Accelerated proportionaldegradation hazards-odds modelin accelerated degradation test

1.Schoolof Reliability and Systems Engineering,Beihang University,Beijing 100191,China; 2.Departmentof Industrial Engineering,Tsinghua University,Beijing 100084,China

Accelerated proportionaldegradation hazards-odds modelin accelerated degradation test

Tingting Huang1,*and Zhizhong Li2

1.Schoolof Reliability and Systems Engineering,Beihang University,Beijing 100191,China; 2.Departmentof Industrial Engineering,Tsinghua University,Beijing 100084,China

An accelerated proportionaldegradation hazards-odds modelis proposed.Itis a non-parametric modeland thus has pathfree and distribution-free properties,avoiding the errors caused by faulty assumptions of degradation paths or distribution of degradation measurements.It is established based on a link function which combines the degradation cumulative hazard rate function and the degradation odds function through a transformation parameter,and this makes the accelerated proportionaldegradation hazards modeland the accelerated proportionaldegradation odds modelspecialcases ofit.Hypothesis tests are discussed,and the proposed modelis applicable when some modelassumptions are satis fi ed.This modelis utilized to estimate the reliability of miniature bulbs under low stress levels based on the degradation data obtained under high stress levels to validate the effectiveness of this model.

accelerated proportional degradation hazards (APDH),accelerated proportionaldegradation odds(APDO),link function,non-parametric model,accelerated degradation test (ADT),reliability estimation.

1.Introduction

As products become more reliable and enduring,itis very dif fi cultto estimate the reliability of the products based on traditionalreliability tests[1].Itis also hard to obtain failure time data of the products through the accelerated life test(ALT).Fortunately,for many products,there is one or more measurements,representing productperformance which degrades with time and thus making degradation test (DT)applicable[2,3].However,in the DT,forsome cases, degradation data ofthe performance parameterof products obtained under normal operating conditions do not have an obvious degradation trend making it hard to estimate the reliability ofthe products.The accelerated degradation test(ADT)makes it possible to predict the reliability of high-reliable and long-lifetime products.In the ADT,products are tested under higher operating stress levels.Degradation data of the performance parameter are monitored and utilized to extrapolate the reliability of the products under normaloperating stress levels.

Accelerated degradation models are used for reliability extrapolation in the ADT.Followed by the categories of approaches to prognosis[4],accelerated degradation models are divided into physics-based models,statistics-based models,and arti fi cial intelligence-based models.Physicsbased models,like Arrhenius modeland Eyring model,derive from physics or chemistry relationships and are used forspeci fi c stress types.For example,the Arrhenius model is used when products are exposed to high temperature[5]. Statistics-based models are based on statistical analysis methods.They are used for data processing when physicsbased models are not applicable.Statistics-based models can be divided into parametric models and non-parametric models[6].In parametric models,the number and meaning of model parameters are pre-determined,thus,a prespeci fi ed degradation path or distribution of degradation measurements is required.In non-parametric models,the number and meaning of model parameters remain fl exible,and thus,the non-parametric modelhas path-free and distribution-free properties.Arti fi cial intelligence-based models,which are proposed and developed in recentyears, are based on arti fi cialintelligenttechniques,such as arti ficial neural networks[7],support vector machine[8],and fuzzy theory[9].

These three types of accelerated degradation models have theirmerits and drawbacks.The physics-based model requires speci fi c physics or chemistry relationships properly describing productfailure mechanisms.However,it is dif fi cult to establish such a relationship for every failure mechanism.The statistics-based modelis more fl exible and can be used in more reliability estimation cases,butitdoes not have any explicit physical meaning.The arti ficialintelligence-based modeldoes notrequire any speci fi c modelas in physics-based and statistics-based approaches. It has the self-training property,however,some obtained data are notfully used as they are used forverifying the established model.Neither does ithave any explicitphysical meaning.

Statistics-based models can be further classi fi ed into parametric models and non-parametric models.A prespeci fi ed degradation path or distribution of degradation measurements is required in a parametric model.However, it might cause comparatively large errors if the assumed path or distribution is notin accordance with the realsituation.There are three approaches used in parametric models:the general degradation path model[10],the timedependentparameter distribution(random process)model [11],and the stochastic process model with independent increment[12-14].

