Parameter estimation for load‐sharing system subject to Wiener degradation process using the expectation‐maximization algorithm

In practice, many systems exhibit load‐sharing behavior, where the surviving components share the total load imposed on the system. Different from general systems, the components of load‐sharing systems are interdependent in nature, in such a way that when one component fails, the system load has to be shared by the remaining components, which increases the failure rate or degradation rate of the remaining components. Because of the load‐sharing mechanism among components, parameter estimation and reliability assessment are usually complicated for load‐sharing systems. Although load‐sharing systems with components subject to sudden failures have been intensely studied in literatures with detailed estimation and analysis approaches, those with components subject to degradation are rarely investigated. In this paper, we propose the parameter estimation method for load‐sharing systems subject to continuous degradation with a constant load. Likelihood function based on the degradation data of components is established as a first step. The maximum likelihood estimators for unknown parameters are deduced and obtained via expectation‐maximization (EM) algorithm considering the nonclosed form of the likelihood function. Numerical examples are used to illustrate the effectiveness of the proposed method.     

A failure mechanism consistency test method for accelerated degradation test

Accelerated degradation test (ADT) is generally used to accelerate degradation processes in products to estimate their lifespan and to assess their reliability in a short period of time. How to perform the failure mechanism consistency test is crucial in the application of the ADT method. Existing failure mechanism consistency test methods assume that degradation rates among individual products are the same. However, these methods do not take degradation dispersions caused by manufacturing technologies into consideration. To address this issue, a failure mechanism consistency test method for ADT based on the activation energy invariant method and the likelihood ratio test is proposed. First, a degradation modeling method for ADT is introduced. Then, the logarithmic maximum likelihood function values of the degradation models are estimated based on the two-step maximum likelihood estimation (MLE) method. Finally, the decision rule is proposed based on the likelihood ratio test. The method mentioned above is, then, used on the real degradation data of carbon-film resistors and bullet O-rings, and its effectiveness is verified. Furthermore, based on the failure mechanism change point in RTV5370 siloxane rubbers, the simulated degradation data are degenerated to compare the proposed method with the method not considering individual differences in different ADT programs and degradation dispersions.     

A two‐stage degradation model considering the stage‐varying of dispersity regulation

In reliability analysis, degradation test has been recognized as an effective method for high reliable products and complex systems when key performance indicators can be observed. Then, a reasonable degradation model becomes a key issue to guarantee a reasonable reliability assessment. Motivated by practical needs, this paper proposes a novel two‐stage degradation model considering the different dispersity regulations corresponding to the two stages. A maximum likelihood estimation (MLE) method for unknown parameters is established, and an initial guess procedure is given to improve the efficiency of optimization algorithm. Then, the reliability inference regarding the product population is discussed. A comprehensive simulation study is conducted to validate the proposed approach where the two‐stage Wiener process model is adopted as a reference for a better understanding. Finally, the constructed model is further verified by two real applications. Comparative results clearly demonstrate the reasonability and effectiveness of the proposed model.     

Two optimized models of life prediction based on luminance degradation for VFD under a conventional life test

Focusing on improving the accuracy of existing life prediction models for optoelectronic products, the three‐parameter Weibull right approximation method (TPWRAM) was employed to substitute exponential function based on the least square method in the analysis and two‐staged methods. Two optimized models were established (Model I and Model II), based on maximum likelihood estimation and the Monte Carlo method, respectively. One group of conventional life tests (CLTs) of vacuum fluorescent display (VFD) were conducted to collect luminance degradation data for each sample, and the two optimized models were applied to achieve VFD life prediction and obtain mean time to failure, median life, and confidence intervals. The results indicate that the CLT test design is correct and feasible, the amount of data on luminance degradation is large, and the test data selection method is reasonable. Model I and Model II optimized by TPWRAM both reflect the VFD luminance variation law well, and the predicted life approaches VFD service life from user feedback, proving that the two models are precise, and thus, can provide technical references for researchers and engineers regarding aspects of life prediction.     

