Generalized degradation reliability model considering phase transition

ZHANG Ao ,WANG Zhihua ,WU Qiong ,and LIU Chengrui

1.School of Aeronautic Science and Engineering, Beihang University, Beijing 100191, China;2.Institute of Spacecraft System Engineering, China Academy of Space Technology, Beijing 100094, China;3.Beijing Institute of Control Engineering, Beijing 100094, China

Abstract: Aiming to evaluate the reliability of phase-transition degrading systems,a generalized stochastic degradation model with phase transition is constructed,and the corresponding analytical reliability function is formulated under the concept of the first hitting time.The phase-varying stochastic property and the phase-varying nonlinearity are considered simultaneously in the proposed model.To capture the phase-varying stochastic property,a Wiener process is adopted to model the non-monotonous degradation phase,while a Gamma process is utilized to model the monotonous one.In addition,the phase-varying nonlinearity is captured by different transformed time scale functions.To facilitate the practical application of the proposed model,identification of phase model type and estimation of model parameters are discussed,and the initial guesses for parameters optimization are also given.Based on the constructed model,two simulation studies are carried out to verify the analytical reliability function and analyze the influence of model misspecification.Finally,a practical case study is conducted for illustration.

Keywords: degradation,phase transition,reliability,stochastic process,nonlinearity.

1.Introduction

Degradation has been recognized as a main cause of failure for highly reliable and long-life products and systems,which accumulates over time and leads to failure once it reaches a certain failure threshold.Therefore, degradationbased reliability modeling has become an effective way in reliability engineering.Since stochastic dynamics and temporal uncertainties are common characteristics involved in degradation procedures, stochastic processes are frequently adopted in degradation modeling including Wiener process [1,2], Gamma process [3,4], and inverse Gaussian (IG) process [5,6].Wiener process has been widely used in non-monotonic degradation modeling,such as hard disk drive (HDD) magnetic heads and light emitting diode (LED) lights by Ye et al.[1], self-regulating heating cables by Whitmore et al.[7], and fatigue crack growth by Li et al.[8]; while Gamma process has been usually applied for monotonic degradation modeling,such as organic coating layer by Nicolai et al.[3] and hermetically sealed circuits by Zhang et al.[9].In addition, IG process has been seldomly utilized because there is no clear physical explanation.Wang et al.[5] and Ye et al.[10,11] tried to use the process to analyze dataset of GaAs laser devices and stress relaxation.The relative research has been well reviewed by Kang et al.[12], Si et al.[13], Ye et al.[14], Zhang et al.[15], Jardine et al.[16], Tsui et al.[17] and others.

From previous literature review, most researches about stochastic-process-based modeling usually assume that a degradation procedure can be governed by a single stochastic process.However, for many degrading systems,two-phase feature widely exists due to the change of internal mechanisms or external environments, and the degradation phase change always happens at the first arriving time when the degradation process achieves a certain threshold.For example, the light display initially experiences a rapid decrease in light intensity until the impurities are burned out completely [18].Moreover, the degradation of a liquid coupling device (LCD) can be divided into two phases: it degrades very fast before a certain vibration level and then slowly deteriorates toward the failure threshold [19].In addition, the vibration signal of a bearing shows an enhanced increasing trend when the micro fatigue crack propagates to a certain level [20].As can be seen from the above examples, phasetransition degradation process is very common in practical engineering.Therefore, it is very meaningful to model this type of degrading systems.

Limited research has focused on modeling this kind of degradation process.Wang et al.found that the LCD data cannot be modeled by a single stochastic process effectively, therefore the reliability based on change-point Gamma-Wiener process in [19] was analysed and separated-phase Gamma process in [21] where the degradation phase was separated by a phase-transition threshold.In addition, there is another type of two-phase degradation model, of which the change point is determined as a certain value or identified by parameter estimation.Most researches are based on the two-phase Wiener process [22–27].Dong et al.[28] further analyzed phase-transition degradation under different types of failure thresholds, and supplemented a new Wiener-Gamma degradation process.They concluded that the crack growths [29] and the degradation of self-healing metalized pulse capacitors [30]are appropriate to be modeled by the Wiener-Gamma process.

However, the above models still have limitations for many circumstances.On the one hand, the stochastic process type is usually unchanged with phase transition in most current two-phase models.However, the degradation monotonicity may change with phase-transition.Therefore, widely-used Wiener process would not be appropriate for monotonous phase, while Gamma process cannot describe non-monotonous phase.As we know, for a faster degradation phase (such as the first phase of LCD degradation and the second phase of crack growth), the degradation is often monotonous, while for a slower degradation phase, the degradation may recover by “self-healing”resulting in a non-monotonous degradation phase.On the other hand, degradation has been always assumed linear with time.However, the nonlinear characteristics are very common [31–33] and phase-transition degradation is no exception.For example, the vibration signals are log-linear with phase-transition, which means that both phases have nonlinear property [34].In addition, capacity degradation of lithium-ion (Li-ion) batteries is stable and linear initially.But, when the capacity degradation achieves a certain value (typically 80 %), it tends to exhibit an exponential decay [35].As we can see, both linear and nonlinear property exist simultaneously.Therefore, the phase-varying properties of temporal stochastics and nonlinearity are common characteristics among the phase-transition degradation procedure.

