Residual life pRediction foR highly Reliable pRoducts with pRioR acceleRated degRadation data

Recently, more and more highly reliable products have emerged in military and aerospace fields. However, it is very time-consuming to obtain enough failure data to evaluate the reliability of highly reliable products under normal stress levels. To resolve this problem, accelerated life test (ALT) is widely adopted to obtain product failure data in an acceptable time period. But for highly reliable products, even accelerated life test is not effective because little failure data can be obtained within limited time and budget. For some highly reliable product, their certain performance indexes will degrade over time. If the performance degradation can be observed, product lifetime information is likely to be extrapolated. Therefore, accelerated degradation test (ADT), under which products are put into accelerated stress levels to accelerate degradation process, has become a popular approach to the lifetime prediction for highly reliable products [13, 21, 26, 32, 33]. However, the aim of ADT is to extrapolate the lifetime information for the population rather than the residual life for an individual. The residual life of a product is defined as the length from the current time to the failure time, and precisely predicting the residual life is important to carry out condition based maintenance (CBM), and prognostics and health management [1, 22]. For an individual with high reliability, the degradation data observed under normal stress levels cannot show a distinct degradation trend, therefore it is difficult to precisely predict the residual life. In order to precisely predict the residual life for an individual with limited real-time degradation data, the prediction methods based on Bayesian inference have been popularly studied [3, 7, 8]. Gebraeel et al [5] developed an exponential degradation model with random parameters to model bearing degradation signals. They assumed the random parameters to obey conjugate prior WAng H-W, Teng K-n. Residual life prediction for highly reliable products with prior accelerated degradation data. eksploatacja i niezawodnosc – Maintenance and Reliability 2016; 18 (3): 379–389, http://dx.doi.org/10.17531/ein.2016.3.9.


Introduction
Recently, more and more highly reliable products have emerged in military and aerospace fields.However, it is very time-consuming to obtain enough failure data to evaluate the reliability of highly reliable products under normal stress levels.To resolve this problem, accelerated life test (ALT) is widely adopted to obtain product failure data in an acceptable time period.But for highly reliable products, even accelerated life test is not effective because little failure data can be obtained within limited time and budget.For some highly reliable product, their certain performance indexes will degrade over time.If the performance degradation can be observed, product lifetime information is likely to be extrapolated.Therefore, accelerated degradation test (ADT), under which products are put into accelerated stress levels to accelerate degradation process, has become a popular approach to the lifetime prediction for highly reliable products [13,21,26,32,33].However, the aim of ADT is to extrapolate the lifetime information for the population rather than the residual life for an individual.
The residual life of a product is defined as the length from the current time to the failure time, and precisely predicting the residual life is important to carry out condition based maintenance (CBM), and prognostics and health management [1,22].For an individual with high reliability, the degradation data observed under normal stress levels cannot show a distinct degradation trend, therefore it is difficult to precisely predict the residual life.In order to precisely predict the residual life for an individual with limited real-time degradation data, the prediction methods based on Bayesian inference have been popularly studied [3,7,8].Gebraeel et al [5] developed an exponential degradation model with random parameters to model bearing degradation signals.They assumed the random parameters to obey conjugate prior

Residual life pRediction foR highly Reliable pRoducts with pRioR acceleRated degRadation data pRognozowanie tRwałości Resztkowej wysoce niezawodnych pRoduktów na podstawie danych histoRycznych z pRzyspieszonych badań degRadacji
To precisely predict the residual life for functioning products is a key of carrying out condition based maintenance.For highly reliable products, it is difficult to obtain abundant degradation data to precisely predict the residual life under normal stress levels.Thus, how to make use of historical degradation data to improve the accuracy of the residual life prediction is an interesting issue.Accelerated degradation testing, which has been widely used to evaluate the reliability of highly reliable products, can provide abundant accelerated degradation data.In this paper, a residual life prediction method based on Bayesian inference that takes accelerated degradation data as prior information was studied.A Wiener process with a time function was used to model degradation data.In order to apply the random effects of all the parameters of a Wiener process, the non-conjugate prior distributions were considered.Acceleration factors were introduced to convert the parameter estimates from accelerated stress levels to normal stress levels, so that the proper prior distribution types of the random parameters can be selected by the Anderson-Darling statistic.A Markov Chain Monte Carlo method with Gibbs sampling was used to evaluate the posterior means of the random parameters.An illustrative example of self-regulating heating cable was utilized to validate the proposed method.

