 - 15 April 2010, Orlando, Florida

AIAA 2010-2764

Copyright © 2010 by Anirban Basudhar and Samy Missoum. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

� Handling of discontinuous and binary responses: This is one of the unique advantages of using theEDSD approach, which arises from the fact that only the classi�cation of responses (e.g. safe andfailed) is required in this method.5 Most other methods are based on the actual response values andare hampered by discontinuities. Binary problems refer to the cases for which there is no quanti�ableresponse value and only the \pass" or \fail" information is available. The EDSD method is naturallysuited to handling such problems.12

� Handling of multiple failure modes: The handling of multiple failure modes is another major advantageof the EDSD approach. The combined limit state for all the failure modes can be represented using asingle SVM, which is re�ned using adaptive sampling.11,13

Construction of the explicit failure boundary using EDSD presents a straightforward way to calculate theprobability of failure using a method such as Monte-Carlo simulations (MCS).5,14 The class of any Monte-Carlo sample (safe or failed) is obtained simply by evaluating the sign of the SVM at that sample. In orderto approximate the boundary, the space is �rst sampled using a design of experiments (DOE);15 each samplecorresponds to a speci�c con�guration of the design and random variables. These samples are then classi�edas safe or failed based on the corresponding simulations or experiments. An explicit failure boundary thatseparates the two classes is then constructed using an SVM.5 The EDSD approach has recently been signi�-cantly improved by implementing an adaptive sampling scheme10,11 which allows one to reduce the numberof function evaluations while obtaining an accurate approximated failure boundary.

However, despite the adaptive sampling, the approximated failure boundary might still introduce error incalculating the probability of failure using MCS. In order to limit the potential consequences of a (locally)inaccurate boundary, a new method to calculate the probability of failure using the SVM boundary is pro-posed. The approach is based on the notion of probabilistic support vector machines (PSVMs)16 that allowone to quantify the probability that the SVM misclassi�es a given sample or any point in the space. Thisprobability of misclassi�cation is combined with the MCS process to obtain a new measure of the probabilityof failure.

The PSVM is constructed based on a sigmoid model16 which provides the probability of misclassi�cation.The parameters of the sigmoid model are found by maximizing the likelihood function. In addition, theproposed approach introduces a distance measure to quantify the sparsity in the space. This enables adi�erent weighing of the indicator function in the MCS. This modi�cation leads to a probability of failurethat is always larger than the one obtained based on basic MCS with a �xed SVM boundary.

The proposed method for calculating the probability of failure using PSVMs is validated against severalanalytical test examples. The examples shown consist of highly nonlinear limit state functions, as well as aproblem with two failure modes. The new calculation of the failure probability is performed concurrently tothe update of the SVM explicit boundary using the adaptive sampling scheme11 as described in the appendix.The probability of failure using the proposed PSVM-based method is compared to the regular SVM-basedmethod and the actual failure probability for each problem.

II. Reliability assessment using explicit failure boundaries

The methodology for reliability assessment using the basic EDSD approach5 is presented in this section. InEDSD, the limit state functions are de�ned explicitly in terms of the design and random variables. Themethod is based on the classi�cation of responses and not their actual values, and has several advantages:

� It can handle problems with discontinuous responses5 as only the classi�cation of the responses isimportant and not their actual values. The classi�cation of samples for discontinuous responses isperformed using a clustering technique, such as K-means clustering.17

� It allows one to handle binary problems with \pass" or \fail" information only.12

� It can easily handle multiple failure modes using a single explicit boundary.11,13

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This section presents an overview of the basic EDSD methodology and the tools used to construct the bound-aries and calculate the probabilities of failure.