The non-parametric model is a path-free and distribution-free model.It avoids the errors caused by faulty assumption of degradation paths or distribution of degradation measurements.In[15]an accelerated degradation modelcalled non-parametric regression accelerated life-stress model was proposed,where it is assumed that various stress levels affect only the degradation rate,but not the shape of the degradation curve.In[16]a proportional wear-out model to analyze accelerated degradation data by non-parametric regression was presented.In[17] a non-parametric accelerated degradation model called (accelerated)proportionaldegradation hazards modelwas presented.This model assumes that stresses have a multiplicative effect on the baseline degradation hazard rate function.

In this accelerated proportional degradation hazards (APDH)model,it is also assumed that the ratio of baseline degradation hazard rate functions atany speci fi c time under differentstress levels is a constant.However,sometimes,it happens that the baseline degradation odds functions ratherthan the baseline degradation hazard rate functions at any speci fi c time under different stress levels are proportional.

This paper establishes a link function which combines the degradation cumulative hazard rate function and the degradation odds function through a transformation parameter.An accelerated proportionaldegradation hazardsodds(APDHO)modelis proposed based on this link function.It makes the APDH,which is proposed based on the degradation cumulative hazard rate function,and the accelerated proportional degradation odds(APDO)model, which can be proposed based on the degradation odds function,special cases of it.The proposed model is validated fi nally by an engineering case of miniature bulbs.

2.The proposed APDHO

In the ADT,the APDH model is proposed based on the degradation cumulative hazard rate function,while the APDO model can be proposed based on the degradation odds function.These two models have the assumption that either degradation cumulative hazard rate functions are proportional(for APDH model)or degradation odds functions are proportional(for the APDO model)at any time underdifferentstress levels.However,there are some cases that neither of the assumptions is violated in engineering applications.The APDHO model is proposed in this session and it makes the two models specialcases of it,thus, it no longer requires deciding which of the two models to use.

A link function is de fi ned and it combines the degradation cumulative hazard rate function and the degradation odds function through a transformation parameter.It is formed as the productof the baseline link function and the effectof stress.The constraints of the link function are discussed before the formation of the baseline link function is given.The likelihood function is then determined, by maximizing which,the estimates of the parameters can be obtained.Hypothesis tests are proposed at the end of this session to presentthe method ofverifying the applicability of this model.

In this paper,it is assumed that there is a degradation measurement that can represent the performance of the product.This degradation measurement increases or decreases monotonously,and the time point when itreaches a pre-speci fi ed threshold is considered as the failure time of the product.The degradation process is irreversible. The failure mechanism of the product remains the same throughoutthe whole degradation process.

2.1 Definition of link function

Followed by the Aranda-Ordaz parameter family[18]and its derivative model in the accelerated life test[19],a link function with time and stress as covariates is de fi ned in(1) in orderto combine the degradation cumulative hazard rate function and the degradation odds function,

where g(x;t,z)is the link function,R(x;t,z)is the reliability function,x is the degradation measurement,t is the time point,z is the stress level,and c is the transformation parameter.

When c→0,gc→0(x;t,z)=-ln R(x;t,z)following the l’Hospital’s rule,then g(x;t,z)becomes a degradationcumulative hazard rate functionΛ(x;t,z);when c=1,yielding to a degradation odds functionθ(x;t,z),where F(x;t,z)is the cumulative distribution function.

The proposed APDHO model is established based on the link function g(x;t,z).As the APDH model is builtbased on the degradation cumulative hazard function Λ(x;t,z),and the APDOmodelcould be builtbased on the degradation odds functionθ(x;t,z),the proposed APDHO model then makes the APDH and APDO models special cases of it.It avoids the estimation errors caused by the improperdecision ofwhich ofthe two other models to use, and as in engineering cases,the assumptions of both models mightbe inviolate.

2.2 Formation of link function

Time t is assumed to have a multiplicative effect on an independentfunction ofdegradation measurement x in the APDHmodel.In addition,stress level z is assumed to have a multiplicative effect on the product of the function of x and the function of t[17].Following these assumptions, link function g(x;t,z)is assumed as

where g0(x;t)is the baseline link function.