基于双方差随机过程的半导体激光器寿命评估

针对高可靠、长寿命半导体激光器的寿命评估问题,提出了基于双方差随机过程的性能退化评估方法。该方法不仅考虑了半导体激光器内部失效机理的固有随机性,还考虑了由人为因素、测量仪器等引起的测量随机误差。首先,建立了半导体激光器性能退化模型及其未知参数的极大似然估计方法。然后,基于首达时的概念给出了失效时间分布函数和概率密度函数的解析表达式,以对半导体激光器的可靠性和寿命进行评估。最后,通过半导体激光器寿命评估工程实例验证了所提出方法的适用性和有效性。结果表明：与现有的性能退化模型相比,所构建模型的拟合效果更好,能够提高寿命评估精度。这可为半导体激光器及其整机系统最优维修决策的制定提供有力支撑。     

Robust Estimation for Weibull Distribution in Partially Accelerated Life Tests with Early Failures

Maximum likelihood estimation (MLE) is a frequently used method for estimating distribution parameters in constant stress partially accelerated life tests (CS‐PALTs). However, using the MLE to estimate the parameters for a Weibull distribution may be problematic in CS‐PALTs. First, the equation for the shape parameter estimator derived from the log‐likelihood function is difficult to solve for the occurrence of nonlinear equations. Second, the sample size is typically not large in life tests. The MLE, a typical large‐sample inference method, may be unsuitable. Test items unsuitable for stress conditions may become early failures, which have extremely short lifetimes. The early failures may cause parameter estimate bias. For addressing early failures in the Weibull distribution in CS‐PALTs, we propose an M‐estimation method based on a Weibull Probability Plot (WPP) framework, which leads a closed‐form expression for the shape parameter estimator. We conducted a simulation study to compare the M‐estimation method with the MLE method. The results show that, with early‐failure samples, the M‐estimation method performs better than the MLE does. Copyright © 2015 John Wiley & Sons, Ltd.     

Remaining Life Prediction Based on Wiener Processes with ADT Prior Information

Precisely predicting the remaining life for an individual plays an important role in condition‐based maintenance, so Bayesian inference method, which can integrate useful data from several sources to improve the prediction accuracy, has became a research hot. Aiming at the situation that accelerated degradation tests have been widely applied to assess the reliability of products, a remaining life prediction method based on Bayesian inference by taking accelerated degradation data as prior information is proposed. A Wiener process with random drift, diffusion parameters is used to model degradation data, and conjugate prior distributions of random parameters are adopted. To solve the problem that it is hard to estimate the hyper parameters from accelerated degradation data using an Expectation Maximization algorithm, a data extrapolation method is developed. With acceleration factors, degradation data are extrapolated from accelerated stress levels to the normal use stress level. Acceleration factor constant hypothesis is used to deduce the expression of acceleration factor for a Wiener degradation model. Besides, simulation tests are designed to validate the proposed method. The method of constructing the confidence levels for the remaining life predictions is also provided. Finally, a case study is used to illustrate the application of our developed method. Copyright © 2015 John Wiley & Sons, Ltd.     

Equivalence analysis of accelerated degradation mechanism based on stochastic degradation models

For an effective accelerated degradation test, it is important to ensure that the degradation mechanism under different stress levels remains unchanged. In this article, we are interested in the equivalence analysis of accelerated degradation mechanism based on degradation data rather than physical or chemical techniques. Under the assumption that products' underlying degradation follows stochastic degradation models, we first introduce the relationship between mechanism equivalence and parameters of stochastic degradation models based on the acceleration factor invariant principle. Then the necessary conditions for mechanism equivalence, which should be satisfied under different stress levels, are derived and tested by the proposed parameter equivalence test method based on the modified Bartlett statistic and T statistic. Next a novel selection method for stochastic degradation models is derived therefrom by comparing the variation of coefficients of acceleration factors. The accuracy of the necessary conditions and the parameter equivalence test method is demonstrated through a simulation study. In addition, an electrical connector example with real stress relaxation data is analyzed to illustrate the proposed method further.     

分布源目标方位估计的降维最大似然估计

介绍了已有的分布源目标方位估计中的最大似然估计(MLE)算法,它是四维非线性最优化问题,文中称之为四维MLE算法,因计算量庞大,同时提出了一种降为三维的MLE算法,简化为三维非线性最优化,称之为三维MLE算法。两种算法均采用牛顿型搜索算法,来搜索未知参数的全局最优点。在单次迭代过程中,三维MLE算法比四维MLE算法减少了51次协方差矩阵求逆和87次矩阵乘法,搜索效率得到提高,并且节省了存储空间。得出了新算法克拉美-罗界的计算公式,其计算量也有所降低。计算机仿真验证,三维MLE算法和四维MLE算法的估计精度相当,新算法在减少计算量的同时并无损失性能,所以实用性和实时性都得到显著提高。     

EM algorithm for estimating the Burr XII parameters with multiple censored data

The two‐parameter Burr XII distribution has been widely used in various practical applications such as business, chemical engineering, quality control, medical research and reliability engineering. In this paper, we present maximum likelihood estimation (MLE) via the expectation–maximization (EM) algorithm to estimate the Burr XII parameters with multiple censored data. We also provide a method that can be used to construct the confidence intervals of the parameters, a method that computes the asymptotic variance and the covariance of the MLE from the complete and missing information matrices. A simulation study is conducted to compare the performance of the MLE via the EM algorithm and the Netwon–Raphson (NR) algorithm. The simulation results show that the EM algorithm outperforms the NR algorithm in most cases in terms of bias and errors in the root mean square. A numerical example is also used to demonstrate the performance of the proposed method. Copyright © 2009 John Wiley & Sons, Ltd.     

