A Bayes nonparametric framework for software-reliability analysis

This paper presents a Bayes nonparametric approach for tracking and predicting software reliability. We use the common assumptions on the software operational environment to get a stochastic model where the successive times between software failures are exponentially distributed; their failure rates have Markov priors. Under these general assumptions we give Bayes estimates of the parameters that assess and predict the software reliability. We give algorithms (based on Monte-Carlo methods) to compute these Bayes estimates. Our approach allows the reliability analyst to construct a personal software reliability model simply by specifying the available prior knowledge; afterwards the results in this paper can be used to get Bayes estimates of the useful reliability parameters. Examples of possible prior physical knowledge concerning the software testing and correction environments are given. The maximum-entropy principle is used to translate this knowledge to prior distributions on the failure-rate process. Our approach is used to study some simulated and real failure data sets     

Bayes predictive analysis of a fundamental software reliabilitymodel

The concepts of Bayes prediction analysis are used to obtain predictive distributions of the next time to failure of software when its past failure behavior is known. The technique is applied to the Jelinski-Moranda software-reliability model, which in turn can show an improved predictive performance for some data sets even when compared with some more sophisticated software-reliability models. A Bayes software-reliability model is presented which can be applied to obtain the next time to failure PDF (probability distribution function) and CDF (cumulative distribution function) for all testing protocols. The number of initial faults and the per-fault failure rate are assumed to be s -independent and Poisson and gamma distributed respectively. For certain data sets, the technique yields better predictions than some alternative methods if the frequential likelihood and U-plot criteria are adopted     

Bayes estimation of hazard and acceleration in accelerated testing

In accelerated life testing, the time transformation function &thetas;(t) is often unknown, even if that function is assumed to be linear. If &thetas;(t) is known, data in the accelerated condition can be adjusted to provide information about the failure time distribution in the use condition. If &thetas;(t) is unknown, the usual estimation procedures require data from the use condition as well as data from the acceleration condition. In this work it is assumed that the uncertainty about &thetas; can be modeled by a prior distribution, chosen from the truncated Pareto family of distributions, and that the uncertainty in λ, the failure rate, can be modeled by a prior distribution from the gamma family. Under these assumptions, the posterior distributions and their first two moments are provided for both λ and &thetas;. Thus, this complete Bayes approach to accelerated life testing with the assumed model allows the adjustment of data taken in the accelerated condition to provide the user with the important estimates in the use condition. The results are illustrated by examples     

基于非下采样Contourlet域高斯尺度混合模型的图像降噪

提出了一种图像去噪方法,将高斯尺度混合(GSM)模型引入非下采样Contourlet变换(NSCT)域,构造了基于NSCT分解系数的邻域模型,并利用Bayes最小均方(BLS)估计进行局部去噪。仿真实验结果表明,通过本文提出的方法能够有效去除高斯噪声,较完整地保持图像中的边缘等细节信息,在峰值信噪比(PSNR)提高与视觉效果上优于其它的去噪方法。实验结果验证了方法的正确性。     

Estimation for a probabilistic stress-strength model

Estimation for an unknown strength distribution is considered in two situations: (1) independent identically distributed stresses are applied to the component until it fails (no cumulative damage); and (2) each applied stress causes damage to the component and damage cumulates until the component fails. Both situations lead to mixtures of probability distributions, with the strength distribution playing the role of the mixing distribution. Based on the observation of cycles to failure of several independent components and on the theory of mixtures of distributions, estimators of the mixing distributions are obtained using linear programming. In particular, the solution to the linear-programming problem yields a probability mass function which approximates the unknown strength distribution. From this estimate of the strength distribution, an estimate of the mean strength of the item can be obtained by the usual computation of the mean of a probability distribution. Hence, the results provide a method of estimating the mean strength, or other parameters of the strength distribution, without requiring observations directly on the strengths of the test components     

A Bayes Explanation of an Apparent Failure Rate Paradox

For the exponential life distribution model and any prior distribution for the failure rate parameter, the predictive distribution has a decreasing failure rate. A Bayes explanation is given of why this is logically reasonable.     

