Using degradation models to assess pipeline life

Longitudinal inspections of thickness at particular locations along a pipeline provide useful information to assess the remaining life of the pipeline. In applications with different mechanisms of corrosion processes, we have observed various types of general degradation paths. We present two applications of fitting a degradation model to describe the corrosion initiation and growth behavior in a pipeline. We use a Bayesian approach for parameter estimation for the degradation model. The failure‐time and remaining lifetime distributions are derived from the degradation model, and we compute Bayesian estimates and credible intervals of the failure‐time and remaining lifetime distributions for both individual segments and for the entire pipeline circuit.     

Bayesian estimation and prediction for the transformed Wiener degradation process

This paper proposes some Bayesian inferential procedures for the transformed Wiener (TW) process, a new degradation process that has been recently suggested in the literature to describe degradation phenomena where degradation increments are not necessarily positive and depend stochastically on the current degradation level. These procedures have been expressly conceived to allow one incorporating into the inferential process the type of prior information, on meaningful physical characteristics of the observed degradation process, that is generally available in practical settings. Several different prior distributions are proposed, each of them reflecting a specific degree of knowledge on the observed phenomenon. Simple strategies for eliciting the prior hyper‐parameters from the available prior information are provided. Estimates of the TW process parameters and some functions thereof are retrieved by adopting a Monte Carlo Markov Chain technique. Procedures that allow predicting the degradation increment, the useful life of a new unit, and the remaining useful life of a used unit are also provided. Finally, an application is developed on the basis of a set of real degradation measurements of some infrared light‐emitting diodes, widely used in communication systems. The obtained results demonstrate the feasibility of the proposed Bayesian approach and the flexibility of the TW process.     

Imperfect repair in degradation processes: A Kijima‐type approach

We consider a Wiener process with linear drift for degradation modeling. Regularly, maintenance actions are carried out which produce a reduction of the degradation level. In this paper, we consider the influence of such maintenance actions to the further development of the degradation process and the resulting lifetime distribution. A connection between virtual age in Kijima‐type models and degradation level in the underlying degradation process is developed. Furthermore, estimators for the process parameters as well as for the degree of repair are developed.     

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.     

The transformed inverse Gaussian process as an age- and state-dependent degradation model

In this paper, a transformed inverse Gaussian (TIG) process is introduced as a new family of monotonic degradation models. Different from most state-of-the-art degradation models, which can only characterize age-dependent performance degradation, the TIG process model is mainly introduced for degradation modelling of industrial products with age- and state-dependent performance degradation. With this new model, promising properties include (1) the modelling capability for characterizing products observed at discrete time points with age- and state-dependent degradation, (2) the mathematical tractability for calculating the reliability function and remaining useful life distribution with high efficiency, and (3) the modelling flexibility of incorporating explanatory variables and random effects for investigating a product population with unit-to-unit heterogeneity. To facilitate the degradation modelling and analysis, methods for parameter estimation and model selection are developed under a coherent Bayesian framework. Simulation studies and real cases are presented to demonstrate the proposed degradation model and the Bayesian methods.     

Estimation of the Failure Rate in a Partially Accelerated Life Test: A Sequential Approach

Abstract

Consider the situation in which a group of units are put on a partially accelerated life test. It is assumed that the lifelengths of the units are independent and exponentially distributed random variables with common failure rate θ, and that θ is the value of a random variable having a gamma distribution. A two‐stage sequential procedure for estimating θ under the squared error loss is proposed. In the first stage, the units are put on the test under normal stress up to time t, where t is determined as a stopping time that minimizes the expected loss plus cost of running the test. In the second stage, the stress is raised to a higher level for those units that did not fail by time t and held constant until they all fail. The accumulated data are then used to estimate θ with the Bayes estimator.     

Optimal plan for Wiener constant-stress accelerated degradation model

This paper explores inferential procedures for the Wiener constant-stress accelerated degradation model under degradation mechanism invariance. The exact confidence intervals are obtained for the parameters of the proposed accelerated degradation model. The generalized confidence intervals are also proposed for the reliability function and pth quantile of the lifetime at the normal operating stress level. In addition, the prediction intervals are developed for the degradation characteristic, lifetime and remaining useful life of the product at the normal operating stress level. The performance of the proposed generalized confidence intervals and the prediction intervals is assessed by the Monte Carlo simulation. Furthermore, a new optimum criterion is proposed based on minimizing the mean of the upper prediction limit for the degradation characteristic at the design stress level. The exact optimum plan is also derived for the Wiener accelerated degradation model according to the proposed optimal criterion. The proposed interval procedures and optimum plan are the free of the equal testing interval assumption. Finally, two examples are provided to illustrate the proposed interval procedures and exact optimum plan. Specifically, based on the degradation data of LEDs, some interval estimates of quantities related to reliability indicators are obtained. For the degradation data of carbon-film resistors, the optimal allocation of test units is derived in terms of the proposed optimal criterion.     