双参数Rayleigh分布环境因子的经验Bayes估计及应用

本文针对Rayleigh分布位置参数已知的情形,给出了Rayleigh分布环境因子的极大似然估计和经验Bayes估计,并将环境因子的估计结果应用于Rayleigh部件的可靠性评估,给出了该部件可靠度函数与失效率的估计。最后的随机模拟例子表明,经验Bayes估计优于极大似然估计,并且在考虑环境因子的情形下,Rayleigh部件可靠性指标的估计优于未考虑环境因子时的估计。     

Degradation modeling and classification of mixed populations using segmental continuous hidden Markov models

With the market demands, the classification for highly reliable products becomes more and more significant. The degradation data can provide information about the degradation states and can be used to classify products to various classes according to the reliability attribute. In this paper, a temporal probabilistic approach, named segmental continuous hidden Markov model (SCHMM), is proposed to tackle the problem of degradation modeling and classification for mixed populations. Separate SCHMMs are built for each class of the mixed populations. The SCHMMs can directly depict the correspondence between actual degradation and the hidden states. A novel method called self‐training algorithm for the preprocessing of the original data from the mixed populations is proposed. Furthermore, the unknown parameters of the SCHMMs are estimated by the maximum likelihood method with the complete degradation data. The root mean square error of the estimated degradation value compared with the actual physical degradation value, as well as Akaike information criterion and Bayesian information criterion, is used for the evolution of the fitting accuracy and the selection of model topologies and discretization methods. Then the maximum posterior probability‐based classification criteria are developed. Degradation tests are designed for the data collection. To obtain the optimal classification policies, a cost function that consists of the degradation test cost and misclassification cost is constructed. A numerical example is used to illustrate the proposed method and demonstrate its advantages by comparing with other classification methods.     

A hierarchical modeling approach to accelerated degradation testing data analysis: A case study

In this case study, we investigate the degradation process of light‐emitting diodes (LEDs), which is used as a light source in DNA sequencing machines. Accelerated degradation tests are applied by varying temperature and forward current, and the light outputs are measured by a computerized measuring system. A degradation path model, which connects to the LED function recommended in Mitsuo (1991), is used in describing the degradation process. We consider variations in both measurement errors and degradation paths among individual test units. It is demonstrated that the hierarchical modeling approach is flexible and powerful in modeling a complex degradation process with nonlinear function and random coefficient. After fitting the model by maximum likelihood estimation, the failure time distribution can be obtained by simulation. Copyright © 2010 John Wiley & Sons, Ltd.     

基于GaAs激光器性能退化的可靠性度量方法

针对传统可靠性度量方法对高可靠性及长寿命的激光产品进行可靠性度量存在因模型假设不准确而出现可靠性度量错误风险的问题,基于性能退化轨迹提出了利用非参数局部线性回归估计对实际的退化模型进行直接估计的方法.该方法在确定实际模型后,利用失效阈值外推获得伪失效寿命时间,进而采用完全寿命时间数据进行可靠性度量.最后通过对GaAs激光器的退化数据进行可靠性验证分析,结果表明此方法提高了可靠性的预测精度,拟合程度高,稳健性好.采用非参数局部线性回归估计方法得到的结果合理、准确.     

Remaining useful life prediction for multivariable stochastic degradation systems with non-Markovian diffusion processes

Multivariable stochastic degradation system (MSDS) is quite common in industries such as blast furnace ironmaking, vehicle transportation, and aerospace manufacturing. Large-scale complex equipments may be affected by multiple factors, resulting in not just a single deteriorating performance characteristic. It is difficult to handle unknown failure structures of practical systems by using traditional univariate degradation modeling methods. A novel health index (HI) is constructed to quantitatively analyze the health state for the overall system. Considering the interaction between internal reactions and external environments, the fractional Brownian motion (FBM), a typical non-Markovian diffusion process, is added for the purpose of reflecting stochastic uncertainties and memory effects. Based on the wavelet estimators and the maximum likelihood estimation (MLE) algorithm, multi-sensor observations of degradation variables are analyzed simultaneously to identify model parameters. A closed-form distribution of system-level remaining useful life (RUL) is obtained with a mild two-layer approximation. Relevant case studies are then handled that adequately demonstrate the effectiveness and the practical utility of the proposed method.     

