Importance Sampling Procedure Research Articles

A single-loop reliability sensitivity analysis strategy for time-dependent rare events with both random variables and stochastic processes

To deal with the time-dependent reliability sensitivity (TDRS) analysis of rare events with both random variables and stochastic processes, a single-loop sampling procedure combined with the cross entropy-based importance sampling strategy (CEIS), namely SCEIS, is derived to lighten the computational burden without loss of precision. Firstly, the TDRS indices are derived to make them implementable by a single-loop importance sampling (IS) procedure. Then, the quasi-optimal IS density is obtained through the extension of the cross-entropy method, which is an adaptive procedure that overcomes the challenges of determining the position and number of design points in traditional IS. Furthermore, the SCEIS can be easily implemented to estimate the TDRS indices for problems with small failure probability. Three examples involving numerical and engineering problems are analyzed to demonstrate the performance of the proposed method. The results obtained from comparing with other existing methods reveal that the proposed method provides satisfactory TDRS analysis for rare events while significantly reducing computational costs.

Unified perspective on exponential tilt and bridge algorithms for rare trajectories of discrete Markov processes.

This article analyzes and compares two general techniques of rare event simulation for generating paths of Markov processes over fixed time horizons: exponential tilting and stochastic bridge. These two methods allow us to accurately compute the probability that a Markov process ends within a rare region which is unlikely to be attained. Exponential tilting is a general technique for obtaining an alternative or tilted sampling probability measure, under which the Markov process becomes likely to hit the rare region at terminal time. The stochastic bridge technique involves conditioning paths towards two endpoints: the terminal point and the initial one. The terminal point is generated from some appropriately chosen probability distribution that covers well the rare region. We show that both methods belong to the class of importance sampling procedures by providing a common mathematical framework of these two conceptually different methods of sampling rare trajectories. We also conduct a numerical comparison of these two methods, revealing distinct areas of application for each Monte Carlo method, where they exhibit superior efficiency. Detailed simulation algorithms are provided.

Two-Phase Adaptive Kriging Model Based Importance Sampling Method for Estimating Time-Dependent Failure Probability

To improve the computational efficiency of the time-dependent failure probability (TDFP) to system with dynamic loads, materials degeneration and other time-varying factors, a two-phase adaptive Kriging model-based importance sampling (IS) method is proposed. The main contribution of this article is that a single Kriging model is established and adaptively refined to gradually approach the optimal IS probability density function (PDF) and estimate TDFP in two sequential phases. In the first phase, a generally approximate IS PDF is constructed by Kriging model based on its prediction characteristic and subsequently updated until convergence. Then, a simple rejection sampling technique is used to robustly generate IS samples for the estimation of TDFP. In the second phase, current Kriging model is subsequently refined for accurately identifying the classification of the IS samples generated in the last phase. Finally, the TDFP can be estimated using the products obtained in the above two phases. As the IS PDF established by Kriging model is used for directly approximating the optimal one, the proposed method has the capacity to deal with the complicated problem containing multiple failure regions. Additionally, since IS procedure can dramatically improve the sampling efficiency, the proposed method is an effective technique to the time-consuming TDFP analysis problems.

Randomly Censored Kumaraswamy Distribution

In this paper, inferential procedures based on classical and Bayesian framework for the Kumaraswamy distribution under random censoring model are studied. We first propose estimators for the distribution parameters, reliability function, failure rate function, and Mean time to system failure based on the maximum likelihood estimation method. Then, we calculate asymptotic confidence intervals for the parameters based on the observed Fisher’s information matrix. Also, for the parameters and reliability characteristics, Bayesian estimates are derived using the importance sampling and Gibbs sampling procedures. Highest posterior density credible intervals for the parameters are constructed using Markov Chain Monte Carlo method. Expected time on test of experiment with random censoring is also calculated. A simulation study is conducted to compare the efficiency of the derived estimates. Finally, the analysis of a real data set is presented for the illustration purpose.

Hamiltonian MCMC methods for estimating rare events probabilities in high-dimensional problems

Accurate and efficient estimation of rare events probabilities is of significant importance, since often the occurrences of such events have widespread impacts. The focus in this work is on precisely quantifying these probabilities, often encountered in reliability analysis of complex engineering systems, based on an introduced framework termed Approximate Sampling Target with Post-processing Adjustment (ASTPA), which herein is integrated with and supported by gradient-based Hamiltonian Markov Chain Monte Carlo (HMCMC) methods. The developed techniques in this paper are applicable from low- to high-dimensional stochastic spaces, and the basic idea is to construct a relevant target distribution by weighting the original random variable space through a one-dimensional output likelihood model, using the limit-state function. To sample from this target distribution, we exploit HMCMC algorithms, a family of MCMC methods that adopts physical system dynamics, rather than solely using a proposal probability distribution, to generate distant sequential samples, and we develop a new Quasi-Newton mass preconditioned HMCMC scheme (QNp-HMCMC), which is particularly efficient and suitable for high-dimensional spaces. To eventually compute the rare event probability, an original post-sampling step is devised using an inverse importance sampling procedure based on the already obtained samples. The statistical properties of the estimator are analyzed as well, and the performance of the proposed methodology is examined in detail and compared against Subset Simulation in a series of challenging low- and high-dimensional problems.