Keywords:
Residual life; Bayesian inference; random parameters; Wiener process; acceleration factor.distributions for mathematical tractability, and obtained the prior distributions of random parameters using the historical degradation data of the population of devices, then predicted the residual life for an individual with the real-time degradation data.Chakraborty et al [2] also studied the exponential degradation model with random parameters, but they assumed the random parameters to obey non-conjugate prior distributions, and adopted the Metropolis-Hasting algorithm to estimate the posterior means of random parameters.Gebraeel et al [6] proposed a Bayesian method, which takes the failure time data as historical information while takes the real-time degradation data of an individual as filed information, to predict the residual life distribution for a rotating machinery.Karandikar et al [10] developed a Bayesian method, which applied Monte Carlo simulations to evaluate the posterior means of random parameters, to predict the residual life for an aircraft fuselage panel.Baraldi et al [1] also developed a Bayesian method, which combines the degradation data of historical equipment and the real-time observing degradation data, to predict the residual life for nuclear power plants.Jin et al [9] presented a Bayesian framework based on a Wiener process with random parameters.The framework utilizes off-line population degradation data and on-line individual degradation data to predict residual life.Wang et al [28] also studied the Bayesian method based on Wiener degradation processes.

Precyzyjne przewidywanie trwałości resztkowej użytkowanego produktu stanowi klucz do prawidłowego utrzymania ruchu w oparciu o bieżący stan techniczny (condition-based maintenance
From the above analysis, it can be concluded that the residual life prediction methods based on Bayesian inference usually take the historical degradation data or failure data under normal stress levels as prior information.However, there is a lack of research on how to use accelerated degradation data as prior information.For some high reliable products, the accelerated degradation data is the only prior information source.Thus, how to make full use of historical accelerated degradation data to predict residual life is an interesting and significant issue.In the paper, we proposed a Bayesian method, which takes accelerated degradation data as prior information while considers the real-time degradation data observed under normal stress level as field information, to improve the prediction accuracy of the residual life for a highly reliable product.
The rest of the paper is organized as follows: Section 2 describes the residual life prediction model based on a Wiener process with random parameters.Section 3 discusses how to evaluate the prior distributions of the random parameters from accelerated degradation data.We deduced the expression of the acceleration factor based on Wiener degradation process, and converted parameter estimates from accelerated stress levels to normal stress level through acceleration factors.Section 4 develops simulation tests to validate the proposed method of converting parameter estimates.Section 5 provides a case study of self-regulating heating cable to investigate the effectiveness of the proposed Bayesian method.Section 6 draws some conclusions.