Second-order general perturbed sweeping process differential inclusion

The paper presents a study of perturbed sweeping process where the moving set depends on both the time and the state. This evolution problem is governed by second-order differential inclusions with an unbounded perturbation. Assuming that such set is prox-regular or subsmooth, we prove the existence of solutions even in the presence of a delay.     

Statistical inference for a Wiener-based degradation model with imperfect maintenance actions under different observation schemes

This paper studies the statistical inference in a degradation model with imperfect maintenance. Technological or industrial devices subject to degradation undergo maintenance actions that reduce their degradation level. The underlying degradation process is a Wiener process with drift. Maintenance effects are assumed to be imperfect, described by an Arithmetic Reduction of Degradation ($AR{D}_1$) model. The system is regularly inspected and the degradation levels are measured. Four different observation schemes are considered so that degradation levels can be observed between maintenance actions as well as just before or just after maintenance times. The paper studies the estimation of the model parameters under the four observation schemes. Maximum likelihood estimators are derived for each scheme. The quality of the estimations is assessed and the observation schemes are compared through an extensive simulation and performance study.     

Peak demand forecasting for a seasonal product using Bayesian approach

An actual demand-forecasting problem of the US apparel dealers is studied. Demand is highly fluctuating during the peak sale season and low prior to the peak season. The model is described by the continuous time stochastic process applying the Bayesian process. The standard gamma distribution is selected for the demand process and an inverse gamma distribution is chosen as the conjugate prior for the model. The choice is supported by the maximum likelihood estimate among a number of non-negative distribution models. The proposed Bayesian models predict the probability of the future demand expressed explicitly conditional on the observed demand prior to the peak season. The data set illustrates partial demand of a seasonal product procured by the US dealers from overseas. In recent years, hazard and operational risks due to weather disasters and equipment shutdowns were felt significantly. These caused supply chain disruption and unrecorded demand. The model is extended to contribute to forecast from an unrecorded data set due to supply disruption. Forecasts are compared with real data and a widely implemented adaptive Holt-Winters (H-W) seasonal forecasting model. Results show that the forecasts calculated by the proposed methods do better than those of the adaptive H-W model.     

Monotonic dynamics of mRNA degradation by two pathways

mRNA degradation plays an important role in gene regulation. However, a defect in mRNA decay is expected to result in an increase in mRNA levels. In this paper, we will first establish a model of mRNA regulation by two pathways denoted by $5''\rightarrow 3''$ and $3''\rightarrow 5''$ for short, where there are two degradation rates $\delta_1$, $\delta_2$ on $5''\rightarrow3''$ pathway and the degradation rate on $3''\rightarrow 5''$ pathway is $\delta_3$. The advantage of this model is that it captures fundamental biochemical reactions in the gene expression process in eukaryotic cells. Then we obtain several basic principles on the monotonicity of the mean level of newly accumulated mRNAs. It is proved that (1) the newly mean level is strictly increasing in $p$ and $\kp$, but is strictly decreasing in $\gm$, where $p, \kp$ and $ \gm$ are the initial activation frequency, the activation rate, and the inactivation rate, respectively; (2) the newly mean level is strictly decreasing in both $\delta_2$ and $\delta_3$, remarkably, is strictly increasing in $\delta_1$ when $\delta_2<\delta_3$ and decreasing when $\delta_2>\delta_3$ and; (3) the newly mean level is strictly increasing in time $t$ when $p<\kp/(\kp+\gm)$. These conclusions not only provide a better understanding on gene expression dynamics but also would be helpful to design reasonable gene expression modules.     

Bayesian planning of step‐stress accelerated degradation tests under various optimality criteria

Step‐stress accelerated degradation testing (SSADT) has become a common approach to predicting lifetime for highly reliable products that are unlikely to fail in a reasonable time under use conditions or even elevated stress conditions. In literature, the planning of SSADT has been widely investigated for stochastic degradation processes, such as Wiener processes and gamma processes. In this paper, we model the optimal SSADT planning problem from a Bayesian perspective and optimize test plans by determining both stress levels and the allocation of inspections. Large‐sample approximation is used to derive the asymptotic Bayesian utility functions under 3 planning criteria. A revisited LED lamp example is presented to illustrate our method. The comparison with optimal plans from previous studies demonstrates the necessity of considering the stress levels and inspection allocations simultaneously.     

An optimal burn‐in policy based on a degradation model

Burn‐in tests help manufacturers detect defective items and remove them before being sold to customers. In a competitive marketplace, cost is a major consideration and not employing a burn‐in test may result in higher and needless expenses. With this in mind, we consider degradation‐based burn‐in tests in which the degradation path follows a Wiener process and weak items are identified when the process crosses a piecewise linear function. We also study linear functions as a special case of such a piecewise linear barrier. Within this setup, we apply a cost model to determine the optimal burn‐in test. Finally, we discuss an illustrative example using GaAs laser degradation data and present an optimal burn‐in test for it.     

