Parameter estimation for partially observable systems subject to random failure

In this paper, we present a parameter estimation procedure for a condition‐based maintenance model under partial observations. Systems can be in a healthy or unhealthy operational state, or in a failure state. System deterioration is driven by a continuous time homogeneous Markov chain and the system state is unobservable, except the failure state. Vector information that is stochastically related to the system state is obtained through condition monitoring at equidistant sampling times. Two types of data histories are available — data histories that end with observable failure, and censored data histories that end when the system has been suspended from operation but has not failed. The state and observation processes are modeled in the hidden Markov framework and the model parameters are estimated using the expectation–maximization algorithm. We show that both the pseudolikelihood function and the parameter updates in each iteration of the expectation–maximization algorithm have explicit formulas. A numerical example is developed using real multivariate spectrometric oil data coming from the failing transmission units of 240‐ton heavy hauler trucks used in the Athabasca oil sands of Alberta, Canada. Copyright © 2012 John Wiley & Sons, Ltd.     

A degradation path-dependent approach for remaining useful life estimation with an exact and closed-form solution

Remaining useful life (RUL) estimation is regarded as one of the most central components in prognostics and health management (PHM). Accurate RUL estimation can enable failure prevention in a more controllable manner in that effective maintenance can be executed in appropriate time to correct impending faults. In this paper we consider the problem of estimating the RUL from observed degradation data for a general system. A degradation path-dependent approach for RUL estimation is presented through the combination of Bayesian updating and expectation maximization (EM) algorithm. The use of both Bayesian updating and EM algorithm to update the model parameters and RUL distribution at the time obtaining a newly observed data is a novel contribution of this paper, which makes the estimated RUL depend on the observed degradation data history. As two specific cases, a linear degradation model and an exponential-based degradation model are considered to illustrate the implementation of our presented approach. A major contribution under these two special cases is that our approach can obtain an exact and closed-form RUL distribution respectively, and the moment of the obtained RUL distribution from our presented approach exists. This contrasts sharply with the approximated results obtained in the literature for the same cases. To our knowledge, the RUL estimation approach presented in this paper for the two special cases is the only one that can provide an exact and closed-form RUL distribution utilizing the monitoring history. Finally, numerical examples for RUL estimation and a practical case study for condition-based replacement decision making with comparison to a previously reported approach are provided to substantiate the superiority of the proposed model.     

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本文考虑了一个其产品保修期内免费小修的退化 生产系统的定期检修策略. 系统的退化过程包括三个状态: 可控制状态, 不可控制状态, 故障状态. 过程呆在可控制状态和不可控制状态的时间假设都服从指数分布. 生产系统在固定的时刻t或发生故障时进行检修, 两者以先发生为准. 本文讨论了使单位产品每周期期望成本最小的最优定期检修时间本文考虑了一个其产品保修期内免费小修的退化生产系统的定期检修策略.系统的退化过程包括三个状态:可控制状态,不可控制状态,故障状态.过程呆在可控制状态和不可控制状态的时间假设都服从指数分布.生产系统在固定的时刻t﹡或发生故障时进行检修,两者以先发生为准.本文讨论了使单位产品每周期期望成本最小的最优定期检修时间t﹡,三种特殊情况显示了最优值t的性质.此外,灵敏性分析和数字实例说明了模型中的参数对最优定期检修策略的影响.     

Efficient Data Augmentation for Fitting Stochastic Epidemic Models to Prevalence Data

Stochastic epidemic models describe the dynamics of an epidemic as a disease spreads through a population. Typically, only a fraction of cases are observed at a set of discrete times. The absence of complete information about the time evolution of an epidemic gives rise to a complicated latent variable problem in which the state space size of the epidemic grows large as the population size increases. This makes analytically integrating over the missing data infeasible for populations of even moderate size. We present a data augmentation Markov chain Monte Carlo (MCMC) framework for Bayesian estimation of stochastic epidemic model parameters, in which measurements are augmented with subject-level disease histories. In our MCMC algorithm, we propose each new subject-level path, conditional on the data, using a time-inhomogenous continuous-time Markov process with rates determined by the infection histories of other individuals. The method is general, and may be applied to a broad class of epidemic models with only minimal modifications to the model dynamics and/or emission distribution. We present our algorithm in the context of multiple stochastic epidemic models in which the data are binomially sampled prevalence counts, and apply our method to data from an outbreak of influenza in a British boarding school. Supplementary material for this article is available online.     