Residual life prediction model
It is well known that the main characteristic of degradation process is uncertain over time, therefore stochastic process models, such as Wiener process, Gamma process, Poisson process, et al, are naturally appropriate to model the evolution of degradation process.Wiener process can characterize both monotonic and non-monotonic degradation data, and describe unit-to-unit variance of products if random parameters are considered.Hence, Wiener process has been widely applied to model stochastic degradation process, see Whitmore and Schenkelberg [31], Wang [29], Peng and Tseng [19], Wang et al [30], Huang et al [4].
In mathematics, a Wiener process that follow a normal distribution as: where µ is a drift parameter, σ σ ( ) Let a constant value l represent the failure threshold, the failure time ξ can be considered to be the time when ( ) X t firstly reaches l.So the failure time ξ can be described as: For a Wiener process, it is well known that the failure time ξ follows an inverse Gaussian distribution.So the probability density function (PDF) of ξ can be expressed as: Given that the degradation  7,24].Therefore, the PDF of RL can be written as: where ' . Furthermore, the expectation of RL can be deduced from: where 1 ( ) − Λ ⋅ denotes the inverse function of ( ) Λ ⋅ .In order to obtain better flexibility and tractability, the form of ( ) Λ ⋅ was specified as Thus, the RL can be evaluated once the estimates of µ σ , ,r are obtained.However, the real-time degradation data of highly reliable product is commonly so little not to be straightforward used to obtain credible estimates µ σ , ,r ˆˆˆ.To resolve the problem, we adopted a Bayesian method to obtain more credible posterior estimates of µ σ , ,r ˆˆˆ by making full use of historical accelerated degradation data.Although the application of Wiener processes in Bayesian inference has been widely studied in literature, most works assume that the random parameters of a Wiener process obey the following conjugate prior distributions: , where IG( ) ⋅ represents an Inverse Gamma distribution, and , , , a b c d are hyper parameters.The above assumption has two limitations.First, the random effect of r can not be taken into account; Second, µ σ , are limited to sciENcE aNd tEchNology obey the above specified distributions, which may not be applicable to some cases.In this paper, the non-conjugate prior distributions of µ σ , ,r are alternatively used in Bayesian inference, and µ σ , ,r are assumed to be mutually independent.Let π µ σ ( , , ) 2 r represent the joint prior PDF, π µ ( ) , π σ ( ) 2 , π ( ) r be the corresponding PDF of µ σ , ,r , respectively, then π µ σ π µ π σ π ( , , ) ( ) ( ) ( ) is the real-time degradation data of an individual under normal stress level and is a likelihood function, then the joint posterior PDF π µ σ ( , , | ) 2 r ∆X can be deduced according to the Bayesian theory [25,34].
The marginal density functions of µ|∆X , σ 2 |∆X , r|∆X can be expressed as: Furthermore, the expectations of of µ|∆X , σ 2 |∆X and r|∆X can be calculated as: Generally, it is difficult to evaluate E( ) r ∆X through direct mathematical calculation.One alternative method is using Markov Chain Monte Carlo (MCMC) simulation with Gibbs sampling [16,20], we implemented the method in WinBUGS.
r ∆X in Eq. ( 4) and Eq. ( 5), the posterior PDF of RL and the posterior expectation of RL can be obtained.There will be more field degradation data available as the individual product works over time.Once new degradation data is available, E(

Evaluate the prior distributions of random parameters
Before predicting RL using Bayesian method, the joint prior PDF π µ σ ( , , ) 2 r need be evaluated from historical accelerated deg- radation data.In this section, we discussed the method of evaluating π µ σ ( , , ) 2 r from historical accelerated degradation data.Firstly, a Wiener process with 3 random parameters were used to model accelerated degradation data.Then, the acceleration factor constant principle was introduced to deduce the corresponding expression of acceleration factor for a Wiener process, and the parameter estimates were converted from accelerated stress levels to normal stress level through acceleration factors.Lastly, the Anderson-Darling goodnessof-fit method was utilized to select the proper distribution types for 3 random parameters.

Estimate parameters from accelerated degradation data
A constant stress accelerated degradation test was conducted to obtain the lifetime information of the population for a kind of highly reliable product.Suppose that 0 S denotes the normal stress level, k S denotes the kth accelerated stress level, ijk X denotes the ith degradation data of the jth product under k S , ijk t represents the corresponding observing time, represents the degradation data increment, and ( ) To obtain maximum likelihood estimates (MLEs) µ σ   

Suppose ( )
, then the acceleration factor of k S relative to 0 S can be defined as: In practice, Eq. ( 15) is rarely applied to evaluate ,0 k AF since 0 t and k t are difficult to accurately obtained.For a Wiener process, a widely adopted assumption is that µ should change with accelerated stress level varying while σ should be invariable, see Padgett and  Tomlinson [17], Park and Padgett [18], Liao and Tseng [11], Lim and = µ µ .However, there are different assumption that both µ and σ will change with accelerated stress level varying, see Whitmore and Schenkelberg [31], Liao and Elsayed [12].Thus, the expression of acceleration factor for a Wiener process needs further research.In this paper, we applied the acceleration factor constant principle to deduce the expression of acceleration factor for a Wiener process.Zhou et al [35] first pointed out the acceleration factor constant principle that ,0 k AF should be a constant which does not change with time 0 , k t t and depends only on From Eq. ( 16), the following identical equation can be deduced: The deducing process was illustrated as: Eq. (20) indicates that both µ and σ should change with accel- erated stress level varying.When we specify ( ) r t t Λ = , the following equation can be deduced: Eq. ( 21) indicates that both µ and σ should change with acceler- ated stress level varying while r should remain invariable.The mathematical relationships between µ , σ and accelerated stresses can be modeled by reaction rate models [15].Assume that temperature T is a stress variable and the Arrhenius relationship is correspondingly selected as the reaction rate model.According to Eq. ( 21), µ , σ 2 and r can be expressed as: where λ λ λ λ , , , are coefficients.
To obtain the MLEs of λ λ λ λ , , , , a likelihood function that incorporates all the accelerated degradation data was constructed: With λ  2 and λ  2 , ,0 k AF was evaluated by: To simply the computation, AF k,0 was represented by ,0 k Z .
Thus, the µ σ 2 under k T can be transformed into the corresponding values under 0 T by: For the denotation convenience, µ  and M denotes the total number of products under the accelerated test.
Substitute Eq. (3) into Eq.( 17), and specify ( ) To ensure ,0 k AF is a constant which does not change with k t , the following relationships must be satisfied: So the following equation can be deduced: sciENcE aNd tEchNology The prior distributions of random parameters µ σ , , 2 r can be ob- tained as µ ηδ ~Wbl( , )