Analysis of the reliability and the maintenance cost for finite life cycle systems subject to degradation and shocks

This paper deals with a deteriorating system subject to two different causes of failure: internal continuous degradation and sudden shocks. The degradation process is modelled using a gamma process. It is assumed that the system fails when the deterioration level reaches a critical threshold. Furthermore, sudden shocks arrive at the system at random times following a non-homogeneous Poisson process. When a sudden shock takes place, the system fails. To control the system reliability, a condition-based maintenance is applied. Under this maintenance policy, availability measures of the system are obtained. It is shown that these measures fulfil Markov renewal equations. A recursive method is developed to compute these measures. Furthermore, the maintenance cost of this system is analysed. Traditionally, the maintenance cost is analysed assuming an infinite time span. However, most systems have a finite life cycle and the application of the asymptotic approach is questionable. In this paper, the maintenance cost is analysed considering a finite life cycle. A recursive method, which combines numerical integration and Monte Carlo simulation, is developed to obtain the expected cost rate in the finite life cycle and its associated standard deviation.     

An influence statistic for linear measurement error models

Detection of multiple outliers or subset of influential points has been rarely considered in the linear measurement error models. In this paper a new influence statistic for one or a set of observations is generalized and characterized based on the corrected likelihood in the linear measurement error models. This influence statistic can be expressed in terms of the residuals and the leverages of linear measurement error regression. Unlike Cook’s statistic, this new measure of influence has asymptotically normal distribution and is able to detect a subset of high leverage outliers which is not identified by Cook’s statistic. As an illustrative example, simulation studies and a real data set are analysed.     

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Step-stress accelerated degradation test (SSADT) is a useful tool for assessing the lifetime distribution of highly reliable products when the available test items are very few. In this paper, we discuss multiple-steps step-stress accelerated degradation models based on Wiener process, and we apply the objective Bayesian method for such analytically intractable models to obtain the noninformative priors (Jefferys prior and two Reference priors). Moreover, we show that their posterior distributions are proper, and we propose Gibbs sampling algorithms for the Bayesian inference based on the Jefferys prior and two Reference priors. Finally, we present some simulation studies to compare the objective Bayesian estimates with the other Bayesian estimate and the maximum likelihood estimates (MLEs). Simulation results demonstrate the superiority of objective Bayesian analysis method.     

Adequacy of finite difference schemes for convection‐diffusion equations

Finite difference schemes for the numerical solution of singularly perturbed convection problems on uniform grids are studied in the limit case where the viscosity and the meshsize approach zero at the same time. The present error estimates are given in terms of order of magnitude in the above limit process and are useful in a priori choosing adequate schemes and meshsizes for boundary‐layer problems and problems with closed characteristics. Published 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 280–295, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/num.10007     

A note on interval estimation of the standard deviation of a gamma population with applications to statistical quality control

Traditional process control charts for a measurement standard deviation are based on the assumption of normality, which may not always be valid. Assuming that measurements follow a gamma distribution, we have obtained an approximate distribution of the sample variance, scaled appropriately. This approximate distribution, which happens to be another gamma model, is used to derive an interval estimate of the population standard deviation. Further, the above approximate gamma model for the sample variance can be used to develop a process control chart as demonstrated by a simulated data set.     

A Flexible Model for Generalized Linear Regression with Measurement Error

This paper focuses on the question of specification of measurement error distribution and the distribution of true predictors in generalized linear models when the predictors are subject to measurement errors. The standard measurement error model typically assumes that the measurement error distribution and the distribution of covariates unobservable in the main study are normal. To make the model flexible enough we, instead, assume that the measurement error distribution is multivariate t and the distribution of true covariates is a finite mixture of normal densities. Likelihood–based method is developed to estimate the regression parameters. However, direct maximization of the marginal likelihood is numerically difficult. Thus as an alternative to it we apply the EM algorithm. This makes the computation of likelihood estimates feasible. The performance of the proposed model is investigated by simulation study.     

roperties and Applications of Alpha Power Gamma Distribution

In this paper, the gamma distribution has been extended by adding an extra shape parameter, we refer to the new distribution as alpha power gamma distribution. It is found that the distribution has a relatively flexible hazard rate function. The properties of the new distribution are studied, including explicit expressions for the $s^{\text{th}}$ raw moments, moment generating function and distributions of order statistics are derived. Also, the integral expressions for the entropy, mean residual life and mean waiting time are obtained. The maximum likelihood estimators of the distribution parameters under complete sample are discussed, the Fisher information matrix is derived. Then, the estimation of the parameters under the general progressive type-II censoring is studied. Finally, the real data set is used to illustrate the practicality of the proposed distribution.