Survey of reliability and availability evaluation of complex networks using Monte Carlo techniques

Generally there are four main difficulties in evaluating complex large-scale system reliability, availability and MTBF: the system structure may be very complex; subsystems may follow various failure distributions; subsystems may conform to arbitrary failure and repair distributions for maintained systems; the failure data of subsystems are sometimes not sufficient, reliability test sample sizes tend to be small. It is difficult and often impossible to obtain s-confidence limits of them by classical statistics. Monte Carlo technique combined with Bayes method is a powerful tool to solve this kind of problems. In this survey, the typical existing Monte Carlo reliability, availability and MTBF simulation procedures, variance reduction methods, and random variate generation algorithms are analyzed and summarized. The advantages, drawbacks, accuracy and computer time of Monte Carlo simulation in evaluating reliability, availability and MTBF of a complex network are discussed. Finally, some conclusions are drawn and a general Monte Carlo reliability and MTTF assessment procedure is recommended.     

Estimating the Number of Items on Life Test

If n items are on life test where n is unknown, and failures are observed either until time T has elapsed or until r failures have occurred, then an estimate of n can be obtained. Both maximum likelihood and Bayes estimates are obtained and both known and unknown failure distributions are considered.     

Reliability analysis of a finite range failure model

The Bayes estimates of reliability and hazard rate functions of the finite range failure model (1) have been developed based on life tests that are terminated at a preassigned time point taking the order of observations into consideration. For the parameter involved, the priors gamma, Weibull and log normal densities have been considered. The importance of the distributions, considered here as priors, in the theory of reliability has led the authors to use them as prior distributions.     

Combining failure rate data from various sources

In this paper a method for pooling failure rate data obtained from different sources is discussed. The pooling is done by the help of a Bayes approach supposing the failure rates follow a common Gamma distribution. We obtain Bayes estimators of the failure rate and its inverse, the mean life time. An asymmetric loss function is used for the Bayes estimator. HPD confidence intervals are obtained for the failure rate.     

Coordinated warranty and burn-in strategies

In product reliability assurance, the warranty and burn-in (W&BI) strategies are usually selected separately, despite the fact that both depend on the early-life failure behavior of the product. This paper treats W&BI strategies together in order to examine the possible benefits of coordinated strategies for product performance management. As these strategies are meaningful only for decreasing hazard-rate systems, a Weibull life distribution is assumed for each system component. A net-profit model that includes an increase in product price as a function of warranty duration is constructed. The model shows how a coordinated W&BI strategy can be selected. The model is quite general and its extension to other cases is explained. A central point that is treated thoroughly is the renewal analysis necessary to determine replacement costs during burn-in and during the warranty period. As part of the analysis, a useful approximation is defined, and efficient optimization routines are identified. An example illustrates the use of analytical methods. The analysis and discussion of the example show that there are advantages in coordinating the selection of W&BI strategies     

Adaptive estimation in linear systems with unknown Markovian noise statistics

The asymptotic behavior of a Bayes optimal adaptive estimation scheme for a linear discrete-time dynamical system with unknown Markovian noise statistics is investigated. Noise influencing the state equation and the measurement equation is assumed to come from a group of Gaussian distributions having different means and covariances, with transitions from one noise source to another determined by a Markov transition matrix. The transition probability matrix is unknown and can take values only from a finite set. An example is simulated to illustrate the convergence.     

A nonparametric-Bayes reliability-growth model

The authors propose a nonparametric reliability-growth model based on Bayes analysis techniques. By using the unique properties of the assumed prior distributions, the moments of the posterior distribution of the failure rate at various stages during a development test can be found. The proposed model is compared with the US Army Material Systems Analysis Activity (AMSAA) model based on relative and mean-square prediction errors. In all but one circumstance, the proposed model performed better than either the AMSAA or nonparametric models. The one exception appears to be when no information about the failure rate is available at the start of test and the actual failure process is nonhomogeneous Poisson, with power-law intensity function, as assumed by the AMSAA model     

