International Conference on Applications of Statistics and Probability in Civil Engineering (ICASP) (12th : 2015)

A nonlinear wavelet density-based importance sampling for reliability analysis Wang, Wei; Dai, Hongzhe Jul 31, 2015

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12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  1 A Nonlinear Wavelet Density-based Importance Sampling for Re-liability Analysis Wei Wang Professor, School of Civil Engineering, Harbin Institute of Technology, Harbin, China Hongzhe Dai Associate Professor, School of Civil Engineering, Harbin Institute of Technology, Harbin, China ABSTRACT: Importance sampling is a commonly used variance reduction technique for estimating reliability of a structural system. The performance of importance sampling is critically dependent on the choice of the sampling density. For the commonly used adaptive importance sampling method, the construction of the sampling density relies on the kernel-based density estimation. However, the choice of the initial bandwidth of the local windows may heavily affect the accuracy of the kernel method, particularly when the number of samples is not very large. To overcome this difficulty, this study de-velops a new adaptive importance sampling method based on nonlinear wavelet thresholding density estimator. The method utilizes the adaptive Markov chain simulation to generate samples that can adaptively populate the important region. The importance sampling density is then constructed using nonparametric wavelet density to implement the importance sampling. The methods takes advantage of the attractive properties of the Daubechies’ wavelet family (e.g., localization, various degrees of smoothness, and fast implementation) to provide good density estimations. Compared with the kernel density estimator, the nonlinear wavelet thresholding density estimator has a high degree of flexibility in terms of convergence rate and smoothness. Moreover, the choice of the initial parameters slightly affects the accuracy of the method. Two examples are given to demonstrate the proposed method.  1. INTRODUCTION In the structural reliability theory, importance sampling method is widely used to calculate the probability of failure, expressed as  d( )[ ( ) 0] ( )d( )ffP I g hh  x xx x xx  (1) where ( )fx x  is the joint probability density func-tion of the d -dimensional random variables  1= , , dX XX , [ ( ) 0]I g x is the indicator function for the failure event ( ) 0g x , and ( )h xis the importance sampling density func-tion(Balesdent et al. 2013). There exits an opti-mal importance sampling density function opt ( )h x   (Melchers 1989). opt ( ) [ ( ) 0] ( ) fh I g f P  xx x x  (2) However, the optimal importance sampling func-tion is unknown as it involves the unknownfP . Therefore nonparametric density estimation techniques are adopted to constructopt ( )h x . This approach first generates samples that are distrib-uted asymptotically according to the optimal im-portance sampling density. Then nonparametric density estimation techniques are used to con-struct the importance sampling density. Ang was the first one who simulated samples in the failure region by Monte Carlo sampling and used these samples to construct the importance sampling density based on kernel method (Ang et al. 1992). Au and Beck used Markov chain simulation to construct a Markov chain whose target distribu-tion is the optimal importance sampling function. Based on the samples generated, the sampling density is constructed using the adaptive kernel 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  2 density estimation (Au and Beck 1999). Kurtz and Song proposed to construct the near-optimal importance sampling density by using Kullback–Leibler cross entropy coupled with a Gaussian mixture kernel (Kurtz and Song 2013). Dai es-tablished an adaptive importance sampling method on the basis of fast Gaussian transform kernel density (Dai et al. 2011). In sum, most of the current practice for constructing the im-portance sampling density relies on the kernel-based density estimation. And the choice of the kernel function and the initial bandwidth of the local windows may heavily affect the accuracy of the kernel method, particularly when the num-ber of samples is not very large (Silverman. 