A localization scheme based on Improving Dynamic Population Monte Carlo Localization method for large‐scale mobile wireless aquaculture sensor networks

AbstractLocalization is one of the essential problems in wireless sensor applications (WSNs). Most range‐free localization schemes for mobile WSNs are based on the Sequential Monte Carlo (SMC) algorithm. Multiple iterations, sample impoverishment and less sample diversity, leading to low localizing efficiency, are the most usual problems demanding to be solved in these SMC‐based methods. An improved localization scheme for mobile aquaculture WSNs based on the Improving Dynamic Population Monte Carlo Localization (I‐DPMCL) method is proposed. A population of probability density functions is proposed to approximate the unknown location distribution based on a set of observations through an iterative mixture importance sampling procedure, accompanied by node dynamic behaviours being analysed quantitatively or definitely. Threefold constrain rules are put forward in the I‐DPMCL scheme to decrease the iteration number and trade off iteration number and enough valid samples to obtain the optimum iteration number. Then, these localization behaviours, especial delay, are predicted based on the statistical point of view. Moreover, performance comparisons of I‐DPMCL with other SMC‐based schemes are also proposed. Simulation results show that delay of I‐DPMCL has some superiority to those of other schemes, and accuracy and energy consumption are improved in some cases of lower mobile velocity.

Bayesian Log-Rank Test

Comparison of two survival curves is a fundamental problem in survival analysis. Although abundant frequentist methods have been developed for comparing survival functions, inference procedures from the Bayesian perspective are rather limited. In this article, we extract the quantity of interest from the classic log-rank test and propose its Bayesian counterpart. Monte Carlo methods, including a Gibbs sampler and a sequential importance sampling procedure, are developed to draw posterior samples of survival functions and a decision rule of hypothesis testing is constructed for making inference. Via simulations and real data analysis, the proposed Bayesian log-rank test is shown to be asymptotically equivalent to the classic one when noninformative prior distributions are used, which provides a Bayesian interpretation of the log-rank test. When using the correct prior information from historical data, the Bayesian log-rank test is shown to outperform the classic one in terms of power. R codes to implement the Bayesian log-rank test are also provided with step-by-step instructions.

Parameter Estimation of Exponentiated Half-Logistic Distribution for Left-Truncated and Right-Censored Data

Left-truncated and right-censored data are widely used in lifetime experiments, biomedicine, labor economics, and actuarial science. This article discusses how to resolve the problems of statistical inferences on the unknown parameters of the exponentiated half-logistic distribution based on left-truncated and right-censored data. In the beginning, maximum likelihood estimations are calculated. Then, asymptotic confidence intervals are constructed by using the observed Fisher information matrix. To cope with the small sample size scenario, we employ the percentile bootstrap method and the bootstrap-t method for the establishment of confidence intervals. In addition, Bayesian estimations under both symmetric and asymmetric loss functions are addressed. Point estimates are computed by Tierney–Kadane’s approximation and importance sampling procedure, which is also applied to establishing corresponding highest posterior density credible intervals. Lastly, simulated and real data sets are presented and analyzed to show the effectiveness of the proposed methods.

An Adaptive-Importance-Sampling-Enhanced Bayesian Approach for Topology Estimation in an Unbalanced Power Distribution System

The reliable operation of a power distribution system relies on a good prior knowledge of its topology and its system state. Although crucial, due to the lack of direct monitoring devices on the switch statuses, the topology information is often unavailable or outdated for the distribution system operators for real-time applications. Apart from the limited observability of the power distribution system, other challenges are the nonlinearity of the model, the complicated, unbalanced structure of the distribution system, and the scale of the system. To overcome the above challenges, this paper proposes a Bayesian-inference framework that allows us to simultaneously estimate the topology and the state of a three-phase, unbalanced power distribution system. Specifically, by using the very limited number of measurements available that are associated with the forecast load data, we efficiently recover the full Bayesian posterior distributions of the system topology under both normal and outage operation conditions. This is performed through an adaptive importance sampling procedure that greatly alleviates the computational burden of the traditional Monte-Carlo (MC)-sampling-based approach while maintaining a good estimation accuracy. The simulations conducted on the IEEE 123-bus test system and an unbalanced 1282-bus system reveal the excellent performances of the proposed method.

