Optimal Importance Sampling Density Research Articles

A novel cross-entropy-based importance sampling method for cumulative time-dependent failure probability function

Cumulative time-dependent failure probability function (C-T-FPF) can reflect the effects of distribution parameters of random inputs and the upper bound of service time interval, which vary in their design domains, on time-dependent failure probability. However, solving C-T-FPF involves a time-consuming triple-layer framework. Thus, we propose an efficient method by combining cross-entropy-based importance sampling (IS) with adaptive Kriging model (CE-IS-AK). The innovations of CE-IS-AK include two aspects. Firstly, we construct a space augmented by both distribution parameters and the upper bound of service time interval, on which the triple-layer framework is decoupled to a single-layer one. And in the augmented space, an optimal IS density is proposed to reduce the required candidate sample size for estimating C-T-FPF. Secondly, we employ Gaussian mixture model (GMM) to approximate the optimal IS density, and the parameters in GMM are determined by minimizing the cross entropy of GMM and the optimal IS density. Moreover, to reduce the model evaluations in the proposed method, Kriging model is adaptively embedded to replace the actual model, and a first failure instant based learning function is proposed to train Kriging model adaptively. Due to the proposed single-layer framework and IS strategy assisted by the Kriging model, the efficiency is greatly improved for estimating C-T-FPF, which is validated by several examples.

Meta model-based importance sampling combined with adaptive Kriging method for estimating failure probability function

The failure probability function (FPF) can provide the entire effect of the distribution parameter in the concerned region on failure probability (FP), which is important for decoupling the optimal parameter search and constraint analysis in the FP-constrained design optimization. To improve the efficiency of solving the FPF, this paper combines meta model-based importance sampling with adaptive Kriging (Meta-IS-AK). First, based on the theoretical optimal importance sampling density (ISD) for estimating the FP (ISD-FP) at the given distribution parameter, the theoretical optimal ISD for solving FPF (ISD-FPF) is constructed using an integral operator. Then, a Kriging model of the performance function is adaptively trained to establish the quasi-optimal ISD-FPF for approximating the optimal performance function, and a simple rejection sampling (SRS) method is designed to extract the samples of the quasi-optimal ISD-FPF. For recognizing the state of the ISD-FPF sample and completing the IS estimation of FPF, the Kriging model is secondly updated to establish ISD-FPF in the sample pool comprising these ISD-FPF samples. Using the proposed method, the whole FPF corresponding to arbitrary distribution parameters can be obtained using one IS reliability analysis based on the constructed ISD-FPF. Four examples and an application illustrate the advantages of the proposed method.

Meta model-based and cross entropy-based importance sampling algorithms for efficiently solving system failure probability function

The multi-mode system failure probability function (SFPF) can quantify how the distribution parameters of the random input vector affect the system safety and decouple the system reliability-based design optimization model. However, for a problem with a time-consuming implicit performance function and rare failure domain, efficiently solving the SFPF remains significantly challenging. Therefore, in this study, two efficient algorithms are proposed, namely, the meta model-based importance sampling and cross entropy-based importance sampling. The contributions of this study are twofold. The first is constructing a single-loop optimal importance sampling density (SL-OISD) method to decouple the double-loop framework for analyzing the SFPF. The second is establishing two methods to efficiently approximate the SL-OISD and complete the SFPF estimation. The first method is based on the meta model of the system performance function, which is abbreviated as SL-Meta-IS. The second method is based on minimizing the cross entropy between the Gaussian mixture density model and SL-OISD, which is abbreviated as SL-CE-IS. To reduce the number of evaluating the system performance function when approximating the SL-OISD, sampling the SL-OISD, and identifying the state of the samples for completing the SFPF estimation, an adaptive Kriging model of the system performance function is introduced into SL-Meta-IS and SL-CE-IS. Owing to decoupling the double-loop framework into a single-loop framework, replacing the time-consuming system performance function with the economic Kriging model, and employing importance sampling variance reduction techniques to address issues related to the rare failure domain, the proposed SL-Meta-IS and SL-CE-IS methods greatly enhance the efficiency of SFPF estimations. The numerical and practical examples demonstrate that the two proposed methods are superior to the existing algorithms; moreover, the efficiency of SL-CE-IS is higher than that of SL-Meta-IS.