II.A. Explicit design space decomposition (EDSD)

The �rst step in the construction of the failure boundaries using EDSD is to sample the design space usinga design of experiments (DOE).15 In order to extract information over the entire space, a uniform designof experiments such as Centroidal Voronoi Tessellations (CVT)18 or Optimal Latin Hypercube Sampling(OLHS)19 can be used. The response of the system is �rst evaluated at selected samples from the DOE.However, instead of approximating the responses, the samples are classi�ed as acceptable or unacceptable.An explicit boundary is then constructed that separates these two classes.5

II.B. Support vector machines (SVM)

In order to construct the explicit boundaries, a machine learning technique known as SVM20{22 is usedin this work. It is a powerful classi�cation tool that can de�ne highly nonlinear failure boundaries thatoptimally separate two classes of samples �1. In the context of reliability assessment, the usual conventionis to consider the �1 class as failure and the +1 class as safe. The equation of an SVM classi�er is:

s(x) = b+

NSVXi=1

�iyiK(xi;x) = 0; (1)

where xi is a sample in the space, �i is the corresponding Lagrange multiplier, K is a kernel function, yi isthe class label corresponding to xi that can take values �1, and b is the bias. NSV is the number of supportvectors, which are the samples lying on s(x) = �1. Once an SVM is trained using the evaluated samples,the class at any point in the space is predicted using the sign of the SVM. The regions of the design spacewith s(x) > 0 are classi�ed as +1 and those with s(x) < 0 are classi�ed as -1.

The kernel function in Equation 1 can have several forms, such as polynomial and Gaussian radial basisfunctions. In this study the Gaussian kernel is used:

K(xi;x) = e�jjx�xijj2

2�2 (2)

An example EDSD using SVM is shown in Figure 1.

SVM boundary

+1 class

(safe)

-1 class

(failed)

x1

x2

Figure 1. Example of EDSD using an SVM boundary.

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II.C. Probability of failure calculation using SVM boundaries

The construction of an explicit SVM boundary allows one to easily calculate the probability of failure. Forexample, Monte-Carlo simulations (MCS)14 can be used with a large number of samples. The class of eachMonte-Carlo sample is determined simply by checking the sign of the SVM and does not require an actualfunction evaluation.5 The probability of failure is given as:

Pf =Nf

NMCS=

1

NMCS

NMCSXi=1

Ig(x); (3)

where Nf is the number of MCS samples classi�ed as failure by the SVM classi�er (Equation 1) and NMCS

is the total number of MCS samples. Ig(x) is an indicator function which is 1 if the MCS sample lies in thefailure domain (s(x) < 0) and 0 otherwise.

In order to increase the accuracy of the probability of failure, an adaptive sampling technique11 is used toconstruct the SVM limit state function. An initial SVM boundary is �rst constructed using a relativelysmall sample size using the method explained in Section II.A. This initial boundary is then re�ned usingthe adaptive sampling scheme11 explained in the appendix. However, there might be error in the calculatedprobability of failure despite the adaptive sampling. In order to take the possibility of such errors intoaccount, a new method to calculate the probability of failure using PSVMs is presented in the followingsections.

III. Probabilistic support vector machines (PSVMs)

Unlike deterministic SVMs, which assign a class �1 to any point in the space, PSVMs provide the probabilitythat a point will belong to the +1 class or the �1 class. Thus, a PSVM also considers the variability in theconstruction of the SVM boundary s(x) = 0. That is, it provides the probability that a given point in thespace will be misclassi�ed by the SVM. An example of misclassi�cation of the space by an SVM is shown inFigure 2.

SVM boundary

actual boundary

safe

failed

x1

x2

Figure 2. Misclassi�cation of the space by an SVM. The shaded yellow regions are classi�ed incorrectly by the SVM.

An extensively used model for PSVM is based on the representation of probabilities using a sigmoid func-tion.16 The probability that a point x belongs to the +1 class is given by:

P (+1jx) =1

1 + eAs(x)+B(4)

The parameters A(A < 0) and B of the sigmoid function are found by maximum likelihood, by solving thefollowing problem:16

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minA;B

�NXi

tilog (pi) + (1� ti) log (1� pi) ; (5)

(6)

where N is the number of training samples, pi = P (1jxi) and ti is given as:

ti =yi + 1

2; (7)

where yi are the class labels. Thus, t = 1 for the +1 samples and t = 0 for the samples belonging to the �1class.