Validation test
A simulation test was developed to validate the conversion method illustrated in section 3.2.The simulated degradation increment 0 0 , ( ) under 0 S was generated from a conditional Wiener process [27,28], that is: where .Then, specify the acceleration factor ,0 k AF to be 0.

AF
, so the result is consistent with Eq. ( 21).

Modeling degradation data
The historical accelerated degradation data of self-regulating heating cable [31] was listed in Table 2.Because the type of heating cable experienced a cure process at the beginning of the degradation test, only the degradation data after the cure process were considered.There were 15 products that were averagely allocated to 3 accelerated temperature levels under a constant-stress ADT.It was assumed that the normal temperature level was 0 448.15KT = . Degradation was measured as the natural logarithm of resistance, and the failure threshold l was specified as ln(2) l = .
However, there was no field degradation data available under 0 T .We simulated the field degradation data which are listed in Table 3.

Determining prior distribution
Eq. ( 14) was used to obtain the MLEs of parameters for each product, and the estimates were listed in Table 4.For a Wiener degradation process, the independent degradation increments ijk x ∆ of each product follows a Normal distribution, written as ( ) ~N 0,1 .If the above relationship holds, the null hypothesis that the degradation process of the jth product under k T obeys a Wiener process is accepted.An Anderson-Darling test with significance level α = 0 05 .validates that the degradation processes of all products obey Wiener processes.  λ  3 10 317 = .
. According to the Eq. ( 23), conversion coefficients were calculated as 1,0 2.944 . Furthermore, we converted the MLEs of parameters under k T to the corresponding values µ  m , σ  m 2 , ˆm r under 0 T , respectively, as listed in Table 5. Anderson-Darling statistic was applied to determining the best fitting distribution.Table 6 shows the values of AD .Since σ  m 2 should be always large than zero, the Normal distribution is excluded.From Table 6, it was concluded that the Weibull distribution was the best fitting distribution type for both µ  m and σ  m 2 , and the Normal distribu-      RL denote the predictions that were evaluated by the method proposed in this paper, the residual life predictions and the confidence intervals of the predictions at several measuring time points were summarized in table 8.The confidence intervals were obtained with 95% confidence level by Bootstrap sampling method.The prediction method for 2 RL was detailed demonstrated in appendix.Figure 4 shows the changing curves of 1 RL , 2 RL ,

3
RL and the PDF of 3 RL .

Analysis and Conclusion
In table 7, the confidence intervals of .It is concluded that the prediction method taking accelerated degradation data as prior information improves the prediction precision of the residual life.
As more field degradation data was available, all the confidence intervals of RL was adopted in the case study.Therefore, it was appropriate and reliable to adopt the prediction method for 3 RL in the case study.

