Real Degradation Data Research Articles

Flexible modelling of a bivariate degradation process with a shared frailty and an application to fatigue crack data

Examples of units with two performance characteristics that degrade over time are ubiquitous in reliability engineering. In this article, we develop a flexible model for bivariate degradation data pertaining to units in which the degradation processes corresponding to the performance characteristics are likely dependent on each other. The proposed model has two features: the degradation processes are marginally modelled by gamma processes, and the dependence between them is modelled by a shared frailty term that is assumed to follow the generalized gamma distribution. We show that this model is far more flexible and efficient than many of the commonly used models for capturing dependence between the performance characteristics. A computational technique for the maximum likelihood estimation, based on Monte Carlo simulation, is developed for the proposed model. Then, the method of estimation is evaluated through an elaborate Monte Carlo simulation study. The joint reliability function of the unit with two performance characteristics and its estimation are also discussed in this general setting. The proposed model is extended to the case of multiple performance characteristics. Finally, a case study is presented in which a real degradation data pertaining to fatigue cracks is analysed through the proposed model to demonstrate its usefulness.

A new Wiener process with bathtub‐shaped degradation rate in the presence of random effects

AbstractThis paper proposes a new time scaled Wiener process with random effects that has been specially designed to allow the description of non‐monotonic degradation phenomena with bathtub shaped degradation rate, here intended as derivative of the mean function. Two different parameterizations of the proposed Wiener process are suggested, and the main features of the process are illustrated and discussed. The maximum likelihood estimation of its parameters is addressed. A failure threshold model is adopted to formulate and estimate the probability distribution function of the remaining useful life, which constitutes the core prognostic tool in condition‐based maintenance. As a motivating example, the proposed model is applied to a set of real degradation data of metal‐oxide‐semiconductor field‐effect transistors, where, as a result of preliminary analyses, some of the process parameters are assumed to be random in order to describe the considerable heterogeneity observed between the degradation paths of different units. Obtained results demonstrate the affordability and utility of the proposed model, especially for the estimation of unit‐specific features performed on the basis of degradation data collected in the early phase of the life of a unit belonging to the considered heterogeneous population.

Likelihood-free Hamiltonian Monte Carlo for modeling piping degradation and remaining useful life prediction using the mixed gamma process

The gamma process is a popular model to characterize the temporal uncertainty in degradation processes. According to this model, degradation occurs in tiny independent increments creating a monotonic non-decreasing trend. Mechanical components or units are often repetitively observed over time for measuring the extent of degradation; apparently, they degrade at different rates although operating in the same environment. The mixed gamma degradation process is suitable for capturing these unexplained differences. However, statistical estimation of its parameters and subsequent prediction of remaining useful life (RUL) turn challenging with limited and noisy measurements.While limited observations bring uncertainties to the model parameters, noisy data turns the likelihood function intractable. In such cases, approximate Bayesian computation (ABC) methods are widely applicable for their ability to bypass the likelihood evaluation and generate samples from an approximate target posterior distribution. The currently available Markov chain Monte Carlo (MCMC) based ABC method shows promising results, although it requires significant thinning due to poor mixing of samples. A new ABC method is derived in this paper which imitates the sampling process of the Hamiltonian Monte Carlo (HMC) method. The proposed ABC-HMC algorithm explores a target probability space more effectively — generating better mixing of samples and requiring less thinning than the MCMC based ABC method. Proof of convergence of ABC-HMC is shown by fulfilling the detailed balance condition. The application of the method is shown using real degradation data of piping components of the heat transport system of a nuclear reactor.

Minimum f-Divergence Estimation With Applications to Degradation Data Analysis

Minimizing the divergence between two probability distributions offers an alternative parameter estimation method. The current literature mainly focuses on minimizing the Kullback-Leibler (K-L) divergence between the true and the proposed models in which the true model is assumed to be known or fixed. In this paper, we propose a parameter estimation method that minimizes the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$f$ </tex-math></inline-formula> -divergence between two probability distributions. The method is suitable for different situations, no matter the true distribution is known or not. The statistical properties of the estimator, including consistency and asymptotic normality, are established. As an illustration, our method is employed to estimate the degradation model, which is a model frequently used to assess the lifetime of highly reliable products. A simulation study and a real degradation data analysis are presented to illustrate the effectiveness of the proposed estimation method.