IV. Reliability assessment using PSVMs

The proposed methodology to perform reliability assessment using PSVMs is presented in this section. Theprobability of misclassi�cation given by the PSVM is combined with the basic MCS to provide a new measureof the probability of failure. This new measure is calculated such that it is always larger than the probabilityof failure obtained using basic MCS with a deterministic SVM boundary.

The total probability of failure that includes the uncertainty in the random variables, as well as the probabilityof misclassi�cation by the SVM is given as:

P totalf =1

NMCS

NMCSXi=1

P (�1jx)

!; (8)

where P (�1jx) is the probability of belonging to the class -1 knowing the position of the sample x. The -1label corresponds to the failure domain. The probability of failure using Equation 8 may either be larger orsmaller compared to the basic MCS estimate using Equation 3. A conservative estimate can be obtained byconsidering the probability of misclassi�cation for MCS samples with s(x) > 0 only:

P totalf =1

NMCS

NMCSXi=1

(�1jx)

!;

(�1jx) =

(1 s(x) � 0

P (�1jx) s(x) > 0(9)

It is natural that Equation 9 would provide a probability of failure that is larger than the one using basicMCS (Equation 3). However, the failure probability estimate using Equation 9 may sometimes becomeover-conservative. The sigmoid PSVM model provides the probability of misclassi�cation based on the SVMvalue at a point; however, this may lead to a non-zero P (�1jx) even for existing +1 training samples orpoints very close to them.

In order to overcome this problem, the region for considering the PSVM outputs (Equation 4) is restrictedin this work. The probability of misclassi�cation by the SVM is considered only in a region �trust thatis de�ned based on a distance measure to quantify the sparsity. The region �trust does not contain anyexisting training samples. The classi�cation provided by the �xed SVM boundary is trusted in the otherregions denoted as trust (i.e. P (�1jx) is either 1 or 0). The region �trust (Figure 3) for considering theprobability of misclassi�cation by the SVM is:

�trust = sd \ (js(x)j < 1 [ s(x)(d+(x)� d�(x) � 0)) ; (10)

where sd is the safe domain based on the �xed SVM, and d+(x) and d�(x) are the distances of x to theclosest +1 and �1 training samples. �trust consists of two kinds of regions in the +1 class - those belongingto the margin of the SVM and those with d+(x) � d�(x).

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-1+1

closer to

-1 sample

but predicted

as +1 by SVMd-1

d+

trust

-1+1

d+

d-1

closer to

-1 sample

but predicted

as +1 by SVM

margin on

+1 side (0 < s(x) < 1)

Figure 3. Region �trust for considering the probability of misclassi�cation by the SVM (yellow shaded region). �trustconsists of the SVM margin in the +1 class, as well as the regions in the +1 class with d+ greater than d�.

The �nal expression for the failure probability measure is:

P totalf =1

NMCS

NMCSXi=1

(�1jx)

!;

(�1jx) =

8><>:1 xi 2 f

0 xi 2 sd � �trust

P (�1jx) xi 2 �trust

(11)

It is clearly observed from Equation 11 that the new probability of failure measure introduces a di�erentweighing factor compared to the binary (0 or 1) indicator function in the case of the basic MCS. As a resultof this weighing factor, the new probability of failure measure is always larger than the one obtained usingthe basic MCS.

V. Examples

Several analytical examples are presented in this section to show the application of the proposed method.The comparison of the new probability measure to the failure probability using a deterministic SVM isprovided for each example. In addition, a comparison to the probability of failure using the analytical limitstate function is also provided. 106 Monte-Carlo samples are used for the calculation of the probability offailure for all the examples.

