Bayesian and likelihood inferences on remaining useful life in two-phase degradation models under gamma process

doi:10.1016/j.ress.2017.11.017

Reliability Engineering & System Safety

Volume 184, April 2019, Pages 77-85

https://doi.org/10.1016/j.ress.2017.11.017 Get rights and content

Highlights

•
Two approaches are presented for the RUL prediction of products with two-phrase degradation.
•
The RUL under gamma process without degradation rate change is considerably underestimated.
•
The RUL prediction for a specific product can be obtained, although the rate change has not occurred.
•
The SEM yields relatively less bias and more reliable interval estimates.
•
The Bayesian approach requires less computational time.

Abstract

Remaining useful life prediction has been one of the important research topics in reliability engineering. For modern products, due to physical and chemical changes that take place with usage and with age, a significant degradation rate change usually exists. Degradation models that do not incorporate a change point may not accurately predict the remaining useful life of products with two-phase degradation. For this reason, we consider the degradation analysis for products with two-phase degradation under gamma processes. Incorporating a probability distribution of the time at which the degradation rate changes into the degradation model, the remaining useful life prediction for a single product can be obtained, even though the rate change has not occurred during the inspection. A Bayesian approach and a likelihood approach via stochastic expectation-maximization algorithm are proposed for the statistical inference of the remaining useful life. A simulation study is carried out to evaluate the performance of the developed methodologies to the remaining useful life prediction. Our results show that the likelihood approach yields relatively less bias and more reliable interval estimates, while the Bayesian approach requires less computational time. Finally, a real dataset on LEDs is presented to demonstrate an application of the proposed methodologies.

Introduction

Due to advance technologies of sensors, there are multiple data sources in play. Therefore, prognostics and system health management (PHM) becomes an important topic in modern reliability engineering for improving the safety and performance of components and systems. In addition, there has been a growing interest of PHM in different fields including electronics, smart grid, nuclear plant, power industry, aerospace and military application, fleet industrial maintenance, and public health management [1]. PHM is a systematic approach for failure prevention by monitoring the health/status of products and systems, predicting failure progression, and mitigating operating risks through repair or replacement [1]. Effective prognostics that can yield an advance warning of impending failure in a system are critical for making maintenance decisions and executing preventive actions prior to failure occurrence to extend system life.

Systems are commonly installed with sensors that are designed to measure system performance. One of the important topics in PHM is degradation data analysis. Many papers have recently appeared on degradation models. Interested readers may refer to articles [2], [3], [4], [5]. Wiener, gamma and inverse Gaussian processes are popular for modeling degradation data in the engineering literature. A Wiener process can represent increase and decrease in the performance of products over time, and thus is useful for some specific datasets. For monotonic degradation data, such as fatigue crack growth rates in metals of steel and aluminum alloy [6], resistance in carbon-film resistors [7], and light emitting diodes (LEDs) [8], [9], gamma and inverse Gaussian processes are more appropriate for the degradation data analysis.

While the application of system health monitoring is established, degradation data that describe quality characteristics over time are collected. In degradation analysis, a failure time can be viewed as the first-passage time of a specified threshold in the degradation process. Furthermore, remaining useful life can be defined as the length of time from the current degradation to a pre-specified threshold, and has been frequently used as one of the vital indexes in PHM. For instance, remaining useful life prediction are valuable for condition-based maintenance policy. Si et al. [10] provided a comprehensive review on the statistical approaches of remaining useful life prediction. Chen and Tsui [11] presented a flowchart of model updating and remaining useful life prediction for condition-based maintenance. However, there is a lack of discussion on remaining useful life prediction under gamma process, probably due to the fact that the prediction has no closed form expression. In this article, an approximation using a two-parameter Birnbaum–Saunders distribution is presented.

In many existing research works on degradation models, it is common to assume that the degradation process is smooth and it does not have any change point. However, this strong assumption may be violated in practice. For instance, in studies of power outputs of lasers [12] and discharge capacity of a Li-ion battery [13], a significant change point is observed in the degradation. The degradation rate often increases due to physical and chemical changes that take place with usage and with age. Interested readers may refer to articles concerning degradation process with a change point such as [11], [12], [14], [15]. However, those existing degradation models may not be appropriate for the analysis of light intensity of LEDs. Specifically, light intensity is usually monotonically decreasing in time. Without loss of physical reality, a gamma process with a change point is proposed in this paper for predicting the remaining useful life of products with two-phase degradation.

In a reliability study of lasers, the amounts of power outputs are measured at several inspection times, however, the time at which the degradation rate changes cannot be observed. To handle the data in presence of missing data, Ng [12] proposed an expectation-maximization (EM) algorithm [16] to find the maximum likelihood estimates. When the likelihood function under the proposed model can be complicated, a stochastic version on the EM algorithm, stochastic EM (SEM) algorithm [17], is an alternative for the parameter estimation. The SEM algorithm has recently been applied to many studies in reliability data analysis [18], [19], [20]. In this paper, the gamma process with a random change point is considered and a SEM algorithm is developed for the parameter estimation. On the other hand, the Bayesian approach is often adopted for parameter estimation. Many researchers applied Bayesian methods to numerous applications in degradation data analysis [4], [11], [15], [21]. However, the choice of prior distributions is always one of the challenging tasks because inaccurate informative priors could seriously bias the parameter estimates, especially in the case of small sample sizes. One of the objectives in this paper is to compare the performance between the likelihood and Bayesian approaches in terms of bias and root mean square error of point estimation, and coverage probability of interval estimation of the remaining useful life.