Conclusions
In this paper, a residual life prediction method for degradation product based on Bayesian inference was proposed.A Wiener process with a time function was used to model degradation data.
Our proposed method made two potential novel contributions.a).Acceleration factor was introduced to transform the test data from accelerated stresses to normal use stress, so that accelerated degradation data can be considered as prior information.b).The non-conjugate prior distributions of random parameters were applied to Bayesian inference, which not only can consider the random effects of all the parameters but also can avoid the potential misspecifications of the parameter distributions when conjugate prior distributions were adopted.
Besides, there are several meaningful workings and conclusions about the study of the paper.
The residual life prediction method based on Bayesian infer-1) ence that integrated the field degradation data and accelerated  degradation data can improve the prediction accuracy and realize real-time updating for the residual life.The acceleration factor constant principle was used to deduce 2) the relationships that the parameters of the Wiener process with a time function should satisfy.It offered a feasible approach to constructing the acceleration models for the parameters of the Wiener process.
The prediction method that adopts non-conjugate prior distri-3) butions can be applied to more cases, since it dose not require parameters to follow the specified distribution types.Moreover, the assumption that parameters are mutually independent facilitates the engineering application of the method.

)
µ|∆X , E( ) σ 2 |∆X and E( ) r|∆X are immediately updated.So the RL of an individual can be real-time predicted.
sciENcE aNd tEchNologyYum[14], et al.Based on the above assumption, the expression of

3 .
033e-2 3.005e-2 3.033e-2 3.045e-2 3.017e-2 σ  m 2 4.500e-5 4.216e-5 4.935e-5 4.827e-5 3.912e-5 tion was the best fitting distribution type for ˆm r .The MLEs of hyper parameters were obtained according to Eq. (26) and Eq.(27).So the prior distributions of 3 random parameters were evaluated from historical accelerated degradation data, they are µ ~( .degradation data was obtained, the MCMC method with Gibbs sampling was used to fit the posterior distributions of random parameters in WinBUGS software.After the 5th field degradation data was got, the fitted posterior distributions were shown in figure 1.From figure 2, it can be seen that all the iterative processes are convergent since the autocorrelation functions quickly tend to 0. The posterior means E( | ) µ ∆X , E( | ) σ 2 ∆X and E( | ) r ∆X were evaluated in WinBUGS, as listed in table 7. Furthermore, with new field degradation data was available, E( | ) µ ∆X , E( | ) σ 2 ∆X and E( | ) r ∆X were corresponding updated, and their changing curves were shown in Figure 3. Let 1 RL denote the predictions that were evaluated from only the field degradation data of an individual, 2 RL denote the predic-

Fig. 3 .Fig. 4 . 2 RL , 3 RL and the PDF of 3 RL
Fig. 3.The changing curves of the posterior means of random parameters

2 RL and 3 RL , the curve of 1 RL 3 RL 3 RL , the curve of 2 RL 2 RL requires µ  z and σ  z 2 to
became smaller.It suggests that the prediction precision was gradually improved, and the width of the PDF curve in figure4became narrower over time also supports the conclusion.Compared with was not smooth and presented comparatively big changes.The reason is that the field degradation data was short and had a nonlinear change.Thus, newly obtained degradation data may significantly changes the former estimates of the parameters, which leaded to the poor regularity of residual life predictions.In comparison, the prediction methods for 2 RL and reduced the impact of the abrupt change of the field degradation and improved the reliability of the prediction results.Com-pared with was more smooth and had smaller changes, because the prediction method for 2 RL didn't consider the random effects of r .The estimates of r obtained from the prior information was used to the prediction model of 2 RL .Thus, 2 RL was more conservative and greatly influenced by the prior information.In contrast, the prediction method for 3 RL took the random effects of 3 parameters into account, which let 3 RL more sensitively reflect the variation of the field degradation data.In addition, the prediction method for characteristics of Normal distribution.The significance level was specified as 0.5, and the information of the goodnessof-fit test was shown in figure5.The null hypothesis was rejected since the values of p were less than 0.05.It would cause the misspecification of distribution types of random parameters if the prediction method for 2

3.3. Select the best fitting distribution types and estimate hyper parameters
4,4, respectively, and calculate the converted degrada-

Table 1 .
The results of simulation test

Table 2 .
Historical accelerated degradation data of heating cables * A period denotes a missing value.

Table 3 .
The simulated field degradation data for an individual heating cable at 0 T

Table 4 .
MLEs of parameters for each product

Table 6 .
The values of AD

Table 7 .
The posterior means of random parameters after the 5th field degradation data was got

Table 8 .
The predictions of residual life (