For each example, the limit state function is approximated using an SVM that is re�ned using an adaptivesampling scheme11 explained in the appendix. The evolution of the probability of failure using the determin-istic SVM and the PSVM is studied during the update. The convergence of the adaptive sampling is basedon the relative change in the probability of failure between successive iterations. The update is stopped ifthe relative change in the probability is less than �0 for d + 1 consecutive iterations, d being the problemdimensionality. The value of �0 is 10�3 for all the examples. For all the examples, the probability of failure iscalculated at the origin. All the variables are assumed to have standard normal distribution with zero meanand unit standard deviation. The sample space for each problem is selected as four standard deviations fromthe mean, i.e. [-4,4].

The following notations are used to present the results:

� P actualf : Failure probability calculated using the analytical limit state function

� PSVMf : Probability of failure calculated using basic MCS with a deterministic SVM boundary.

� PPSVMf : New failure probability measure calculated using a PSVM.

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� N : Number of training samples to construct an SVM.

� �SVMfinal : Final relative percentage error in the probability of failure based on a deterministic SVM.

� �PSVMfinal : Final relative percentage error in the probability of failure based on a PSVM.

V.A. Example 1: two-dimensional problem with non-convex limit state function

The actual limit state function for this example is a non-convex cubic function with two random variablesx1 and x2. The failure domain is de�ned as:

f = (5x1 + 10)3

+ (5x2 + 9:9)3 � 18 � 0 (12)

The probability of failure calculated using the analytical function is 0:0057. The limit state function isapproximated using an SVM. The initial SVM constructed using 20 CVT samples is updated using adaptivesampling. The �nal SVM boundary is constructed using 87 samples. The probabilities of failure calculatedusing the SVM at various stages of the update are given in Table 1. These values are compared to thePSVM-based probability of failure measure (Equation 11), which is always comparatively more conservative.It is also observed for this problem that the probability of failure calculated using the deterministic SVM isless than P actualf . On the contrary, the �nal PSVM-based probability measure is larger than P actualf for this

problem. In the general case, the PSVM-based measure may be less than P actualf . However, it is always moreconservative than the failure probability based on the deterministic SVM. The evolution of the probabilityof failure during the update is plotted in Figure 5. The actual limit state function, the SVM approximationand the various P (�1jx) contours using the PSVM are plotted in Figure 4.

−4 −2 0 2 4−4

−3

−2

−1

0

1

2

3

4

x1

x2

actual boundarySVM boundarysafefailed

−4 −2 0 2 4−4

−3

−2

−1

0

1

2

3

4

x1

x2

actual boundarySVM boundarysafefailedP(−1|x) = 0.4P(−1|x) = 0.3P(−1|x) = 0.2P(−1|x) = 0.1

Figure 4. Two-dimensional cubic limit state function. The left �gure shows the actual limit state function and the �nalSVM obtained with the update. The right �gure shows the various contours of P (�1jx).

N PSVMf PPSVMf �SVM �PSVM

20 0:0047 0:0102 �17:4% +78:4%

45 0:0073 0:0101 +27:3% +77:4%

70 0:0058 0:0070 +1:3% +21:9%

87 0:0054 0:0064 �4:8% +12:8%

Table 1. Final results of the update for example 1.The value of Pactualf is 0:0057.

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20 35 55 70 80 870.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

N (samples)

Pf

P actualf

P SV Mf

P P SV Mf

Figure 5. Example 1. Variation of the probability of failure measures during the update. The PSVM-based measure(blue) is always more conservative than the one using a deterministic SVM (black).

V.B. Example 2: two-dimensional problem with two failure modes

The failure domain for this example is de�ned using two failure modes:

f =

(�8(x1 � 2) + x2

2 � 0

x2 � tan( �12 )(x1 + 2) + 4 � 0(13)

The limit state function combining the two failure modes is shown in Figure 6. The �nal SVM approximationis also plotted on the same �gure. The probability of failure P actualf using the actual limit state functionis calculated as 0:0183. The probabilities of failure calculated using the deterministic and the probabilisticSVM are listed in Table 2. The values of the failure probabilities during the update are plotted in Figure 7.Similar to the Example 1, the probability of failure using the PSVM-based method is always larger than thefailure probability using the deterministic SVM.