Optimal economic production quantity policy for imperfect process with imperfect repair and maintenance

This study integrates maintenance and production programs with the economic production quantity (EPQ) model for an imperfect process involving a deteriorating production system with increasing hazard rate: imperfect repair and rework upon failure (out of control state). The imperfect repair performs some restorations and restores the system to an operating state (in-control state), but leaves its failure until perfect preventive maintenance (PM) is performed. There are two types of PM, namely imperfect PM and perfect PM. The probability that perfect PM is performed depends on the number of imperfect maintenance operations performed since the last renewal cycle. Mathematical formulas are obtained for deriving the expected total cost. For the EPQ model, the optimum run time, which minimizes the total cost, is discussed. Various special cases are considered, including the maintenance learning effect. Finally, a numerical example is presented to illustrate the effects of PM, setup, breakdown and holding costs.     

A parametric lifetime model for a multicomponents system with spatial interactions

In this paper, we consider a system made of n components displayed on a structure (eg, a steel plate). We define a parametric model for the hazard function, which includes covariates and spatial interaction between components. The state (nonfailed or failed) of each component is observed at some inspection times. From these data, we consider the problem of model parameter estimation. To achieve this, we suggest to use the SEM algorithm based on a pseudo‐likelihood function. A definition for the time‐to‐failure of the system is given, generalizing the classical cases. A study based on numerical simulations is provided to illustrate the proposed approach.     

基于最优估计的数据融合理论

本文提出了一种最优加权的数据融合方法，分析了最优权值的分配原则；给出了多源信息统一的线性融合模型，使其表示不受数据类型和融合系统结构的限制，并指出在噪声协方差阵正定的前提下，线性最小方差估计融合和加权最小二乘估计融合是等价的；介绍了数据融合中的Bayes极大后验估计融合方法，给出了利用极大后验法进行传感器数据融合的一般表示公式；最后以两传感器数据融合为例，证明了利用Bayes极大后验估计进行两传感器数据融合所得到的融合状态的精度比相同条件下极大似然估计得到的精度要高，同时它们均优于任一单传感器局部估计精度。     

Nonparametric estimation of interval-censored failure time data in the presence of informative censoring

Nonparametric estimation of a survival function is one of the most commonly asked questions in the analysis of failure time data and for this, a number of procedures have been developed under various types of censoring structures (Kalbfleisch and Prentice, 2002). In particular, several algorithms are available for interval-censored failure time data with independent censoring mechanism (Sun, 2006; Turnbull, 1976). In this paper, we consider the interval-censored data where the censoring mechanism may be related to the failure time of interest, for which there does not seem to exist a nonparametric estimation procedure. It is well-known that with informative censoring, the estimation is possible only under some assumptions. To attack the problem, we take a copula model approach to model the relationship between the failure time of interest and censoring variables and present a simple nonparametric estimation procedure. The method allows one to conduct a sensitivity analysis among others.     

Parameter estimation for time varying nonlinear circuit from state analysis and simulation

This paper develops a parameter estimation technique for a nonlinear circuit. The nonlinear circuit is represented by a state space model and perturbation theory is applied to obtain the approximate analytical solution for the state vector. The state model is assumed to be slowly time varying so that the parameter vector is constant over different time slots. The expressions obtained for the state vector are matched with the noisy data using the gradient algorithm and hence the parameter vector is estimated. Simulations are based on discretization of the state space model using MATLAB.     