1986).  As a nonparametric density estimation method with good localization and asymptotic property, the wavelet density estimation outper-forms the kernel density estimator in terms of global approximation (Hardle et al. 1997). This paper presents a new wavelet thresholding densi-ty-based adaptive importance sampling scheme. The proposed method involves the generation of samples that can adaptively populate the im-portant region by the adaptive Markov chain simulation, and the construction of importance sampling density by nonlinear wavelet threshold-ing density estimator, followed by importance sampling to evaluatefP . Numerical examples are given to demonstrate the application and efficiency of the proposed method. 2. SAMPLE GENERATION: ADAPTIVE MARKOV CHAIN SIMULATION Since the closed-form expression for the optimal importance sampling function is unknown, the direct Monte Carlo sampling procedure cannot be used to generate the samples. Therefore, the adaptive Markov chain simulation is proposed to generate the samples that cover the region of most interest (Metropolis et al. 1953). Let ()  be the target distribution density function. Suppose that at time 1t   the states  0 t-1, ,x x have been sampled, a candidate point y  is then sampled from a proposal distri-bution  0 t-1, ,q  | x x . The candidate point y  is accepted with a probability    t-1min 1,      yx,  (3) and is rejected with the remaining probability  1  , i.e., t t-1x = x .  The proposal distribution can be chosen as a multivariate normal distribution with a mean at the point t-1x  and a covariance matrix tC  given by   0 t-1= Cov , , It d d dS S x xC  , (4) where the subscript d  represents the dimension of the random variable, dS  is a scaling parame-ter which was suggested as 22.4dS d (Gelman AG 1996),   is a positive constant that can be chosen very small, Id  denotes a d -dimensional identity matrix, and  0 t-1Cov , ,x x is the em-pirical covariance matrix for the samples 0 t-1, ,x x (Haario et al. 2001). To start the adaptive Markov chain simulation, an arbitrary, strictly positive definite initial covariance matrix 0C  is selected using prior knowledge. Note that 1tC  and tx satisfy the recursion equations:    1T T Tt-1 1 t t1=1 It tdt t t dttStt   tx x x x x xC C  (5) and  t t-1 t11 1tt t  x x x.  (6) where   01 1 kk ik    ix x. The aforementioned adaptive Markov chain simulation lends itself to simulating a sample from ()opth  by assuming ()opth  is the target dis-tribution of the adaptive Markov chain. Suppose that the Markov chain is in statet-1x , then a can-12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  3 didate point y  is generated from the Gaussian proposal distribution with a mean of t-1x  and a covariance matrixtC , which is computed recur-sively from Eq.(5). The acceptance probability   is given by:      t-10min 1, g ff       xx1 y yx.  (7) Note that the evaluation of Eq.(7) only re-quires the ratio of the target distribution be-tween consecutive states, and fPis not needed. The first sample in the failure domain can be generated as a point simulated according to()opth , or be assigned more efficiently based on engineering judgment (Au and Beck 1999). Alt-hough the adaptive Metropolis algorithm is non-Markovian, it has been shown that it has the right ergodic properties, and converges correctly to the target distribution. It has also been shown that the adaptive Metropolis algorithm often achieves a faster convergence than the classical Metropo-lis algorithm (Haario et al. 2001).  3. SAMPLING DENSITY ESTIMATION: WAVELET EXPANSION 3.1. Multiresolution wavelet analysis and Wave-let density estimation Methodology Wavelets involve a new family of basis functions that can be used to approximate or express other functions. This section gives a brief introduction to the basic concept of multiresolution analysis and wavelet expansions. Detailed information can be found, for example, in (Chui and K. 1992; Daubechies 1992) . We introduce wavelets using the multireso-lution framework developed in (Mallat 1989). Let  and  denote the set of real numbers and integers, respectively.  