−4 −2 0 2 4−4

−3

−2

−1

0

1

2

3

4

x1

x2

actual boundarySVM boundarysafefailed

−4 −2 0 2 4−4

−3

−2

−1

0

1

2

3

4

x1

x2

actual boundary

SVM boundary

safe

failed

P(−1|x) = 0.4

P(−1|x) = 0.3

P(−1|x) = 0.2

P(−1|x) = 0.1

Figure 6. Two-dimensional problem with two failure modes. The left �gure shows the actual limit state function andthe �nal SVM obtained with the update. The right �gure shows the various contours of P (�1jx).

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20 35 50 70 870

0.01

0.02

0.03

0.04

0.05

0.06

N (samples)

Pf

P actualf

P SV Mf

P P SV Mf



20 0:0300 0:0610 +64:2% +233:4%

45 0:0173 0:0221 �5:6% +21:0%

70 0:0179 0:0228 �2:0% +24:3%

87 0:0178 0:0206 �2:9% +12:5%


V.C. Example 3: three-dimensional problem with non-convex limit state function

This problem is an extension of the Example 1 to three random variables. The failure domain (Figure 8) isgiven as:

f = (5x1 + 10)3

+ (5x2 + 9:9)3

+ (5x3 + 5)3 � 18 � 0 (14)

The value of P actualf for this example, calculated using MCS based on the analytical function, is 0:0039.In order to approximate the failure boundary, the initial SVM is constructed using 40 CVT samples. It isthen updated using the adaptive sampling scheme and the �nal SVM is constructed using 192 samples. Theprobabilities of failure at di�erent stages of the adaptive sampling are shown in Figure 9. The numericalvalues are provided in Table 3. As seen for the previous examples, the PSVM-based failure probability inthis case also is always more conservative than the deterministic SVM. This is a direct consequence of thedi�erent weighing factor for MCS in Equation 11 that takes into account the probability of misclassi�cationby the SVM.


40 0:0056 0:0115 +42:4% +194:0%

90 0:0046 0:0060 +16:9% +52:3%

140 0:0045 0:0053 +15:8% +36:3%

192 0:0038 0:0042 �3:19% +6:3%


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−4−2

02

4

−4

−2

0

2

4

−4

−3

−2

−1

0

1

2

3

4

x2

x1

x3

SVM boundaryactual boundarysafefailed

Figure 8. Three-dimensional example. Actual limit state function and the �nal SVM approximation.

40 70 100 120 140 170 1920

0.002

0.004

0.006

0.008

0.01

0.012

N (samples)

Pf


V.D. Discussion on the results

The examples shown in the preceding sections present a comparison of the probability of failure estimatesbased on deterministic and probabilistic SVMs. Number of similarities are seen in the results for the threeexamples. It is observed that although the use of adaptive sampling signi�cantly increases the accuracy ofthe approximation, there is still some error in the calculation of the failure probability. This is precisely thereason for the development of the PSVM-based method in this work. The possibility of such errors can beavoided only if a large number of samples are used, which may not be practical in the general case due tothe associated costs. However, the use of the PSVM-based method helps to avoid the consequences of theinaccuracies in the SVM, as it also includes the probability of misclassi�cation by the deterministic SVM.It re�nes the probability of failure by introducing a di�erent weighing factor for the MCS based on the ade-quacy of the data. As a result of the weighing factor, the probability of failure is always more conservativethan the one using the deterministic SVM. This is clearly seen from Figures 5,7 and 9.