The formation of the effectof stress z,i.e.e?βz,is defi ned based on physics-based accelerated models,such as the Arrhenius modeland the inverse power law model.

p(x)and q(t)are determined based on the following consideration.For Weibull distribution,the cumulative hazard rate function of a random variable x is given asΛ(x)=xmη?m,where m is the shape parameter and ηis the scale parameter.Thus,the degradation cumulative hazard rate function of a random variable x with covariate t can be described asΛ(x;t)=xm[ε(t)]?m,where m is the shape parameterandε(t)is the scale parametervarying with covariate t.

For log-logistic distribution,the odds function of a random variable x isθ(x)=xpλ?p,where p is the shape parameter andλis the scale parameter.Thus,the degradation odds function of a random variable x with covariate t can be described asθ(x;t)=xp[ε(t)]?p,where p is the shape parameter andε(t)is the scale parameter varying with covariate t.

Degradation cumulative hazard rate functionΛ(x;t)= g0c→0(x;t)for the Weibull distribution and degradation odds functionθ(x;t)=g0c=1(x;t)forthe log-logistic distribution satis fi es the assumption that the time has a multiplicative effect on an independent function of a degradation measurement.Thus,p(x)and q(t)are determined based on the formation ofΛ(x;t)for the Weibull distribution andθ(x;t)for the log-logistic function as p(x)=xγand q(t)=ε(t)?γ.

The link function g(x;t,z)is then de fi ned as

whereγ＞0 andβ＞0 are unknown parameters,ε(t) is a function of time t and it represents the characteristic value of degradation measurement x attime t under stress level z.

2.3 Constraints on link function

2.3.1 Degradation percentage

To unify the degradation paths of monotonously increasing and decreasing degradation processes and eliminate the inconsistency of initial values of different test units, x is rede fi ned as the degradation percentage.For the monotonously decreasing degradation process,

The change of degradation percentage x with time t under differentstress levels are shown in Fig.1.

Fig.1 Degradation percentage curves under different stress levels

Thus,for both monotonously increasing and decreasing degradation processes,degradation percentage x increases with the time,and when itreaches a threshold,the testunit is considered failed.

2.3.2 Constraints

From(1)and(4),the reliability function is expressed as

The complementary reliability function is de fi ned as

Itis then given as

Based on(9)and the property of the complementary reliability function,the following equations must be satisfi ed,

The relationship between g(x;t,z)easily expressed based on(1)and(9)as follows:

Equation(10)could then resultin

Considering g(x;t,z)=p(x)q(t)e?βz,it can be deduced that p(0)=0,p(∞)=∞,q(0)=∞and q(∞)=0.

After the transformation of x from the degradation measurement to the degradation percentage,the degradation curves become monotonously increasing lines pass through the origin(for the monotonously increasing degradation process,degradation percentages are assumed to remain the same trend after they reach one,the extreme threshold,as reliability becomes zero from then).The constraints of p(x)=xγas p(0)=0 and p(∞)=∞are then satis fi ed.As q(t)=ε(t)?γ,ε(t)has the constraints thatε(0)=0 andε(∞)=∞in order to satisfy the constraints on q(t).ε(t)should be assumed as a function of t thatsatis fi es the constraints on it.

2.4 Establishment of log-likelihood function

Based on(7),the probability density function is described as

The log-likelihood function is de fi ned as the logarithm of the probability density function

By maximizing the log-likelihood function as in(14), the estimates of modelparameters are obtained.

2.5 Hypothesis tests

In the APDHO model,g(x;t,z)is de fi ned as g(x;t,z)= p(x)q(t)e?βz.Thus,it is assumed that the ratio of either degradation cumulative hazard rate functions or degradation odds functions at different times under each stress levelis a constant(Assumption 1).Moreover,it is also assumed thatthe ratio of either degradation cumulative hazard rate functions ordegradation odds functionsatany time under differentstress levels is a constant(Assumption 2).

The APDHO model is applicable when both assumptions are satis fi ed.It might cause errors to use this model when any ofthe two assumptions is violated.Thus,hypothesis tests are required to be performed in orderto verify the applicability of this model.