Reliability analysis method of coupling optimal importance sampling density and multi-fidelity Kriging model

The commonly used reliability analysis approaches for Kriging-based models are usually conducted based on high-fidelity Kriging models. However, high-fidelity surrogate models are commonly costly. Therefore, in order to balance the calculation expense and calculation time of the surrogate model, this paper proposes a multi-fidelity Kriging model reliability analysis approach with coupled optimal important sampling density (OISD+MFK). First, the MEI learning function is proposed considering the training sample distance, model computation cost, expected improvement function, and model relevance. Second, a dynamic stopping condition is proposed that takes into account the failure probability estimation error. Finally, the optimal importance sampling density is incorporated into the reliability analysis process, which can effectively reduce failure probability estimation error. The results of the study show that the approach proposed in this paper can reduce the calculation cost while outputting relatively accurate failure probability evaluation results.

A review and assessment of importance sampling methods for reliability analysis

This paper reviews the mathematical foundation of the importance sampling technique and discusses two general classes of methods to construct the importance sampling density (or probability measure) for reliability analysis. The paper first explains the failure probability estimator of the importance sampling technique, its statistical properties, and computational complexity. The optimal but not implementable importance sampling density, derived from the variational calculus, is the starting point of the two general classes of importance sampling methods. For time-variant reliability analysis, the optimal but not implementable stochastic control is derived that induces the corresponding optimal importance sampling probability measure. In the first class, the optimal importance sampling density is directly approximated by a member of a family of parametric or nonparametric probability density functions. This approximation requires defining the family of approximating probability densities, a measure of distance between two probability densities, and an optimization algorithm. In the second class, the approximating importance sampling density has the general functional form of the optimal solution. The approximation amounts to replacing the limit-state function with a computationally convenient surrogate. The paper then explores the performances of the two classes of importance sampling methods through several benchmark numerical examples. The challenges and future directions of the importance sampling technique are also discussed.

Series System Reliability of Uncertain Linear Structures under Gaussian Excitation by Cross Entropy–Based Importance Sampling

We present an adaptive importance sampling (IS) method to estimate the reliability of linear structures with parameter uncertainties that are subjected to Gaussian process excitation. Structural failure is defined as a union of multiple first-passage failure events. The main contribution is the introduction of an efficient IS density for the uncertain structural parameters. This density is determined by minimizing the cross-entropy (CE) between the theoretically optimal IS density of the structural parameters and a chosen parametric family of probability distributions. The CE minimization procedure requires evaluation of the system failure probability conditional on fixed values of the uncertain parameters. A closed-form analytical approximation of this conditional failure probability was derived based on an upper bound on the out-crossing rate. Finally, a joint IS density of the random excitation and the uncertain structural parameters was introduced to estimate the series system failure probability involving parameter uncertainties. The accuracy and efficiency of the proposed method was demonstrated by means of two examples: a Gaussian white noise–excited two-story linear shear frame; and a six-story, three-bay moment-resisting steel frame subjected to a filtered nonstationary Gaussian excitation.

Cross entropy-based importance sampling for first-passage probability estimation of randomly excited linear structures with parameter uncertainty

We propose an efficient importance sampling (IS) method for estimating the first-passage probability of linear structures with uncertain parameters and subjected to Gaussian process excitations. The method evaluates the reliability through integrating the conditional first-passage probability given the uncertain structural parameters. We develop an adaptive IS strategy to efficiently perform this integration based on an IS density that is constructed using the cross entropy (CE) method. The CE method determines the IS density by adaptively minimizing the Kullback–Leibler divergence between the theoretically optimal sampling density and a chosen parametric family of probability distributions. The CE optimization problem is solved for a series of target densities that gradually approach the optimal IS density of the structural parameters. To define the intermediate densities, a smoothening of the conditional first-passage probabilities is employed. Once the IS density of the uncertain structural parameters is obtained, an effective IS density of the random excitations conditional on the structural parameters is introduced to estimate the failure probability of the structure. Unlike other tailored methods for solving this problem, the proposed IS approach does not require any prior analysis of the dynamic system and can be applied as a black-box method. Numerical examples demonstrate that the proposed method can calculate the first-passage probability with remarkable efficiency.

