Cross entropy-based importance sampling using Gaussian densities revisited

•We compare two distributions as sampling densities in the cross entropy method.•A modified expectation-maximization algorithm is proposed for fitting the parameters.•The DBSCAN algorithm is used to estimate the number of Gaussians in the mixture.•We compare the performance of the two densities in t...

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Published in:	Structural safety Vol. 76; pp. 15 - 27
Main Authors:	Geyer, Sebastian, Papaioannou, Iason, Straub, Daniel
Format:	Journal Article
Language:	English
Published:	Amsterdam Elsevier Ltd 01.01.2019 Elsevier BV
Subjects:	Adaptive sampling Algorithms Clustering Component reliability Computer simulation Cross entropy method Density Divergence Entropy Entropy (Information theory) Expectation maximization Failure analysis Gaussian mixture Importance sampling Markov chains Mathematical models Normal distribution Optimization Probabilistic models Probability Reliability analysis Reliability engineering Sampling Sequential importance sampling Simulation Statistical analysis Structural reliability System reliability Sequential importance sampling Simulation Cross entropy method Expectation maximization Reliability analysis Gaussian mixture Importance sampling
ISSN:	0167-4730, 1879-3355
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Abstract	•We compare two distributions as sampling densities in the cross entropy method.•A modified expectation-maximization algorithm is proposed for fitting the parameters.•The DBSCAN algorithm is used to estimate the number of Gaussians in the mixture.•We compare the performance of the two densities in the cross entropy method. The computation of the probability of a rare (failure) event is a common task in structural reliability analysis. In most applications, the numerical model defining the rare event is nonlinear and the resulting failure domain often multimodal. One strategy for estimating the probability of failure in this context is the importance sampling method. The efficiency of importance sampling depends on the choice of the importance sampling density. A near-optimal sampling density can be found through application of the cross entropy method. The cross entropy method is an adaptive sampling approach that determines the sampling density through minimizing the Kullback-Leibler divergence between the theoretically optimal importance sampling density and a chosen parametric family of distributions. In this paper, we investigate the suitability of the multivariate normal distribution and the Gaussian mixture model as importance sampling densities within the cross entropy method. Moreover, we compare the performance of the cross entropy method to sequential importance sampling, another recently proposed adaptive sampling approach, which uses the Gaussian mixture distribution as a proposal distribution within a Markov Chain Monte Carlo algorithm. For the parameter updating of the Gaussian mixture within the cross entropy method, we propose a modified version of the expectation-maximization algorithm that works with weighted samples. To estimate the number of distributions in the mixture, the density-based spatial clustering of applications with noise (DBSCAN) algorithm is adapted to the use of weighted samples. We compare the performance of the different methods in several examples, including component reliability problems, system reliability problems and reliability in varying dimensions. The results show that the cross entropy method using a single Gaussian outperforms the cross entropy method using Gaussian mixture and that both distribution types are not suitable for high dimensional reliability problems.
AbstractList	The computation of the probability of a rare (failure) event is a common task in structural reliability analysis. In most applications, the numerical model defining the rare event is nonlinear and the resulting failure domain often multimodal. One strategy for estimating the probability of failure in this context is the importance sampling method. The efficiency of importance sampling depends on the choice of the importance sampling density. A near-optimal sampling density can be found through application of the cross entropy method. The cross entropy method is an adaptive sampling approach that determines the sampling density through minimizing the Kullback-Leibler divergence between the theoretically optimal importance sampling density and a chosen parametric family of distributions. In this paper, we investigate the suitability of the multivariate normal distribution and the Gaussian mixture model as importance sampling densities within the cross entropy method. Moreover, we compare the performance of the cross entropy method to sequential importance sampling, another recently proposed adaptive sampling approach, which uses the Gaussian mixture distribution as a proposal distribution within a Markov Chain Monte Carlo algorithm. For the parameter updating of the Gaussian mixture within the cross entropy method, we propose a modified version of the expectation-maximization algorithm that works with weighted samples. To estimate the number of distributions in the mixture, the density-based spatial clustering of applications with noise (DBSCAN) algorithm is adapted to the use of weighted samples. We compare the performance of the different methods in several examples, including component reliability problems, system reliability problems and reliability in varying dimensions. The results show that the cross entropy method using a single Gaussian outperforms the cross entropy method using Gaussian mixture and that both distribution types are not suitable for high dimensional reliability problems. •We compare two distributions as sampling densities in the cross entropy method.•A modified expectation-maximization algorithm is proposed for fitting the parameters.•The DBSCAN algorithm is used to estimate the number of Gaussians in the mixture.•We compare the performance of the two densities in the cross entropy method. The computation of the probability of a rare (failure) event is a common task in structural reliability analysis. In most applications, the numerical model defining the rare event is nonlinear and the resulting failure domain often multimodal. One strategy for estimating the probability of failure in this context is the importance sampling method. The efficiency of importance sampling depends on the choice of the importance sampling density. A near-optimal sampling density can be found through application of the cross entropy method. The cross entropy method is an adaptive sampling approach that determines the sampling density through minimizing the Kullback-Leibler divergence between the theoretically optimal importance sampling density and a chosen parametric family of distributions. In this paper, we investigate the suitability of the multivariate normal distribution and the Gaussian mixture model as importance sampling densities within the cross entropy method. Moreover, we compare the performance of the cross entropy method to sequential importance sampling, another recently proposed adaptive sampling approach, which uses the Gaussian mixture distribution as a proposal distribution within a Markov Chain Monte Carlo algorithm. For the parameter updating of the Gaussian mixture within the cross entropy method, we propose a modified version of the expectation-maximization algorithm that works with weighted samples. To estimate the number of distributions in the mixture, the density-based spatial clustering of applications with noise (DBSCAN) algorithm is adapted to the use of weighted samples. We compare the performance of the different methods in several examples, including component reliability problems, system reliability problems and reliability in varying dimensions. The results show that the cross entropy method using a single Gaussian outperforms the cross entropy method using Gaussian mixture and that both distribution types are not suitable for high dimensional reliability problems.
Author	Straub, Daniel Geyer, Sebastian Papaioannou, Iason
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Snippet	•We compare two distributions as sampling densities in the cross entropy method.•A modified expectation-maximization algorithm is proposed for fitting the... The computation of the probability of a rare (failure) event is a common task in structural reliability analysis. In most applications, the numerical model...
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SubjectTerms	Adaptive sampling Algorithms Clustering Component reliability Computer simulation Cross entropy method Density Divergence Entropy Entropy (Information theory) Expectation maximization Failure analysis Gaussian mixture Importance sampling Markov chains Mathematical models Normal distribution Optimization Probabilistic models Probability Reliability analysis Reliability engineering Sampling Sequential importance sampling Simulation Statistical analysis Structural reliability System reliability
Title	Cross entropy-based importance sampling using Gaussian densities revisited
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