极大似然高分辨波达估计新方法

极大似然估计器是波达方向估计中公认的最佳估计器,但是计算量很大。为了解决极大似然估计器由于进行多维格形搜索而带来的计算量大的不足,将粒子滤波方法与极大似然估计相结合,提出了一种基于粒子滤波的极大似然波达方向估计器(Maximum Likelihood DOA Estimator Based on Particle Filtering,简称MLE-PF)。研究结果表明,MLE-PF不但保持了原极大似然估计方法的优良性能,大大减小了计算量,计算复杂度由O(LK)降至O(K×Ns),而且在低信噪比时也具有比MUSIC以及MiniNorm方法更加优越的估计性能。     

A comparison of three estimators of the Weibull parameters

Using mean square error as the criterion, we compare two least squares estimates of the Weibull parameters based on non‐parametric estimates of the unreliability with the maximum likelihood estimates (MLEs). The two non‐parametric estimators are that of Herd–Johnson and one recently proposed by Zimmer. Data was generated using computer simulation with three small sample sizes (5, 10 and 15) with three multiply‐censored patterns for each sample size. Our results indicate that the MLE is a better estimator of the Weibull characteristic value, θ, than the least squares estimators considered. No firm conclusions may be made regarding the best estimate of the Weibull shape parameter, although the use of maximum likelihood is not recommended for small sample sizes. Whenever least squares estimation of both Weibull parameters is appropriate, we recommend the use of the Zimmer estimator of reliability. Copyright © 2001 John Wiley & Sons, Ltd.     

Analyzing Degradation by an Independent Increment Process

There is no denying that degradation test is a widely used technique to assess the life‐time information for complex systems and highly reliable products. In this paper, motivated by laser degradation data, valve recession data and independent increment process theory, we propose an independent increment random process method, in which linear mean and standard deviation functions are used to describe the degradation procedure. A one‐stage maximum likelihood estimation of parameters is established. Product's mean‐time‐to‐failure and percentile of the failure time distribution are also derived. The proposed method is illustrated and verified in practice degradation analyzing for the two motivating data sets. It shows that the model can provide reasonable results. Copyright © 2013 John Wiley & Sons, Ltd.     

A new approach to MLE of Weibull distribution with interval data

We study the two-parameter maximum likelihood estimation (MLE) problem for the Weibull distribution with consideration of interval data. Without interval data, the problem can be solved easily by regular MLE methods because the restricted MLE of the scale parameter β for a given shape parameter α has an analytical form, thus α can be efficiently solved from its profile score function by traditional numerical methods. In the presence of interval data, however, the analytical form for the restricted MLE of β does not exist and directly applying regular MLE methods could be less efficient and effective. To improve efficiency and effectiveness in handling interval data in the MLE problem, a new approach is developed in this paper. The new approach combines the Weibull-to-exponential transformation technique and the equivalent failure and lifetime technique. The concept of equivalence is developed to estimate exponential failure rates from uncertain data including interval data. Since the definition of equivalent failures and lifetimes follows EM algorithms, convergence of failure rate estimation by applying equivalent failures and lifetimes is mathematically proved. The new approach is demonstrated and validated through two published examples, and its performance in different conditions is studied by Monte Carlo simulations. It indicates that the profile score function for α has only one maximum in most cases. Such good characteristic enables efficient search for the optimal value of α.     

Inference for a geometric‐poisson‐Rayleigh distribution under progressive‐stress accelerated life tests based on type‐I progressive hybrid censoring with binomial removals

Based on failures of a parallel‐series system, a new distribution called geometric‐Poisson‐Rayleigh distribution is proposed. Some properties of the distribution are discussed. A real data set is used to compare the new distribution with other 6 distributions. The progressive‐stress accelerated life tests are considered when the lifetime of an item under use condition is assumed to follow the geometric‐Poisson‐Rayleigh distribution. It is assumed that the scale parameter of the geometric‐Poisson‐Rayleigh distribution satisfies the inverse power law such that the stress is a nonlinear increasing function of time and the cumulative exposure model for the effect of changing stress holds. Based on type‐I progressive hybrid censoring with binomial removals, the maximum likelihood and Bayes (using linear‐exponential and general entropy loss functions) estimation methods are considered to estimate the involved parameters. Some point predictors such as the maximum likelihood, conditional median, best unbiased, and Bayes point predictors for future order statistics are obtained. The Bayes estimates are obtained using Markov chain Monte Carlo algorithm. Finally, a simulation study is performed, and numerical computations are performed to compare the performance of the implemented methods of estimation and prediction.