2L  denotes the set of square-integrable one-dimensional function ( )f x . The wavelet representation approximates any function  2f L by a sequence of func-tions jf  which are smoother than f and which can be characterized by their sampling on the lattice -2 j . A multiresolution analysis (or ap-proximation) of  2L  consists of a nested se-quence  ,jV j  of closed subspaces  2L  and there also exists a function 0V  such that the sequence   ,x k k    is an orthonor-mal basis of the space 0V . The function  is the scaling function. Define     2, 2 2 ,j jj k x x k    (8) The variable j  determines the amount of varia-ble scaling or dilation, and k  represents shift or translation. The family   , ,j k x k   spans the thj  scale of the multiresolution analysis and forms an orthonormal basis forjV .When the scale increases from j  to 1j  , the approxima-tion at the finer level is obtained by adding some details about f . These details can be modeled at the scale j  by the orthogonal complement of jV  in 1jV . Let jW be this orthogonal complement of jV  in 1jV  , we get another sequence  ,jW j  of closed mutually orthogonal sub-space of  2L .Let     2, 2 2 ,j jj k x x k    (9) then the family   , ,j k x k   is an orthogonal basis for jW  and is an orthonormal basis for 2L . The spaces  ,jW j decompose the function into its smooth and detail parts. Any function  2f L  can be represented in a wavelet series as       0 00, , , , .j k j k j k j kk j j kf x c x d x       (10) The range of k  will be discussed later in Section 3.3. The first part of Eq. (10) is the pro-jection of f  on the coarse approximation space 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  4 0jV, and the oscillating features are approximat-ed in fine details by the second part. The scaling coefficients ,j kc  and wavelet coefficients ,j kd  are defined as        , , ,, , ,,,j k j k j kj k j k j kc f f x x dxd f f x x dx      (11) Let 1, , NX X  be independent identically distributed samples from X  with a probability density function f . Using an orthogonal wavelet basis, the wavelet representation of f  is given  by Eq.(10) The scaling coefficients ,j kc  and wavelet coefficients ,j kd  can be estimated by     , ,1, ,11ˆ ,1ˆ .Nj k j k iiNj k j k iic XNd XN  (12) And one can estimate f  by       11 0 00, , , ,ˆ ˆˆ ,jj j k j k j k j kk j kf x c x d x    (13) where 1 0j j . Nonlinear wavelet thresholding density estimator can be yeilded by adopting thresholding shrinkage method (Donoho et al. 1996),      10 00, , , ,ˆ ˆˆ ,jj k j k j k j kk j kf x c x d x      (14) where ,ˆ j kd denotes the wavelet coefficients after  thresholding operation, and the soft thresholding function can be chosen as follows(Walter and Shen 1994)    , , ,ˆ ˆsignj k j k j kd d d     (15) where  max ,0x x  , and thresholding value   is given by(Mallat 1989)    ,, ˆ0.6 0.8 max j kj k d   (16) 3.2. Multivariate wavelet density estimation The one-dimensional multiresolution analysis described above can be readily extended to high-er dimensional case using the tensor product of one dimensional multiresolution analysis (Vida-kovic 1999). The resulting d -dimensional multi-resolution analysis corresponds to one d -variate scaling function       11, , ,dd iiix x x   (17) and d -dimensional wavelet functions    l x          11, , ,dld iiix x x   (18) with    or , but not all   .Any function  2f L  can be represented as          0 002 1, , , ,1,dl lj j j jj j lf c d     k k k kk kx x x (19)  where    1 1, , , , , dd dx x k k  x k,and             0 002,12,12 2 ,2 2 ,dj d jj i iiidjd jj i iiix kx k    kkxx  (20) with    or , but not all   . Multivariate wavelet density estimation is straightforward generalization of the aforemen-tioned univariate density estimation. Let   : 1, ,i i NX be a sample of d -dimensional random variables X  with probability density function f , and       1 , ,i ii dX XX , the wave-let estimator of f  is 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  5          0 010, ,2 1, ,1ˆ ˆˆ ,dj jjl lj jj j lf cd k kkk kkx xx  (21) where  1, , dx xx ,  1, , dk kk , and 0 ,j kand  0 ,lj k are given by Eq.