The rest of this paper is organized as follows. In Section 2, a two-phase degradation model is formulated. For the purpose of PHM, the remaining useful life prediction is also presented in Section 2. In Section 3, a likelihood approach via SEM algorithm and a Bayesian approach are developed for estimation of model parameters. The maximum likelihood and the Bayesian inferences on the remaining useful life for a single product is presented in Section 4. In Section 5, a simulation study is carried out to evaluate the performance of the developed methodologies for different sample sizes and different numbers of inspections. In Section 6, the developed methodologies are illustrated by applying to analyze a real dataset on light emitting diodes (LEDs). Finally, in Section 7, some concluding remarks are made wherein some further issues of interest are pointed out.

Section snippets

A two-phase degradation model

A two-phase degradation model is motivated by data consisting of outputs of light intensity of twelve LEDs [22]. Fig. 1 displays the outputs of light intensity of the LEDs. It can be seen that the degradation rate changed around 500 hours. We can also observe that the outputs degraded rapidly in the initial phase and then degraded gradually. As the outputs are monotonically decreasing in time, it is therefore reasonable to formulate the outputs by using a two-phase degradation model under gamma

Estimation of model parameters

In this section, we propose the likelihood and Bayesian approaches to estimate the model parameters in the two-phase degradation model described in Section 2

Inference on remaining useful life

With the sequences of the model parameter vector and the rate change times based on either the likelihood approach via SEM algorithm or the Bayesian approach, we can generate sequences of the degradation and the corresponding expected remaining useful life using the following steps:

C1.
Set $d_{i} = - \ln (δ) - G_{i, m} .$ When $τ_{i}^{[b]} < t_{m},$ compute the expected remaining useful life $μ_{ℓ_{i}}^{[b]} = ℓ (t_{m}, Y_{i} (t_{m}), δ | T = τ_{i}^{[b]})$ by using (7), and generate a remaining useful life $ℓ_{i}^{[b]}$ from Birnbaum–Saunders distribution with shape $κ_{i} = \sqrt{β /}$

Simulation study

In this section, an extensive Monte Carlo simulation study is conducted to examine the performance of the proposed point and interval estimation methods in terms of their biases, root mean square errors (RMSE), coverage probabilities (CP) and computational time, for different sample sizes $n = (12, 30)$ and different numbers of inspection $m = (10, 20) .$ Suppose that the degradation of each item was measured at $(m + 1)$ different times, $t_{j} = 5 j, j = 0, 1, \dots, m .$ A failure is defined when the percentage of

Illustrative example

As an illustration, we perform the degradation analysis on the light intensity data of twelve LEDs presented in Fig. 1 using the proposed two-phase degradation model. For the Bayesian approach, we assume that the prior distributions π(θ_g) and π(θ_τ) are gamma distributed with shape parameters (2.54, 0.51, 1.95) and (56, 1.6), and scale parameters (0.01, 0.01, 0.01) and (10, 1), respectively. In Step B1 of the Bayesian computation, $θ_{g}^{*}$ and $θ_{τ}^{*}$ are sampled from gamma distributions with expected

Concluding remarks

In this paper, we proposed a two-phase degradation model that incorporates the random rate change time in a gamma process. Both likelihood and Bayesian approaches are developed for parameter estimation and the remaining useful life prediction for an individual item under the proposed two-phase degradation model. The proposed model and methodologies are applied to analyze the data of outputs of light intensity of 12 LEDs and subsequently make statistical inference on the remaining useful life of

Acknowledgement

We express our sincere thanks to the anonymous reviewers for their constructive comments and useful suggestions on the original manuscript which led to this improved version. The authors thank the National Natural Science Foundation of China (Project No. 11471275), the Research Grants Council Theme-Based Research Scheme (Project No. T32-101/15-R) and the Early Career Research Fund from the Research Grants Council of Hong Kong (Project No. 28300114) for funding this research. Ng is supported by

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Bayesian and likelihood inferences on remaining useful life in two-phase degradation models under gamma process

Highlights

Abstract

Introduction

Section snippets

A two-phase degradation model

Estimation of model parameters

Inference on remaining useful life

Simulation study

Illustrative example

Concluding remarks

Acknowledgement

Reliab Eng Syst Saf

Eur J Oper Res

Reliab Eng Syst Saf

Reliab Eng Syst Saf

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

Eur J Oper Res

Accelerated degradation tests: modeling and analysis

Technometrics

Stochastic modelling and analysis of degradation for highly reliable products (with discussion)

Appl Stoch Models BusInd

Bivariate analysis of incomplete degradation observations based on inverse Gaussian processes and copulas

IEEE Trans Reliab

Covariates and random effects in a gamma process model with application to degradation and failure

Lifetime Data Anal

Stochastic degradation models with several accelerating variables

IEEE Trans Reliab

Accelerated degradation analysis for the quality of a system based on the Gamma process

IEEE Trans Reliab

Inference on remaining useful life under gamma degradation models with random effects

Statistical Modeling for Degradation Data, 253–266

Condition monitoring and remaining useful life prediction using degradation signals: revisited

IIE Trans

An application of the EM algorithm to degradation modeling

IEEE Trans Reliab

State-of-the-art of battery state-of-charge determination

Meas Sci Technol