Efficient Importance Sampling in Quasi-Monte Carlo Methods for Computational Finance

We consider integration with respect to a $d$-dimensional spherical Gaussian measure arising from computational finance. Importance sampling (IS) is one of the most important variance reduction techniques in Monte Carlo (MC) methods. In this paper, two kinds of IS are studied in randomized quasi-MC (RQMC) setting, namely, the optimal drift IS (ODIS) and the Laplace IS (LapIS). Traditionally, the LapIS is obtained by mimicking the behavior of the optimal IS density with ODIS as its special case. We prove that the LapIS can also be obtained by an approximate optimization procedure based on the Laplace approximation. We study the promises and limitations of IS in RQMC methods and develop efficient RQMC-based IS procedures. We focus on how to properly combine IS with conditional MC (CMC) and dimension reduction methods in RQMC. In our procedures, the integrands are first smoothed by using CMC. Then the LapIS or the ODIS is performed, where several orthogonal matrices are required to be chosen to reduce the effective dimension. Intuitively, designing methods to determine all these optimal matrices seems infeasible. Fortunately, we prove that as long as the last orthogonal matrix is chosen elaborately, the choices of the other matrices can be arbitrary. This helps to significantly simplify the RQMC-based IS procedure. Due to the robustness and the superiority in efficiency of the gradient principal component analysis (GPCA) method, we use the GPCA method as an effective dimension reduction method in our RQMC-based IS procedures. Moreover, we prove the integrands obtained by the GPCA method are statistically equivalent. Numerical experiments illustrate the superiority of our proposed RQMC-based IS procedures.

Fuzzy importance sampling method for estimating failure possibility

To improve the computational efficiency of the fuzzy simulation in estimating failure possibility of the structure containing fuzzy uncertainty, a fuzzy importance sampling method is proposed in this paper. In the proposed method, the optimal importance sampling density for estimating failure possibility is deduced as the product of the indicator function related to failure domain and the joint membership function of fuzzy model inputs at first. Subsequently, the Markov chain simulation and an adaptive kernel sampling method are employed to generate a group of samples asymptotically following the optimal importance sampling density. Based on this group of samples, an approximate expression of the optimal importance sampling density expression is constructed. Finally, the failure possibility can be efficiently estimated by using a small number of samples generated by the approximate optimal importance sampling density. Results of three examples demonstrate that the proposed method is a more efficient and robust method for estimating the failure possibility estimation compared with the original fuzzy simulation method.

A novel dual-stage adaptive Kriging method for profust reliability analysis

Profust reliability analysis (RA) is established on probability assumption for the model inputs and fuzzy state assumption, and it is a useful tool to measure the safety degree of the structure with fuzzy failure state. In order to overcome the inefficiency of the existing AK-MCS method in profust RA for the structure with small profust failure probability, a novel dual-stage adaptive Kriging (DS-AK) method is proposed. Firstly, to improve the sampling efficiency, the importance sampling (IS) is used to estimate the profust failure probability where the optimal importance sampling density (OISD) is derived in this paper. As the analytical expression of the OISD cannot be acquired, the Kriging model for the true performance function is employed to build an approximate OISD. Then, the profust failure probability can be expressed as the product of the augmented profust failure probability and the correction factor, which are estimated in two different stages of the proposed DS-AK method. In the first stage of the DS-AK method, a Kriging model is constructed and updated to obtain the approximate OISD and generate the IS samples, and the augmented profust failure probability can be estimated as a byproduct. In the second stage of the DS-AK method, the current Kriging model is continuously updated to accurately predict the membership functions of the fuzzy failure domain at the IS samples generated in the first stage, from which the correction factor can be calculated efficiently. Finally, the fuzzy failure probability can be estimated as the product of augmented failure probability estimate obtained in the first stage and the correction factor estimate acquired in the second stage. Results of the validation cases demonstrate the accuracy, efficiency and robustness of the proposed method in estimating profust failure probability.