Multivariate statistical aggregation and dimensionality reduction techniques to improve monitoring and maintenance in railways: The wheelset component

Reliable monitoring and assessment of wear evolutions are critical for performing effective railway maintenance. Several characteristics and variables are used to quantify a worn condition of railway wheelsets. To measure all these wear quantities, emerging inspection technologies are being designed with increasingly complex architectures, working mechanisms and associated high costs. Moreover, data-driven models to support condition-based maintenance to the wheelset easily increase their complexity when too many variables are taken into account and may not provide a straightforward guideline to maintenance decision-makers. The purpose of this paper is to reduce the complexity when describing the wear level, by applying multivariate statistical techniques to real degradation data from railway wheelsets. Several wheelset condition variables and their relationships are analysed. Variables are synthetized through a principal component analysis (PCA) where the varimax rotation effect can be observed. A cluster analysis, which uses the principal components, allows identifying characteristics that lead to different wear evolutions. A strong correlation between the flange thickness and flange slope in the wear process is identified. Differences in wear trajectories between motor and trailer wheelsets are strongly significant. The findings are expected to support the improvement of state monitoring techniques and predictive maintenance optimization models.

A failure mechanism consistency test method for accelerated degradation test

AbstractAccelerated 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.

Analysis of correlation between real degradation data and a carbonation model for concrete structures

Over the past few decades, climate change has become a significant problem, the ramifications of which are of concern around the world. Global warming is a long-term process caused by phenomena such as increased levels of carbon dioxide (CO2) concentration in the atmosphere, increases in rainfall, changes in relative humidity and changes in temperature. Thus, this will aggravate the degradation process of reinforced concrete (RC) structures, leading to the decrease of durability, safety and serviceability. Regarding the durability of concrete structures, carbonation-induced corrosion is definitely a significant and costly source of degradation, directly related to climatic parameters such as CO2, temperature and relative humidity. In Paraguay, the lack of control during the construction of buildings often leads to weak structures highly vulnerable to the attack of harmful external agents. Therefore, this paper seeks to provide an analysis of the correlation between real degradation data and a numerical carbonation model for concrete structures in the country. According to the Intergovernmental Panel on Climate Change (IPCC) reports, climatic parameters will increase in the future. For this reason, an analysis of the decreasing performance of these structures caused by climate change will be addressed in this paper. This research aims to choose the most accurate mathematical model that can be applied in Paraguay for the prediction of the service life of concrete structures subjected to carbonation-induced corrosion. For that purpose, parameters and variables of the selected model have been defined and subsequently validated through carbonation tests results carried out in a set of RC structures built in the country. The results of this paper should be considered as a first stage for the further formulation of maintenance strategies, which is highly necessary for Paraguay.

A perturbed gamma degradation process with degradation dependent non‐Gaussian measurement errors

AbstractThis paper proposes and illustrates a new perturbed gamma degradation process where the measurement error is modeled as a non‐Gaussian random variable that depends stochastically on the actual degradation level. The expression of the likelihood function for a generic set of noisy degradation measurements is derived, and the expression of the remaining useful life distribution of a degrading unit that fails when its degradation level exceeds a given threshold limit is formulated. A particle filter method is suggested, which allows one to compute the likelihood function and to estimate the remaining useful life distribution in a quick yet efficient manner. In addition, a closed‐form approximation of the perturbed gamma process is proposed to use in the special, yet meaningful, case where the standard deviation of the measurement error depends linearly on the actual degradation level. Finally, an applicative example is discussed, where the parameters of the perturbed gamma process, the remaining useful life distribution, and the mean remaining useful life of the degrading units are estimated from a set of noisy real degradation data.

The transformed gamma process for degradation phenomena in presence of unexplained forms of unit‐to‐unit variability

AbstractThe transformed gamma process is a suitable model for degradation phenomena where damages accumulate gradually over time in a sequence of tiny increments. Attractiveness of the transformed gamma process mainly lies in the fact that it provides a relatively easy way for dealing with phenomena in which the degradation increments over disjoint time intervals are not independent. The transformed gamma process is also a very flexible model. In fact, it is indexed by 2 functions, the “age function” and the “state function,” whose mathematical form can be chosen ad hoc for modeling the dependence of the future degradation increment of a unit on its current age and state, respectively. In this paper, the transformed gamma process is adopted to describe the degradation paths of degrading units in the presence of an unexplained form of unit‐to‐unit variability. The degradation path of each unit is described via a transformed gamma process. Heterogeneity among paths of different units is accounted for by assuming that the scale parameters of the age and state functions vary randomly from unit to unit. Under these assumptions, a quite mathematically tractable model is obtained. The main properties of the proposed model are discussed, and inferential procedures based on the maximum likelihood criterion are implemented. A simple test is presented to check the goodness of fit of the proposed model. Three applicative examples, based on real degradation data, are developed.

Feature Selection Method for Life Prediction in Multiple Degradation Unit: Generalized Rank Mutual Information

Many industries are making efforts to minimize the losses caused by shutdown of manufacturing facilities and to set an optimal maintenance schedule. In this context, prognostics, which predict remaining useful life (RUL) based on information extracted from sensory signals, have attractedattention. There are three methods to perform life prediction: physics-based, data-based, and hybrid. However, data-driven methods are the only way to apply them to a complex industrial facility. By assuming multiple degradation unit data, we can extract various features from the data and select the best feature to create a health index(HI). In this study, we propose a new method for the feature selection step that greatly determines the performance of RUL prediction. Proposed algorithm can automatically select features that are monotonic and have a consistent level of value in normal and failure zone. We validate our method using real degradation data acquired from bearing life testbed.