Inference of partially observed causes for failure of Lomax competing risks model under type-II generalized hybrid censoring scheme

Time-to- failure under different causes of failure is known as a competing risks model. Practice, competing risks data can be appeared in different applications such as engineering fields or biological and medical lifetime studies as well as other related areas. Also, the causes of failure, which are competing may be partially observed. In this paper, we adopted the competing risks model with partially observed causes of failure when the latent failure times follow Lomax life distribution under type-II generalized hybrid censoring scheme. The maximum likelihood estimators of the model parameters with the associated confidence intervals are discussed. Moreover, Bayes estimators under importance sampling procedure with probability credible intervals are developed. The results are discussed using both real and simulated data sets for illustration purposes. Finally, the Monte Carlo simulation experiments are performed to assess and compare the different proposed methods with some brief comments.

Estimation of the Inverse Weibull Distribution Parameters under Type-I Hybrid Censoring

The hybrid censoring is a mixture of type-I and type-II censoring schemes. This paper presents the statistical inferences of the inverse Weibull distribution parameters when the data are type-I hybrid censored. First, we consider the maximum likelihood estimates of the unknown parameters. It is observed that the maximum likelihood estimates can not be obtained in closed form. We further obtain the Bayes estimates and the corresponding highest posterior density credible intervals of the unknown parameters under the assumption of independent gamma priors using the importance sampling procedure. We also compute the approximate Bayes estimates using Lindley's approximation technique. The performance of the Bayes estimates have been compared with maximum likelihood estimates through the Monte Carlo Markov chain techniques. Finally, a real data set have been analysed for illustration purpose.

Inference for Inverse Power Lomax Distribution with Progressive First-Failure Censoring.

This paper investigates the statistical inference of inverse power Lomax distribution parameters under progressive first-failure censored samples. The maximum likelihood estimates (MLEs) and the asymptotic confidence intervals are derived based on the iterative procedure and asymptotic normality theory of MLEs, respectively. Bayesian estimates of the parameters under squared error loss and generalized entropy loss function are obtained using independent gamma priors. For Bayesian computation, Tierney–Kadane’s approximation method is used. In addition, the highest posterior credible intervals of the parameters are constructed based on the importance sampling procedure. A Monte Carlo simulation study is carried out to compare the behavior of various estimates developed in this paper. Finally, a real data set is analyzed for illustration purposes.

Inference on Nadarajah–Haghighi distribution with constant stress partially accelerated life tests under progressive type-II censoring

This paper presents methods of estimation of the parameters and acceleration factor for Nadarajah–Haghighi distribution based on constant-stress partially accelerated life tests. Based on progressive Type-II censoring, Maximum likelihood and Bayes estimates of the model parameters and acceleration factor are established, respectively. In addition, approximate confidence interval are constructed via asymptotic variance and covariance matrix, and Bayesian credible intervals are obtained based on importance sampling procedure. For comparison purpose, alternative bootstrap confidence intervals for unknown parameters and acceleration factor are also presented. Finally, extensive simulation studies are conducted for investigating the performance of the our results, and two data sets are analyzed to show the applicabilities of the proposed methods.

Classical and Bayesian Inference for a Progressive First-Failure Censored Left-Truncated Normal Distribution

Point and interval estimations are taken into account for a progressive first-failure censored left-truncated normal distribution in this paper. First, we derive the estimators for parameters on account of the maximum likelihood principle. Subsequently, we construct the asymptotic confidence intervals based on these estimates and the log-transformed estimates using the asymptotic normality of maximum likelihood estimators. Meanwhile, bootstrap methods are also proposed for the construction of confidence intervals. As for Bayesian estimation, we implement the Lindley approximation method to determine the Bayesian estimates under not only symmetric loss function but also asymmetric loss functions. The importance sampling procedure is applied at the same time, and the highest posterior density (HPD) credible intervals are established in this procedure. The efficiencies of classical statistical and Bayesian inference methods are evaluated through numerous simulations. We conclude that the Bayes estimates given by Lindley approximation under Linex loss function are highly recommended and HPD interval possesses the narrowest interval length among the proposed intervals. Ultimately, we introduce an authentic dataset describing the tensile strength of 50mm carbon fibers as an illustrative sample.

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.

Efficient Computation of Extreme Excursion Probabilities for Dynamical Systems through Rice's Formula

We develop a novel computational method for evaluating the extreme excursion probabilities arising from random initialization of nonlinear dynamical systems. The method uses a Markov chain Monte Carlo or a Laplace approximation approach to construct a biasing distribution that in turn is used in an importance sampling procedure to estimate the extreme excursion probabilities. The prior and likelihood of the biasing distribution are obtained by using Rice's formula from excursion probability theory. We use Gaussian mixture biasing distributions and approximate the non-Gaussian initial state distribution by the method of moments to circumvent the linearity and Gaussianity assumptions needed by excursion probability theory. The method requires the tangent linear model of the nonlinear dynamical system to be bounded. Additionally, our method works best when the uncertainty-induced trajectories of the nonlinear dynamical system are small perturbations around a mean linear trajectory and thus when the target system exhibits only mild nonlinearity. We demonstrate the effectiveness of this computational framework for nonlinear dynamical systems of up to 100 dimensions.