Profust reliability of a gracefully degradable system

Research in traditional reliability theory is based mainly on probist reliability, which uses a binary state assumption and classical reliability distributions. In the present paper the binary state assumption has been replaced by a fuzzy state assumption, thereby leading to profust reliability estimates of a powerloom plant, which is modelled as a two unit gracefully degradable system. Results of Bowles and Palaez [Application of fuzzy logic to reliability engineering, Proc. IEEE 83(3) (1995) 435–449] have been deduced as a particular case of results presented here. It is also recognized that estimation of system parameters such as failure rates, is vital in reliability estimation. Available methods for such estimation do not cover the underlying uncertainty in the failure data collection involving human judgment, evaluation and decision. In this paper we introduce a new approach based on fuzzy set theory to estimate such system parameters.     

特殊状态特殊修理的可修系统可靠性和诊断策略

本研究了特殊状态需要特殊修理的可修系统的可靠性和诊断策略。假设系统有三种运行状态：正常状态、异常状态、故障状态，有些异常状态和故障状态需要特殊的修理，系统处于哪个状态需要诊断才能知道。每当系统开始正常工作状态后，每隔一段随机时间T进行一次诊断，直到系统故障或被诊断为异常。利用概率分析和向量马尔科夫过程方法，求得了系统的可靠性指标并研究了最优诊断策略。     

Asymptotics for non-parametric likelihood estimation with doubly censored multivariate failure times

This paper considers non-parametric estimation of a multivariate failure time distribution function when only doubly censored data are available, which occurs in many situations such as epidemiological studies. In these situations, each of multivariate failure times of interest is defined as the elapsed time between an initial event and a subsequent event and the observations on both events can suffer censoring. As a consequence, the estimation of multivariate distribution is much more complicated than that for multivariate right- or interval-censored failure time data both theoretically and practically. For the problem, although several procedures have been proposed, they are only ad-hoc approaches as the asymptotic properties of the resulting estimates are basically unknown. We investigate both the consistency and the convergence rate of a commonly used non-parametric estimate and show that as the dimension of multivariate failure time increases or the number of censoring intervals of multivariate failure time decreases, the convergence rate for non-parametric estimate decreases, and is slower than that with multivariate singly right-censored or interval-censored data.     

A novel optimal preventive maintenance policy for a cold standby system based on semi-Markov theory

A novel optimal preventive maintenance policy for a cold standby system consisting of two components and a repairman is described herein. The repairman is to be responsible for repairing either failed component and maintaining the working components under certain guidelines. To model the operational process of the system, some reasonable assumptions are made and all times involved in the assumptions are considered to be arbitrary and independent. Under these assumptions, all system states and transition probabilities between them are analyzed based on a semi-Markov theory and a regenerative point technique. Markov renewal equations are constructed with the convolution of the cumulative distribution function of system time in each state and corresponding transition probability. By using the Laplace transform to solve these equations, the mean time from the initial state to system failure is derived. The optimal preventive maintenance policy that will provide the optimal preventive maintenance cycle is identified by maximizing the mean time from the initial state to system failure, and is determined in the form of a theorem. Finally, a numerical example and simulation experiments are shown which validated the effectiveness of the policy.     

Sub-optimal constant-volume control for open channel networks

A state variable mathematical model for use in the synthesis of automatic control systems for open-channel networks is presented. The system considered here consists of n-cascaded reaches joined by control gates.The linear time invariant model consists of a controllable and observable representation where the state variables are the stored water volume variations in each reach and the control signals are the variations of the control gates opening sections. The model derives, through appropriate simplifications, from a more complex one in terms of transfer functions which was derived by linearizing the Saint-Venant equations.The problem of a linear quadratic optimal regulator is formulated in classical terms for the canal system and the constant-volume control laws obtained for the simplified model have been imposed on the complex one: such a control is therefore to be considered sub-optimal.The results of a digital simulation of the controlled system behaviour indicate that the system operates with practically constant stored water volumes in each reach and that such behaviour is fairly close to that of a pressure-water pipe system.     

The signal model: A model for competing risks of opportunistic maintenance

This paper presents a competing risks reliability model for a system that releases signals each time its condition deteriorates. The released signals are used to inform opportunistic maintenance. The model provides a framework for the determination of the underlying system lifetime from right-censored data, without requiring explicit assumptions about the type of censoring to be made. The parameters of the model are estimated from observational data by using maximum likelihood estimation. We illustrate the estimation process through a simulation study. The proposed signal model can be used to support decision-making in optimising preventive maintenance: at a component level, estimates of the underlying failure distribution can be used to identify the critical signal that would trigger maintenance of the individual component; at a multi-component system level, accurate estimates of the component underlying lifetimes are important when making general maintenance decisions. The benefit of good estimation from censored data, when adequate knowledge about the dependence structure is not available, may justify the additional data collection cost in cases where full signal data is not available.     