Optimal Design for Destructive Degradation Tests With Random Initial Degradation Values Using the Wiener Process

This study investigates modeling, estimation and optimization of destructive degradation tests (DDTs) for highly reliable products with random initial degradation values. It is common to observe that the degradation paths of distinct products start from different values specified by a random variable. The random initial value introduces additional uncertainties to the degradation of the product. In this study, Wiener-process-based degradation models are developed for products with random initial values. We first consider a DDT without stress acceleration. In a DDT, the measurement of the degradation destroys a test unit and, thus, only one measurement is available for each unit. Closed-form maximum likelihood (ML) estimators are derived. Then, an accelerated DDT (ADDT) is considered. Based on these results, we investigate optimal designs of both DDT and ADDT with the objective of minimizing the asymptotic variance of the estimated $p$ th-quantile of the failure time distribution under use conditions. The optimal test plans have to be obtained through a numerical approach. Optimality of the plans is verified by the general equivalence theorem. An adhesive bond example with real degradation data is analyzed to show the performance of the proposed methods.

A switching diffusion model for lifetime estimation in randomly varying environments

This article presents a switching diffusion model for estimating the useful lifetime of a component that operates in a randomly varying environment. The component’s degradation process is unobservable; therefore, a signal of degradation is observed to estimate the environment parameters using a Markov chain Monte Carlo statistical procedure. These parameter estimates serve as key inputs to an analytical stochastic model that approximates the first passage time of the degradation process to a critical threshold. Several numerical examples involving simulated and real degradation data are presented to illustrate the quality of these approximations.

Approaches for component degradation modelling in time-varying environments with application to residual life prediction

This article explores two approaches for modelling the effect of time-varying environmental conditions on the degradation processes of systems subject to linear/exponential degradation, and derives the procedures for applying the approaches to real-time residual life prediction. The first approach, namely the look-up table (LUT) approach, accesses the effect of environmental conditions empirically from historical data without artificially imposing any additional term on the classical stochastic linear degradation model, provided that the historical training data covers all possible finite environmental conditions. The second approach, namely the multiplicative acceleration effect (MAE) approach, regards that the environmental condition has a multiplicative acceleration/deceleration effect on component degradation rate, and is applicable to the case when the historical data only partially covers the possible finite/infinite environmental conditions. In approach development, the parameter estimation algorithms are explicitly derived so that the historical training samples with time-varying environmental conditions can be used to estimate the prior distributions of component degradation rate. Several numerical experiments designed based on real degradation data of ball grid array solder joints are conducted to evaluate the proposed approaches. The experimental results verify the superiority of the proposed approaches over the classical approach which simply ignores the effect of environmental conditions. The LUT approach slightly outperforms the MAE approach when they are both applicable. On the other hand, the MAE approach still performs well in presence of new environmental conditions.

Using degradation data to assess reliability: a case study on train wheel degradation

AbstractDegradation experiments are usually used to assess the lifetime distribution of highly reliable products, which are not likely to fail under the traditional life tests or accelerated life tests. In such cases, if there exist product characteristics whose degradation over time can be related to reliability, then collecting ‘degradation data’ can provide information about product reliability. In general, the degradation data are modeled by a nonlinear regression model with random coefficients. If we can obtain the estimates of parameters under the model, then the failure‐time distribution can be estimated. In order to estimate those parameters, three basic methods are available, namely, the analytical, numerical and the approximate. They are chosen according to the complexity of the degradation path model used in the analysis. In this paper, the numerical and the approximate methods are compared in a simulation study, assuming a simple linear degradation path model. A comparison with traditional failure‐time analysis is also performed. The mean‐squared error of the estimated 100pth percentile of the lifetime distribution is evaluated for each one of the approaches. The approaches are applied to a real degradation data set. Copyright © 2008 John Wiley &amp; Sons, Ltd.

Lifetime Distribution Based Degradation Analysis

The methods commonly used for degradation analysis deduce the lifetime distribution of a product from the distribution of the random parameters in the degradation path model for the product. This approach requires a functional form of the joint distribution of the random parameters, which poses certain practical difficulties. In this paper, we propose an alternative approach which makes inference directly on the lifetime distribution itself. In the proposed approach, degradation data are first used to derive predictive intervals of individual lifetimes. Then an imputation algorithm is invoked to obtain the estimate of the lifetime distribution. The approach has the following advantages: 1) the adequacy of the assumption on the lifetime distribution can be easily examined, 2) the estimated lifetime distribution has a closed form, and 3) the procedure can be more easily implemented. A simulation study is reported to demonstrate the efficiency of the proposed approach. The approach is applied to two real degradation data sets, and compared with Lu-Meeker's two stage method in the applications.