(20). The empirical scaling coefficient and wavelet coefficient are computed as                 1, ,11 21 11, ,11 21 1ˆ2ˆ2 2Nij k j kidNijd jm mmi mNl l ij k j kidNijd jm mmi mc NN X kd NN X k      XX  (22) where    or  , but not all   . Note that the d -dimensional scaling function is the prod-uct of the scaling functions on each dimension. Similar threshold shrinkage operation as Eq.(15) can be applied to wavelet coefficients  ,ˆ lj kd to ob-tain the nonlinear wavelet density estimator. 3.3. Computation of the wavelet estimator There are several practical considerations when using wavelets for density estimation: (1) select-ing wavelet family, and (2) finding the range of 1j and k  for  ,j k x and  ,j k x  in Eq.(10). 1j  can be determined by the scalogram of the densi-ty f ,    2,, j kkf jf .  (23) The scalogram describes the energy distribution at various scale j  of the density f . The scalo-gram can be obtained by empirical scalogram      2 1 2,1ˆdlj kl kj d    (24) The optimal level 1j  is the smaller one at which the energy distribution of the density f  at two successive scales increases exponentially. Another issue is to determine the range of the translation index k  for both of the scaling function and the wavelet function. For Db#q, the support of    2, 2 2j jj k x x k  is  0,2 1q  , therefore,  2 12 2j jk q kx   .  (25) Assume that the range of sample is the in-terval  ,a b , one can calculate the values of k  for which the support of corresponding functions ,j k  intersects  ,a b . Therefore, the range of k  is  2 2 1 2j ja q k b         ,  (26) in which y    denotes the smallest integer which is larger than or equal to y  and y    is the larg-est integer which does not exceed y . For the multivariate wavelet density estimator, if the range of the data is    1 1, ,d da b a b  , the range of  1, , dk kk  is  2 2 1 2j ji i ia q k b         .  (27) Likewise, the range of  1, , dk kk for the wavelet functions is   2 2 1j ji i ia q k b q            (28) 4. NONLINEAR WAVELET THRESHOLD-ING DENSITY-BASED ADAPTIVE IM-PORTANCE SAMPLING The procedure of the proposed methodology can be summarized as follows. 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  6 Step 1: Generate N samples, whose target distri-bution is the opt ( )h x , using the adaptive Markov chain simulation procedure pre-sented in Section2. Step 2: Based on the samples generated in Step 1, construct the wavelet sampling density  fˆ x  in Eqs.(10) or (21) by nonlinear wavelet thresholding density estimator. Step 3: Use the nonlinear wavelet thresholding density estimator constructed in Step 2 as the importance sampling density, and perform importance sampling simula-tion. Assuming that N  Markov chain samples are generated in Step 1 and M  samples are used in the importance sampling process (Step 3), the total number of function evaluations of the limit state is N M . For most reliability analysis of structures of practical interest, majority of the computational cost is expended on the multiple evaluations of the limit state function. The CPU time needed for the wavelet density estimation can be negligible in comparison with that of per-forming multiple limit state analyses. Therefore, the total number of function calls of limit state is used as the measure of the computational cost. 5. EXAMPLES Two examples from literature were selected to demonstrate the proposed method. The multivar-iate normal distribution was used as the proposal distribution for the adaptive Markov chain simu-lations. The Daubechies wavelet with 4 vanish-ing moments (Db#4) was used in the wavelet density estimation. The efficiency of the wavelet density estimation is examined through compari-son with the classical kernel density-based im-portance sampling method. In the following dis-cussion, Wavelet-based IS denotes the proposed method, while Kernel-based IS represents the importance sampling using the kernel density estimation. 5.1. Example 1: a series system  The first example is a series system with two branches,(Au and Beck 1999; Dubourg et al. 2013) 421 121 23 exp( ) min 10 58X XXgX X                 x  where 1X  and 2X  are independent standard normal variables. The system has two design points,  0,4  and  2.83,2.83 , respectively.   (a) Kernel sampling density  (b) Wavelet sampling density Fig. 1 Importance sampling density for Example 1 Fig. 1(a) and (b) plot the contours of the sampling density constructed by the nonlinear wavelet and kernel method, respectively, using 300 samples generated by the adaptive Markov chain simulation. The generated samples are clustered around the region that contributes most to the probability of failure. Using these points, both methods can construct the sampling density that reveals the import regions near the limit state surface. This example suggests that the wavelet thresholding density can be used as an alternative -6 -4 -2 0 2 4 61357X1X2  limit state functionsamples generated by Metropolisdesign point-6 -4 -2 0 2 4 61357X1X2  limit state functionsamples generated by Metropolisdesign point12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  7 to the kernel density for importance sampling. Since the constructed wavelet density has a shape similar to the optimal sampling density, a failure region of almost any shape can be han-dled without performing preliminary component reliability analysis for each limit state.  Table 1 compares the failure probabilities from Wavelet-based IS and Kernel-based IS. In both methods, 300 samples ( 300N  ) were gen-erated by the adaptive Markov chain simulation, and another 500 samples ( 500M  ) were used in the subsequent importance sampling simula-tion. Therefore, the computational costs of the two methods are comparable. The ‘exact’ proba-bility of failure was found to be 56.81 10 using 610 Monte Carlo simulations. It can be seen that the Wavelet-based IS is more accurate than the Kernel-based IS, due to high degree of flexibility and adaptivity of the nonlinear wavelet thrshold-ing density estimator and the dependence on the choice of initial parameters of the kernel-based IS. Table 1 Reliability results of example 1 Methods No. samples Pf Error(%) Monte Carlo 106 6.8110-5 - wavelet N=300, M=500 7.4110-5 8.81 kernel N=300, M=500 7.7410-5 13.66 Table 2 Reliability results for different choices of N  No. samples Wavelet-based IS Pf        Error(%) Kernel-based IS    Pf         Error(%) N=100,M=500 55.39 10   20.85 58.55 10   25.55 N=200,M=500 55.94 10   12.78 55.48 10   19.53 N=300,M=500 57.41 10    8.81 57.74 10   13.66 N=400,M=500 57.26 10    6.61 57.60 10   11.60 N=500,M=500 57.19 10   5.58 56.24 10   8.37 The results of two importance sampling methods with different values of N , while keep-ing that 500M   unchanged, are tabulated in Table 2. It can be seen that the proposed method is more accurate than the kernel method, particu-larly when the number of adaptive Markov chain samples is relatively small, which can be at-tributed to the fast convergence rate of the non-linear wavelet thresholding density estimator. With the increasing number of the Markov chain samples (i.e., 500N  ), the performance of the kernel method is improved. This suggests that the kernel method has a relatively high demand to the number of pre-samples, resulting in the computational inefficiency. 5.2. Example 2:Noisy limit state function A system with noisy limit state function was chosen to investigate the robustness of the pro-posed method (Kurtz and Song 2013). The limit state function is defined by 222 1 1( )= ( ) 0.001 sin(100 )iig b X X e X     x  where iX  are independent standard normal vari-ables. Parameters ,b  and e  are constants, with 5b  , 5  and 0.1e  . This limit state corre-sponds to two design points and the noisy term makes it difficult to search the design points ac-curately.  Table 3 compares the failure probabilities from Wavelet-based IS and Kernel-based IS. Re-sults show that the proposed method is relatively insensitive to the noisy term and the relative er-ror is about half of the traditional Kernel method, demonstrating the noisy reduction effect of the nonlinear thresholding rules.  