Bayes computation for reliability estimation

Bayes estimation of complicated functions requires simpler estimation techniques due to the mathematical difficulties involved in the classical Bayes approach. Bayes estimation enjoys many approximation techniques and computational methods like Metropolis, and Gibbs sampler. Bayes estimation of the reliability of a mixture inverse Gaussian distribution requires a numerical approach since the calculations are immensely difficult from the exact Bayes point of view. Lack of full conditional prior distributions for all 3 parameters of this particular case makes the use of Gibbs sampler inefficient. Application of the rejection method, however, is reasonable since it is very simple to implement without any constraints on the prior distributions or on the hyper-parameters     

The Prediction and Measurement of System Availability: A Bayesian Treatment

For any physical system there is always some degree of uncertainty regarding the values of the parameters governing the performance of that system. Uncertainties in the values of the failure rate ? and the repair rate ? reflect themselves in an uncertainty in the value of the point availability, A = ?/(? + ?). Treating these uncertain parameters as random variables, exact expressions for the mean, variance, and distribution of the point availability are derived by combining the distributions of the failure and repair rates. Hence we can construct estimates and confidence statements for the availability which are consistent with the equivalent statements on the failure-rate and repair-rate parameters. Exact mean and variance results are also provided for mission, transient, and other time-dependent availability expressions. The acquisition of failure and repair data introduces additional information as well as additional uncertainties. A Bayes transformation is developed which utilizes the two data sets to readily convert prior estimates and confidence statements on the availability into posterior versions. This particular paper is restricted to a basic model involving an alternating sequence of independent exponentially distributed operational and repair intervals with the respective rate parameters described by distinct gamma distributions. For this model the point availability proves to have an Euler distribution.     

Expert judgment in maintenance optimization

A comprehensive method for the use of expert opinion for obtaining lifetime distributions required for maintenance optimization is proposed. The method includes procedures for the elicitation of discretized lifetime distributions from several experts, the combination of the elicited expert opinion into a consensus distribution, and the updating of the consensus distribution with failure and maintenance data. The development of the method was prompted by the lack of statistical training of the experts and the high demands on their time. The use of a discretized life distribution provides more flexibility, is more comprehendible by the experts in the elicitation stage, and greatly reduces the computation in the combination and updating stages. The methodology is Bayes, using the Dirichlet distribution as the prior distribution for the elicited discrete lifetime distribution. Methods are described for incorporating information concerning the expertise of the experts into the analysis     

Neural network for demixing super-Gaussian signals

A new method for separating linear mixtures of statistically independent signals with super-Gaussian probability distributions, using a simple neural network, is proposed. The procedure is based on geometric properties, and it is shown that the maxima of the mixed density distribution belong to straight lines, the direction vectors of which, when taken as columns of a matrix, comprise a demixing matrix. The results obtained with synthetic mixtures of real speech signals are shown     

Empirical-Bayes Estimation of Mean Life for a Censored Sample, Constant Hazard Rate Model

The estimation of mean life based on a censored sample from a population with a constant failure rate is considered in an empirical Bayes situation. Bayes risks of empirical Bayes estimates for various numbers of past samples are compared with the Bayes risk of the sample mean, and the minimum Bayes risk. A Monte Carlo simulation shows that the Bayes risk of the empirical Bayes estimator is very close to the minimum Bayes risk for as few as 5 past estimates of the mean life. Morever, it is better than the uniformly minimum variance unbiased estimator (the sample mean) in terms of the Bayes risk. Effects of misclassification of the prior are also considered.     

An Extended Periodic Imperfect Preventive Maintenance Model With Age-Dependent Failure Type

This paper presents a generalized periodic imperfect preventive maintenance (PM) model for a system with age-dependent failure type. The imperfect PM model proposed in this study incorporates improvement factors vis-À-vis the hazard-rate function, and effective age. As failures occur, the system experiences one of the two types of failure: type-I failure (minor), and type-II failure (catastrophic). Type-I failures are rectified with minimal repair. In a PM period, the system is preventively maintained following the occurrence of a type-II failure, or at age $T$ , whichever takes place first. At the $N$th PM, the system is replaced. An approach that generalizes the existing studies on the periodic PM policy is proposed. Taking age-dependent failure type into consideration, the objective consists of determining the optimal PM & replacement schedule that minimize the expected cost per unit of time, over an infinite horizon.