Improved cross entropy-based importance sampling with a flexible mixture model

The probability of a rare event or failure event is defined through a potentially high-dimensional integral, whose integration domain is often only known point-wise in terms of the outcome of a numerical model. The probability of failure can be estimated efficiently through importance sampling (IS), provided that an effective IS density is chosen. The cross entropy (CE) method is an adaptive sampling approach that determines the IS density through minimizing the Kullback–Leibler divergence between the theoretically optimal IS density and a chosen parametric family of distributions. We propose an improved version of the classical CE method that introduces a smooth transition to make better use of the samples from intermediate sampling levels for fitting the sought IS density. The improved CE method is combined with a novel flexible parametric distribution model that is able to handle low- and high-dimensional problems as well as problems with multimodal failure domains. A set of numerical examples demonstrate that the proposed approach performs consistently better than the classical CE method in various problem settings.

Cross-entropy based importance sampling for stochastic simulation models

To efficiently evaluate system reliability based on Monte Carlo simulation, importance sampling is used widely. The optimal importance sampling density was derived in 1950s for the deterministic simulation model, which maps an input to an output deterministically, and is approximated in practice using various methods. For the stochastic simulation model whose output is random given an input, the optimal importance sampling density was derived only recently. In the existing literature, metamodel-based approaches have been used to approximate this optimal density. However, building a satisfactory metamodel is often difficult or time-consuming in practice. This paper proposes a cross-entropy based method, which is automatic and does not require specific domain knowledge. The proposed method uses an expectation–maximization algorithm to guide the choice of a mixture distribution model for approximating the optimal density. The method iteratively updates the approximated density to minimize its estimated discrepancy, measured by estimated cross-entropy, from the optimal density. The mixture model’s complexity is controlled using the cross-entropy information criterion. The method is empirically validated using extensive numerical studies and applied to a case study of evaluating the reliability of wind turbine using a stochastic simulation model.

An efficient exponential twisting importance sampling technique for pricing financial derivatives

This paper develops an efficient Exponential Twisting Importance Sampling technique for variance reduction when it is used to price financial derivatives. A general class of exponential change of measure is well explored. This is the generalization of the proportional exponential twisting function, , discussed as an Example 4.6.2 in Glasserman’s book (2004). Then a new framework to find the optimal importance sampling density function is proposed based on the Cauchy-Schwartz Inequality and the Least Square Approach. In the special Gaussian case, the theory of our general exponential change of measure means finding the new drift vector and covariance matrix simultaneously. The method proposed by the paper has little smoothness requirements for the payoff functions and doesn’t rely on the initial values. It is illustrated that this method is high efficient for pricing financial derivatives, such as Asian options, Straddle options, Volatility derivatives and the pricing under the stochastic volatility models. Furthermore, this method can be extended to more general importance sampling densities such as non-Gaussian or multi-modal distributions.

Cross entropy-based importance sampling using Gaussian densities revisited

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.

Reliability sensitivity estimation with sequential importance sampling

In applications of reliability analysis, the sensitivity of the probability of failure to design parameters is often crucial for decision-making. A common sensitivity measure is the partial derivative of the probability of failure with respect to the design parameter. If the design parameter enters the definition of the reliability problem through the limit-state function, i.e. the function defining the failure event, then the partial derivative is given by a surface integral over the limit-state surface. Direct application of standard Monte Carlo methods for estimation of surface integrals is not possible. To circumvent this difficulty, an approximation of the surface integral in terms of a domain integral has been proposed by the authors. In this paper, we propose estimation of the domain integral through application of a method termed sequential importance sampling (SIS). The basic idea of SIS is to gradually translate samples from the distribution of the random variables to samples from an approximately optimal importance sampling density. The transition of the samples is defined through the construction of a sequence of intermediate distributions, which are sampled through application of a resample-move scheme. We demonstrate effectiveness of the proposed method in estimating reliability sensitivities to both distribution and limit-state parameters with numerical examples.

An adaptive importance sampling method with a Kriging metamodel to calculate failure probability

A Markov chain simulation was performed to extract points in a failure region. A Kriging metamodel was constructed to approximate a limit state based on the points extracted by the Markov chain simulation. A kernel sampling density was constructed to approximate the optimal importance sampling density. The points extracted in the failure region by the Markov chain simulation were assumed as a mean of each kernel. An importance sampling method was applied to calculate the failure probability. In the importance sampling method, points are extracted from the kernel in the vicinity of a limit state. Considering the statistical distance as well as the learning function, additional experimental points were selected for the kriging metamodel. A stable numerical calculation method was applied to find the parameters of the kernel sampling density. The completeness of the Kriging metamodel was evaluated on the basis of possible changes in failure probability.