It is also observed that the di�erence between the probability of failure measures using the deterministic

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and probabilistic SVMs reduces in general with the progress of the update. This is a result of the reductionof the probability of misclassi�cation due to the addition of more information. In theory, the two measuresshould be the same if the sampling is su�cient to provide a zero probability of misclassi�cation by theSVM. However, in practice, very often the number of samples is limited due to the cost of each sampleevaluation. In such cases, the use of the proposed PSVM-based method allows one to avoid the consequencesof the errors due to inadequate data. Also, it is observed for all the examples that the probability of failurechanges drastically in the initial stages of the adaptive sampling and number of samples are required for smallincremental changes towards the latter stages. Since the PSVM-based method also considers the probabilityof misclassi�cation by the SVM, it may not be necessary to push the SVM update to such high accuracy.

VI. Conclusion

This paper presents a method to calculate the probability of failure using PSVMs. This method providesa new measure of the probability of failure by introducing a di�erent weighing factor for MCS, comparedto the traditional binary indicator function. Since the weighing factor includes the probability of misclassi-�cation by the SVM, the proposed method allows one to re�ne the probability of failure by accounting forthe inaccuracy of the SVM due to inadequate data. The new probability of failure measure based on thePSVM is such that it is always more conservative than the probability of failure using the deterministic SVM.

Further work is being performed to improve the methodology presented in this paper. Methods to improvethe adaptive sampling scheme are being explored to further reduce the number of samples. Also, alternatemethods to de�ne the region �trust for considering the probability of misclassi�cation are being explored.In addition, an alternative PSVM model is also being developed.

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17Hartigan, J.A. and Wong, M.A. A k-means clustering algorithm. Applied Statistics, 28:100{108, 1979.18Romero, Vincente J., Burkardt, John V., Gunzburger, Max D., and Peterson, Janet S. Comparison of pure and \latinized"centroidal voronoi tesselation against various other statistical sampling methods. Journal of Reliability Engineering and SystemSafety, 91(1266{1280), 2006.19Liefvendahl, M. and Stocki, R. A study on algorithms for optimization of latin hypercubes. Journal of Statistical Planningand Inference, 136:3231{3247, 2006.20Shawe-Taylor, J. and Cristianini, N. Kernel Methods For Pattern Analysis. Cambridge University Press, 2004.21Tou, J.T. and Gonzalez, R.C. Pattern Recognition Principles. Addison-Wesley, 1974.22Vapnik, V.N. Statistical Learning Theory. John Wiley & Sons, 1998.

Appendix A: Adaptive sampling scheme for the re�nement of theSVM failure boundary approximation

The adaptive sampling scheme used to re�ne the SVM boundary for the calculation of the probability offailure is presented in this section. The algorithm consists of three major sample selection steps - primary,secondary and tertiary. Each of the samples is evaluated only if the maximum possible change in the failureprobability due to the evaluation is greater than a threshold. The maximum change is evaluated by assumingboth the possibilities, of having +1 or �1 class at the new sample, and comparing the corresponding changesin the probability of failure. The details of the three steps are provided below.

1. Step 1: Primary sample and additional sampleThe step 1 consists of the selection of a primary sample xp on the SVM boundary by maximizing thedistance to the closest training sample:

maxx

jjx� xnearestjj

s(x) = 0 (15)

Here, xnearest is the training sample closest to the new sample. The decision to evaluate xp is takenbased on the maximum possible change in the failure probability. Also, an additional sample xpais selected for the case when xp is evaluated and found to provide a critical change in the failureprobability. The additional sample xpa is selected within a hypersphere centered at xp:

minx

ycs(x)

s:t: jjx� xcjj � Rycs(x) � 0 (16)

Here, xc represents the center of the hypersphere, which in this case is xp, and yc is the class label(�1) of xc. The radius R of the hypersphere is calculated as half of the distance from xc to the closestsample from the opposite class. The sample xpa is evaluated only if the maximum possible relativechange in the failure probability due to its evaluation is greater than a threshold �2.