Hypothesis test 1The link function g(x;t,z)is defi ned as g(x;t,z)=p(x)q(t)e?βz,and thus,the ratio of g(x;t,z)at two differenttimes under one speci fi ed stress level z is

When c→0,gc→0(x;t,z)=Λ(x;t,z),it can be deduced from(15)that

whereλ(x;t,z)is the degradation hazard rate function.

Exchanging the numeratorwith the denominatoron two sides of(16),

Itresults in the following equation

When c=1,gc=1(x;t,z)=θ(x;t,z),it can be deduced from(15)that,

Exchanging the numeratorwith the denominatoron two sides of(19),

Itresults in the following equation

Hypothesis test 1 can then be performed to verify whether(18)or(21)is satis fi ed.

Hypothesis test 2The link function g(x;t,z)is defi ned as g(x;t,z)=p(x)q(t)e?βz,and thus,the ratio of g(x;t,z)atone speci fi ed time t under two differentstress levels is

Following the same methods as in hypothesis test 1,it can be deduced from(22)that,when c→0,

and when c=1,

Hypothesis test 2 can then be performed to verify whether(23)or(24)is satis fi ed.

Itcan be concluded thatthe APDHOmodelis applicable when both assumptions are satis fi ed.

3.Case study

3.1 Accelerated degradation test system

An accelerated degradation testsystem is established in order to obtain degradation data of miniature bulbs under several stress levels.The failure mechanism of miniature bulbs is mainly the non-uniform evaporation of the fi lamentowing to localdefects[20].The voltage is chosen as the stress type.The ADT system mainly consists of power supply,bulb circuit board,I/O connector,data acquisition board,PC,data recording software,miniature bulbs and resistors as shown in Fig.2. The powersupply provides the voltage as the stress over the bulbs which are connected with resistors in series on the bulb circuit board.The voltages over the resistors are acquired through the data acquisition board and the I/O connector,and then recorded in PC by the data recording software.

Fig.2 Accelerated degradation test system for miniature bulbs

3.2 Data collection

3.2.1 Degradation data

Degradation tests are performed understress levels 8 V,7.5 V and 7 V.Totally 14 testunits are tested undereach ofthe three stress levels.Degradation tests are operated until all the test units fail.Degradation paths of voltage degradation measurements under the three stress levels are shown in Fig.3.

Fig.3 Degradation paths of voltage degradation measurements under stress levels 8 V,7.5 V and 7 V

With the degradation of the miniature bulbs,the voltages over the resistors decrease and fi nally drop to a certain value and mostly remain constant,when the bulbs fail.Voltage degradation measurements are recorded every minute and then transformed into voltage degradation percentages using(5).The voltage degradation percentage right before the time point when a bulb fails is considered as the threshold and it is set as 0.1 based on history data.

3.2.2 Failure time data

Totally 13 failure time data under stress levels 6.7 V,6.5 V and 6.3 V are collected for the validation of the proposed APDHO model.Failure time data under the three stress levels are listed in Table 1.

Table 1 Failure time data

3.3 Degradation data pre-processing

Voltage degradation percentages are sampled every 30 min,70 min and 100 min under stress levels 8 V, 7.5 V and 7 V respectively so that the obtained 33 data points cover 70%-90%of the time span from the teststart time point to the fi rst observed failure time point of the bulbs.Thus,the assumed censoring time points for the three stress levels are 961 min,2 241 min and 3 201 min respectively.A moving average method is used to eliminate the noise of the sampled data and the chosen spans are 3,5 and 7 forstress levels 8 V,7.5 V and 7 V.Two data points at each end under each stress level are excluded in estimations.

3.4 Hypothesis tests

Each pair of the three stress levels 8 V,7.5 V and 7 V are combined for hypothesis tests.For each combination, three equally spaced time points are selected and reliability estimates of voltage degradation percentages at these time points under the two stress levels are calculated by the Kaplan-Meier estimator.The logarithm of degradation cumulative hazard rate functionΛ(x;t,z)and the logarithm of degradation odds functionθ(x;t,z)at the three time points under the two stress levels are then evaluated based on the reliability estimates.