混合指数分布的参数估计

混合指数分布是寿命数据分析中一个非常重要的统计模型\bd 但是利用正规的统计方法如矩估计、极大似然估计等估计模型的参数往往比较困难\bd 本文应用EM算法详细研究了混合指数分布在正常工作条件下和在进行恒加应力加速寿命实验条件下, 在完全数据场合、I-型截尾和II-型截尾场合的参数估计问题\bd 模拟说明利用EM算法来估计混合指数分布是一种非常有效的方法.     

Secure state estimation for complex networks with multi-channel oriented round robin protocol

This paper focuses on the secure state estimation problem for complex networks (CNs) which are compromised by deception attack and constrained with limited communication resource. Firstly, a multi-channel oriented round robin (RR) protocol is proposed to schedule the data transferred over the communication network consisting of multiple transmission channels. The extended RR protocol can not only avoid the data collision caused by limited communication resource, but also fully utilize the sliced network bandwidth. Then, a state estimation error model is constructed by further considering the influence of deception attack. Following the model, efficient state estimators are designed based on analyzing the sufficient conditions that assuring the stability of the formulated state estimation error system. Finally, numerical results are presented to validate the theoretical outcomes.     

Analyzing continuous state space failure time processes: Two further results*

In a recent paper, Petersen (1988) considered a continuous state space failure time process. The central result provided in that paper was that the destination‐specific rate of transition of the process can be specified in two steps. First, one specifies the overall rate at which a change occurs. Then, one specifies the probability density function of the destination state, given that a transition occurred. This two‐step property was used in deriving the likelihood of the data and was exploited for purposes of estimation. The overall rate of transition can be estimated from the data on durations between changes in the dependent variable. The density for the new value of the dependent variable, given a change, can be estimated from the data on the values of the dependent variable after the change.

This paper extends these results in two ways. First, it is shown that one can derive the likelihood of the process directly from the destination‐specific rate of transition, without going through its decomposition into the overall rate times the density of the destination state, given a transition. Once the likelihood is derived, estimation is comparatively straightforward. Second, it is shown how one can derive, at each point in time, a more standard regression function for the continuous dependent variable, where its value is expressed in terms of its conditional mean plus an error term.     

Markovian Imperfect Software Debugging Model and Its Performance Measures

Abstract

This paper considers a Markovian imperfect software debugging model incorporating two types of faults and derives several measures including the first passage time distribution. When a debugging process upon each failure is completed, the fault which causes the failure is either removed from the fault contents with probability p or is remained in the system with probability 1 ? p. By defining the transition probabilities for the debugging process, we derive the distribution of first passage time to a prespecified number of fault removals and evaluate the expected numbers of perfect debuggings and debugging completions up to a specified time. The availability function of a software system, which is the probability that the software is in working state at a given time, is also derived and thus, the availability and working probability of the software system are obtained. Throughout the paper, the length of debugging time is treated to be random and thus its distribution is assumed. Numerical examples are provided for illustrative purposes.     

线性定常系统特征模型的特征参量辨识

说明线性定常系统特征模型的特征参量是一组由高阶线性定常系统的相关信息压缩而成,于是不能简单的作为与状态无关的慢时变参数来处理. 基于特征建模思想,建立了线性定常系统特征模型的特征参量与子空间方法之间的联系,给出了一种该特征模型的特征参量 的合成辨识算法.同时证明了在用于子空间辨识的样本量充分大和用于状态估计的时间充分长的情况下, 特征参量的估计值与真值之间的误差达到充分小. 最后,对于一个六阶的单输入单输出线性定常系统的仿真例子,对投影的带遗忘因子最小二乘算法和合成辨识算法进行了比较,验证了合成辨识算法的有效性.