Table 3  Reliability results of example 2 Methods No. samples Pf Error(%) Monte Carlo 3.6105 3.0710-3 - wavelet N=500,M=500 3.3310-3 8.47 kernel N=500,M=500 2.5110-3 18.24 Comparison between the nonlinear Wave-let-based IS and the Kernel-based IS with differ-ent combination N and M ( 500N M  )is giv-en in Table 4. It is shown that the nonlinear Wavelet-based IS is more accurate than the ker-nel method, particularly when the number of adaptive Markov chain samples is relatively 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  8 small, illustrating the advantage of the efficiency of the wavelet thresholding estimator. It's should be noted that too small M (number of IS samples) may reduce the accuracy of the proposed method, even if the constructed sampling density is ap-proaching to the optimal sampling density.  Table 4 Reliability results for different choices of N  and M  No. samples  Wavelet-based IS     Pf          error(%) Kernel-based IS    Pf        error(%) N=100,M=400 3.9610-3   28.99 4.4610-3  45.82 N=200,M=300 3.8310-3     24.76 1.9810-3  35.55 N=300,M=200 2.4410-3   20.52 2.0610-3  32.90 N=400,M=100 1.8710-3   39.09 4.1210-3  34.20 6. CONCLUSIONS A new adaptive importance sampling method has been developed using adaptive Markov chain simulation and nonlinear wavelet density estima-tion. Wavelet density estimator has remarkable advantages, including different degrees of smoothness, localization, and fast implementa-tion; it may approximate the near-optimal sam-pling density more efficiently than the classical kernel method. Two examples have been ana-lyzed to demonstrate the efficiency and accuracy of the proposed method. In both examples, the proposed method gave good results with reason-able computational effort. The error is less than or about 10% when 1000 samples (including both Markov chain samples and subsequent im-portance sampling samples) were used. It was also found that the proposed method is more ac-curate than the conventional kernel density-based importance sampling method.  7. REFERENCES  Ang, G. L., Ang, A. H. S., and Tang, W. H. (1992). "Op-timal Importance-Sampling Density Estimator." Journal of Engineering Mechanics, 118(6), 1146-1163. Au, S. K., and Beck, J. L. (1999). "A new adaptive im-portance sampling scheme for reliability calcula-tions." Structural Safety, 21(2), 135 - 158. Balesdent, M., Morio, J., and Marzat, J. (2013). "Kriging-based adaptive Importance Sampling algorithms for rare event estimation." Structural Safety, 44, 1-10. Chui, and K., C. (1992). An introduction to wavelets, Aca-demic Press. Dai, H., Zhao, W., and Wang, W. (2011). "An efficient adaptive importance samping method for structural reliability analysis." Lixue Xuebao/Chinese Journal of Theoretical and Applied Mechanics, 43(6), 1133-1140. Daubechies, I. (1992). Ten Lectures on Wavelets, Society for Industrial and Applied Mathematics. Donoho, D. L., Johnstone, I. M., Kerkyacharian, G., and Picard, D. (1996). "Density estimation by wavelet thresholding." The Annals of Statistics, 24(2), 508-539. Dubourg, V., Sudret, B., and Deheeger, F. (2013). "Meta-model-based importance sampling for structural re-liability analysis." Probabilistic Engineering Me-chanics, 33, 47-57. Gelman AG, R. G. G. W. (1996). "Efficient Metropolis jumping rules.", Oxford University Press, Oxford, 599–608. Haario, H., Saksman, E., and Tamminen, J. (2001). "An Adaptive Metropolis Algorithm." Bernoulli, 7(2), 223-242. Hardle, W., Kerkyacharian, G., Picard, D., and Tsybakov, A. B. (1997). Wavelets, Approximation And Statis-tical Application, Chapman and Hall, Berlin-Paris. Kurtz, N., and Song, J. (2013). "Cross-entropy-based adap-tive importance sampling using Gaussian mixture." Structural Safety, 42, 35-44. Mallat, S. G. (1989). "A theory for multiresolution signal decomposition: the wavelet representation." IEEE Transactions on Pattern Analysis and Machine In-telligence, 11(7), 674-693. Mallat, S. G. (1989). "Multiresolution Approximations and Wavelet Orthonormal Bases of L2(R)." Transac-tions of the American Mathematical Society, 315(1), 69-87. Melchers, R. E. (1989). "Importance sampling in structural systems." Structural Safety, 6(1), 3 - 10. Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H., and Teller, E. (1953). "Equation of State Calculations by Fast Computing Machines." Journal of Chemical Physics(No.6), 1087-1092. Silverman., B. W. (1986). "Density estimation for statistics and data analysis.", Chapman and Hall/CRC, 175. Vidakovic, B. 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