For hypothesis tests under stress levels 8 V,7.5 V and 7 V,there are originally 14 data for each of the time points 211,421 and 631.The logarithm of degradation cumulative hazard rate functionΛ(x;t,z)and the logarithm of degradation odds functionθ(x;t,z)atthe three time points are calculated based on corresponding reliability values. The Kaplan-Meier method is used for calculating reliability values.Thus,when allthe 14 testunits fail,the reliability is equalto zero,which can neither be in the logarithm function,as in the calculating logarithm of the degradation cumulative hazard rate function,nor be in the denominator,as in the calculating logarithm of the degradation odds function.Thus,only 13 data points appear in fi gures showing hypothesis test results.The results are shown in Fig.4-Fig.9.

Fig.4 lnΛ(x;t,z)under stress levels 8 V and 7.5 V

Fig.5 lnθ(x;t,z)under stress levels 8 V and 7.5 V

Fig.6 lnΛ(x;t,z)under stress levels 8 V and 7 V

Fig.7 lnΛ(x;t,z)under stress levels 7.5 V and 7 V

Fig.8 lnθ(x;t,z)under stress levels 8 V and 7 V

Fig.9 lnθ(x;t,z)under stress levels 7.5 V and 7 V

For each pair of the three stress levels,it can be seen from these fi gures that for each stress level,the curves of different time points do not cross,but they are approximately parallelto each other(Hypothesis 1).Similar situations can be observed for each time pointunder different stress levels(Hypothesis 2).Assumptions of the APDHO model can be regarded satis fi ed from an engineering perspective.

3.5 Functions specification

The sampled voltage degradation percentages under three stress levels 8 V,7.5 V and 7 V are shown in Fig.10.They representthe degradation paths of voltage degradation percentages.Voltage degradation percentages as shown in the

fi gure are used in the extrapolation of the reliability under low stress levels.

Fig.10 Degradation paths of voltage degradation percentages under stress levels 8 V,7.5 V and 7 V

ε(t)re fl ects the characteristic value of voltage degradation percentage x at time t,and thus the formation of ε(t)should be speci fi ed based on the degradation path of voltage degradation percentage x.Linear polynomial, quadratic polynomial and power functions are used to fi t the degradation paths.The linearpolynomialand quadratic polynomial have smaller errors than power functions in curve fi tting.As there is no big difference of errors between linear polynomialand quadratic polynomial,the linear polynomialis chosen to fi t the degradation path as it has fewer parameters.The formation ofε(t)is de fi ned as ε(t)=at,where a is an unknown parameter.Substitute ε(t)with at in(9)and(14),the complementary reliability function and the log-likelihood function are then expressed as

3.6 Reliability estimation

Maximizing the log-likelihood function as(26),the estimates of the parameters are obtained.A pattern search method is used to solve the non-linear optimization problems.The pattern search method is a kind of directsearch method,which does not need to calculate or approximate derivatives and does not enforce a notion of suf fi cientdecrease while still guarantee global convergence[21].The initialvalues ofthe pattern search method are fi rstsetas the uniformly distributed values in a bounded feasible ranges of each parameter,and then,they are reduced to smaller ranges orchanged to anotherbounded feasible range based on the previous optimization results.The intervaldistance between the uniformly distributed values is big atthe start, and changes to smaller values step by step untilan acceptable interval distance is achieved.Estimates of modelparameters are listed in Table 2.Time points are divided by 1 000 in the calculations to fi t the tolerance of the pattern search method.

Table 2 Estimates of parameters

Complementary reliability estimates under stress levels 6.7 V,6.5 V and 6.3 V are calculated by substituting the estimates of parameters and stress levels in(25).The Kaplan-Meier estimator is used to evaluate the reliability underthese three low stress levels based on the failure time data under the corresponding stress level for comparison. The estimation results are shown in Fig.11-Fig.13.The mean squared error(MSE)is used to indicate the differences between the estimated reliability under stress levels 6.7 V,6.5 V and 6.3 V using the APDHO modelbased on degradation data under stress levels 8 V,7.5 V and 7 V, and calculate the reliability by the Kaplan-Meier estimator based on failure time data under stress levels 6.7 V,6.5 V and 6.3 V.The MSE results are given in Table 3.