Cross-entropy method for estimation of posterior expectation in Bayesian VAR models

ABSTRACTIn this article, an importance sampling (IS) method for the posterior expectation of a non linear function in a Bayesian vector autoregressive (VAR) model is developed. Most Bayesian inference problems involve the evaluation of the expectation of a function of interest, usually a non linear function of the model parameters, under the posterior distribution. Non linear functions in Bayesian VAR setting are difficult to estimate and usually require numerical methods for their evaluation. A weighted IS estimator is used for the evaluation of the posterior expectation. With the cross-entropy (CE) approach, the IS density is chosen from a specified family of densities such that the CE distance or the Kullback–Leibler divergence between the optimal IS density and the importance density is minimal. The performance of the proposed algorithm is assessed in an iterated multistep forecasting of US macroeconomic time series.

On the quantification and efficient propagation of imprecise probabilities resulting from small datasets

Abstract This paper addresses the problem of uncertainty quantification and propagation when data for characterizing probability distributions are scarce. We propose a methodology wherein the full uncertainty associated with probability model form and parameter estimation are retained and efficiently propagated. This is achieved by applying the information-theoretic multimodel inference method to identify plausible candidate probability densities and associated probabilities that each method is the best model in the Kullback-Leibler sense. The joint parameter densities for each plausible model are then estimated using Bayes’ rule. We then propagate this full set of probability models by estimating an optimal importance sampling density that is representative of all plausible models, propagating this density, and reweighting the samples according to each of the candidate probability models. This is in contrast with conventional methods that try to identify a single probability model that encapsulates the full uncertainty caused by lack of data and consequently underestimate uncertainty. The result is a complete probabilistic description of both aleatory and epistemic uncertainty achieved with several orders of magnitude reduction in computational cost. It is shown how the model can be updated to adaptively accommodate added data and added candidate probability models. The method is applied for uncertainty analysis of plate buckling strength where it is demonstrated how dataset size affects the confidence (or lack thereof) we can place in statistical estimates of response when data are lacking.

A new maximum entropy-based importance sampling for reliability analysis

Importance sampling can be highly efficient if a good importance sampling density is constructed. Although the parametric sampling densities centered on the design points are often good choices, the determination of the design points can be a difficult and inefficient task itself, especially for problems with multiple design points, or highly nonlinear limit state functions. This paper introduces a nonparametric importance sampling method based on the Markov chain simulation and maximum-entropy density estimation (MEDE). In the proposed method, Markov chain simulation is utilized to generate samples that distribute asymptotically to the optimal importance sampling density. A nonparametric estimation of the optimal importance sampling density is then obtained using the MEDE technique. The conventional MEDE method is difficult for multi-dimensional problems as it needs to solve a set of simultaneous nonlinear integral equations. This paper developed a new MEDE technique for multivariate dataset. The method starts with using histogram to approximate a density. The multi-dimensional histogram is converted into a series of one-dimensional conditional PDFs in each dimension and the density is reconstructed by means of orthogonal expansion. Thus, the solution of MEDE is converted to a set of coefficients of the Legendre polynomials. The new importance sampling method is illustrated and compared with the classical kernel-based importance sampling using a number of numerical and structural examples.

Sequential importance sampling for structural reliability analysis

This paper proposes the application of sequential importance sampling (SIS) to the estimation of the probability of failure in structural reliability. SIS was developed originally in the statistical community for exploring posterior distributions and estimating normalizing constants in the context of Bayesian analysis. The basic idea of SIS is to gradually translate samples from the prior distribution to samples from the posterior distribution through a sequential reweighting operation. In the context of structural reliability, SIS can be applied to produce samples of an approximately optimal importance sampling density, which can then be used for estimating the sought probability. The transition of the samples is defined through the construction of a sequence of intermediate distributions. We present a particular choice of the intermediate distributions and discuss the properties of the derived algorithm. Moreover, we introduce two MCMC algorithms for application within the SIS procedure; one that is applicable to general problems with small to moderate number of random variables and one that is especially efficient for tackling high-dimensional problems.