The motivation behind the selection of samples in step 1 is given below:

� Sampling of regions with high probability of misclassi�cation: this is achieved by sampling on theSVM boundary.

� Sampling of sparse regions: this is achieved through maximization of the distance to the closestexisting sample.

� Checking for the possibility of further re�nement of the boundary locally if the primary sampleprovides a critical change in the probability of failure - this is achieved by selecting an additionalsample xpa in the vicinity of xp.

The summary of step 1 is given in Figure 10.

2. Step 2: Secondary sampleStep 2 consists of the selection of a secondary sample xs that does not lie on the SVM boundary.

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Select primary sample

(xp)

Evaluate primary sample

and and reconstruct the SVM.

Calculate ΔPf,xp

Maximum

possible change

ΔPf ,xp>δ1

Go to step 2

Checking maximum possible change

(two cases):

yp = +1 yp = -1

larger change

ΔPf,xp> δ1

Select additional sample xpa

Evaluate the additional sample xpa

and reconstruct the SVM.

Maximum

possible change

ΔPf,xpa> δ2

smaller change

px px

px

pax

new SVM

boundary

further modified

SVM boundary

Go to step 2

Go to step 2

yes

yes

yes

no

no

no

Figure 10. Summary of the step 1 of the adaptive sampling scheme.

The motivation for this step is to remove a phenomenon referred to as the \locking of SVM". In theSVM locking phenomenon, the rate of convergence of the SVM limit state function to the actual onebecomes very slow, which cannot be resolved by the evaluation of primary samples only. Therefore, asecondary sample is intended to overcome this issue.

The secondary sample is selected using Equation 16. However, the center xc in this case is a supportvector. The support vector is selected such that it provides the maximum radius R. In other words,the support vector with largest distance to the closest sample from the opposite class is selected. Thesample xs is evaluated if the maximum possible relative change in the failure probability due to theevaluation is greater than the threshold �2. The summary of step 2 and the selection of the secondarysample to remove locking are shown in Figure 11.

3. Step 3: Tertiary sampleIn step 3, a tertiary sample xt is selected at the most probable point (MPP) on the SVM boundary,which is the closest point to the mean that lies on the boundary. The objective of this step is to ensureat least �rst order accuracy of the failure probability estimate. It should be noted, however, that this

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Select secondary sample

(xs)

Maximum

possible change

ΔPf,xs> δ2

Go to step 3

Evaluate secondary sample

and reconstruct the SVM

actual limit state

SVM boundary

SVM boundary after the evaluation of xp

pxnew SVM boundary

(negligible change due to

the evaluation of xp)

SVM locking

SVM boundary after the evaluation of xs

Removal of SVM locking using secondary sample xS

sx

yes

no

Figure 11. Summary of the step 2 of the adaptive sampling scheme. The �gures on the left show the SVM lockingphenomenon and its removal using a secondary sample.

is achieved only at the end of the update.

minx

jjx� x0jj

s:t: s(x) = 0 (17)

where x0 is the mean at which probability of failure is calculated. The tertiary sample xt is evaluatedonly if the maximum possible relative change in the failure probability is greater than �3.

The convergence criterion for the adaptive sampling is based on the relative change in the probability offailure between successive iterations. Each iteration consists of the three sample selection steps mentionedabove. The update is stopped if the relative change in the probability is less than �0 for d + 1 consecutiveiterations, where d is the number of parameters in the problem.

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Select tertiary sample

(xt)

Maximum

possible change

ΔPf,xt> δ3

Evaluate tertiary sample

and reconstruct the SVM

SVM boundary

SVM boundary after the evaluation of xt

new SVM boundary

Selection of a tertiary sample at the MPP

Go to step 1

0

1

k

f

k

f

k

f

P

PP

txt

x

mean

Check convergence

yes

no

no

Figure 12. Summary of the step 3 of the adaptive sampling scheme. The tertiary sample is selected at the MPP, whichis the closest point to the mean that lies on the SVM boundary.

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