Fig.11 Reliability estimates under stress level6.7 V

Fig.12 Reliability estimates under stress level6.5 V

Fig.13 Reliability estimates under stress level6.3 V

Table 3 Estimate differences

It can be concluded from Fig.11-Fig.13 and Table 3 that reliability estimates of miniature bulbs under stress level 6.7 V using the APDHO model based on the degradation data under the three high stress levels are close to the estimates using the Kaplan-Meier estimator based on the failure time data under stress level 6.7 V.The reliability estimates using APDHOunder6.5 V and 6.3 Vbecome further from those using the Kaplan-Meier estimator.This is due to that the estimation error increases when the extrapolation span increases.In addition,limited data also lead to estimation errors.It causes estimation error using the Kaplan-Meier estimator,so thatthe estimates mightbe away from the real values,and it also leads to estimation errors when using APDHO.Considering these factors,the proposed APDHO is effective.

The estimate of the parameter c approaches one,and it could be concluded that the degradation odds function, ratherthan the degradation cumulative hazard rate function is proportionalin this case.Thus,itis more properly to use the APDO modelfor this case than the APDH model.The assumptions of both models are satis fi ed,and itis dif fi cult to choose which model to use in advance.The proposed APDHO modelmakes itpossible to letthe data choose the modelfor itself and itcould dealwith more cases than the othertwo models when c is between zero and one.

4.Conclusions

A non-parametric model called APDHO model for the ADT is proposed in this paper.As a non-parametric model, ithas path-free and distribution-free properties.Itdoes not require the assumption of either degradation path or distribution of degradation measurements.Thus,itavoids the errors caused by the faulty assumption ofdegradation path or distribution of degradation measurements.

The proposed APDHO model is established based on a link function which combines the degradation cumulative hazard rate function and the degradation odds function through a transformation parameter.Thus,the APDHO model makes the APDH model and the APDO model which are established and could be established based on the degradation cumulative hazard rate function and the degradation odds function,respectively,specialcases of it.

The link function is formed as the time and the stress have a multiplicative effecton an independentfunction of the degradation measurement.Thus,the proposed model has the assumptions thatthe ratio of either the degradation cumulative hazard rate function or the degradation odds function at differenttime points under each stress level is a constant and ratio of either the degradation cumulative hazard rate function or the degradation odds function at any time point under different stress levels is a constant. The proposed model is only applicable when both of the assumptions are satis fi ed.

The proposed model is utilized to extrapolate the reliability of miniature bulbs under three low stress levels based on the degradation data obtained from degradation tests under three high stress levels.The results show that reliability estimates under the three low stress levels using the APDHO model based on the degradation data under the three high stress levels match wellwith those using the Kaplan-Meier estimator based on failure time data under the corresponding three low stress levels.The effectiveness of the proposed modelis validated.Itis necessary to mention that based on the characteristics of this model,more than one kind ofdegradation data are required ateach time pointundereach stress level.

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Biographies

Tingting Huang was born in 1981.She is an assistant professor for the School of Reliability and Systems Engineering,Beihang University,China. She worked as a postdoctoral for the Department of Industrial Engineering,Tsinghua University in 2011.She recieved her Ph.D.from the School of Reliability and Systems Engineering,Beihang University in 2010.She recieved her M.S.degree from the Department of Industrial and Systems Engineering,Virginia Tech in 2014.She was a visiting scholar in the Department of Industrial and Systems Engineering,Rutgers University,USA in 2008.Her research interests are accelerated life testing,accelerated degradation testing and other reliability and environment testing technology.Herrecentwork is on proportionalhazards-proportionalodds modelbased accelerated degradation testing.

E-mail:htt@buaa.edu.cn

Zhizhong Li was born in 1969.He is a full professor for the Department of Industrial Engineering,Tsinghua University,China.He received his B.S.,M.S.and Ph.D.degrees in manufacturing engineering and automation from Tsinghua University in 1993,1995,and 1999,respectively.His current research areas include ergonomics issues in safetycritical systems,human error,3D anthropometry, and occupationalsafety.

E-mail:zzli@tsinghua.edu.cn

10.1109/JSEE.2015.00046

Manuscriptreceived March 04,2014.

*Corresponding author.

This work was supported by the postdoctoral funding at Tsinghua University.