Inference for Kumaraswamy Distribution Based on Type I Progressive Hybrid Censoring

In this paper, we investigate the estimation problems of unknown parameters of the Kumaraswamy distribution under type I progressive hybrid censoring. This censoring scheme is a combination of progressive type I and hybrid censoring schemes. We derive the maximum likelihood estimates of parameters using an expectation-maximization algorithm. Bayes estimates are obtained under different loss functions using the Lindley method and importance sampling procedure. The highest posterior density intervals of unknown parameters are constructed as well. We also obtain prediction estimates and prediction intervals for censored observations. A Monte Carlo simulation study is performed to compare proposed methods and one real data set is analyzed for illustrative purposes.

Reliability and reliability sensitivity analysis of structure by combining adaptive linked importance sampling and Kriging reliability method

The application of reliability analysis and reliability sensitivity analysis methods to complicated structures faces two main challenges: small failure probability (typical less than 10−5) and time-demanding mechanical models. This paper proposes an improved active learning surrogate model method, which combines the advantages of the classical Active Kriging – Monte Carlo Simulation (AK-MCS) procedure and the Adaptive Linked Importance Sampling (ALIS) procedure. The proposed procedure can, on the one hand, adaptively produce a series of intermediate sampling density approaching the quasi-optimal Importance Sampling (IS) density, on the other hand, adaptively generate a set of intermediate surrogate models approaching the true failure surface of the rare failure event. Then, the small failure probability and the corresponding reliability sensitivity indices are efficiently estimated by their IS estimators based on the quasi-optimal IS density and the surrogate models. Compared with the classical AK-MCS and Active Kriging – Importance Sampling (AK-IS) procedure, the proposed method neither need to build very large sample pool even when the failure probability is extremely small, nor need to estimate the Most Probable Points (MPPs), thus it is computationally more efficient and more applicable especially for problems with multiple MPPs. The effectiveness and engineering applicability of the proposed method are demonstrated by one numerical test example and two engineering applications.

Analysis of Reliability Characteristics of Bathtub-Shaped Distribution Under Adaptive Type-I Progressive Hybrid Censoring

In reliability and life testing experiments, obtaining complete data still consumes lots of time, financial and human supports. A censoring scheme which can have balance between the total testing time, the used number of units and cost in life tests is desirable. Nevertheless, with short product development times, life testing experiments must be performed with severe time constraints, which makes failure-censored schemes such as type-II censoring and progressive type-II censoring schemes are no longer applicable in many real life scenarios. Therefore, an adaptive type-I progressive hybrid censoring scheme has been shown to be useful in this case, which assures the termination of the life testing experiment at a predetermined time and results in a higher efficiency estimations. In this paper we consider the estimation problem of parameters, reliability and hazard rate functions of a two-parameter bathtub-shaped distribution under adaptive type-I progressive hybrid censoring scheme using the maximum likelihood and Bayesian estimation methods. Lindley’s approximation and an importance sampling procedure under the assumption of independent gamma priors are used to obtain the Bayes estimators. Approximate confidence, two parametric bootstrap and highest posterior density intervals are also obtained for the unknown parameters, reliability and hazard rate functions. The approximate confidence intervals of the reliability and hazard rate functions are obtained by estimating their variances using delta method. A simulation study is conducted to compare the performance of the different estimators in terms of their mean squared errors and interval lengths. Finally, two real data sets are analyzed to show the applicability of the different estimations. The data analysis showed that the adaptive type-I progressive hybrid censoring scheme is flexible and very useful in terminating the experiment when the time is the main concern to the experimenter.

Statistical Inference on the Shannon Entropy of Inverse Weibull Distribution under the Progressive First-Failure Censoring

Entropy is an uncertainty measure of random variables which mathematically represents the prospective quantity of the information. In this paper, we mainly focus on the estimation for the parameters and entropy of an Inverse Weibull distribution under progressive first-failure censoring using classical (Maximum Likelihood) and Bayesian methods. For Bayesian approaches, the Bayesian estimates are obtained based on both asymmetric (General Entropy, Linex) and symmetric (Squared Error) loss functions. Due to the complex form of Bayes estimates, we cannot get an explicit solution. Therefore, the Lindley method as well as Importance Sampling procedure is applied. Furthermore, using Importance Sampling method, the Highest Posterior Density credible intervals of entropy are constructed. As a comparison, the asymptotic intervals of entropy are also gained. Finally, a simulation study is implemented and a real data set analysis is performed to apply the previous methods.