A generalized phase-transition degradation model and corresponding reliability function are formulated and discussed in this study.Compared with existing relative works, the phase-varying stochastic property and nonlinearity are considered simultaneously; that is, both the type of stochastic process and the time scale function may change with phase transition.To enhance the practical application significance of the model, the phase model determination and model parameter estimation are also discussed.In addition, simulation study is adopted to verify the reasonability of the reliability function and the impact of model misspecification.Finally, an LCD dataset is revisited to demonstrate the effectiveness and applicability of the generalized model.

The remainder of the paper is organized as follows:Section 2 introduces a generalized phase-transition model.Section 3 formulates a general expression of reliability function.Section 4 discusses the phase model determination, the parameter estimation and the initial guesses for parameter optimization.The reliability function verification and model misspecification analysis based on simulation are presented in Section 5.In Section 6, an LCD degradation dataset with phase-transition is revisited by a comparative study.Conclusions are given in Section 7.

2.Generalized phase-transition degradation model

A phase-transition stochastic degradation process{X(t)|t≥0} consists of two phases separated by a phasetransition threshold ω.Before it first reaches ω, the degradation process can be described by a stochastic process denoted asX1(t).After that, the degradation process is governed by another stochastic process denoted asX2(t).tωis a transition time point betweenX1(t) andX2(t).A product fails when its degradation processX(t)first reaches a predefined failure thresholdD.A typical degradation procedure is illustrated in Fig.1.

Fig.1 Illustration of phase-transition degradation procedure

Based on the above description, we further define a generalized phase-transition degradation model as

wheretωis the degradation phase transition time.It is defined as the first hitting time (FHT) when degradation valueX(t) first exceeds the phase transition threshold ω.tωcan be expressed as follows:

Therefore,X(0) is the initial degradation amount whent=0, andX(tω) is the initial value of second phase at change pointtω.It is obvious thattωis a random variable depending onX1(t).To facilitate the derivation of the reliability function in Section 3, the cumulative distribution function (CDF) and probability density function (PDF) oftωare denoted asFtω(u) andftω(u), respectively.Specific forms and parameters of them depend on the type ofX1(t), which will be disscussed in Section 3.

It is worth noticing thatX1(t) andX2(t) can involve two stochastic processes.In the current study, two commonly adopted stochastic processes with independent increments are focused, including Wiener process and Gamma process.Wiener process is applicable for nonmonotonous degradation procedures, and Gamma process is suitable for modeling monotonous degradation paths.

Besides, transformed time scales have been widely applied in stochastic degradation modeling to describe the nonlinearity of the degradation path, which is always denoted as Λ(t).The power law form, i.e., Λ(t)=tbhas been most commonly adopted.If Λ(t)=t, the stochastic process model (Wiener process and Gamma process)illustrates a direct linear mechanism.In the proposed model, transformed time scale functions can be linear or nonlinear for both phases, denoted as Λ1(t) and Λ2(t).

As stated above, a Wiener processwith a transformed time scale can be expressed as

whereB(t) is a standard Brownian motion, and µ and σ denote the drift and the diffusion coefficients, respectively.

To sum up, bothX1(t) andX2(t) can be regarded as Wiener process or Gamma process with a transformed time scale, i.e.,X1(t)=µ1Λ1(t)+σ1B(Λ1(t)) orX1(t)∼Ga(α1Λ1(t),β1) andX2(t−tω)=µ2Λ2(t−tω)+σ2B(Λ2(t−tω))orX2(t−tω)∼Ga(α2Λ2(t−tω),β2).

Based on the properties previously described, the constructed degradation modelM0encompasses the following forms that have been studied in the literature as limiting situations:

(i)X1(t) andX2(t) are different stochastic processes(such as Wiener process or Gamma process), but the time scale Λ1(t) and Λ2(t) illustrate a same function form.In this case (1) becomes a two-phase model which can consider the phase-varying stochastic property and cannot consider the phase-varying nonlinearity.Special cases have been studied, including Gamma-Wiener process [19]and Wiener-Gamma process [28].We denote this type of degradation models asM1.

(ii) BothX1(t) andX2(t) are the same stochastic processes, and Λ1(t)=Λ2(t)=t, the degradation procedure becomes a simple two-phase and linear stochastic process.Only different model parameters for the two phases are involved as distinctions.If both phases are Wiener processes, (1) becomes the widely studied Wiener-Wiener process model [23–26,36].If both phases are Gamma process, the degradation procedure is a monotonous twophase process [21].Here, this type of degradation models is denoted byM2.

In summary,M1is a type of special cases forM0without considering the phase-varying nonlinearity; whileM2is a simpler type which does not incorporate the phase-varying stochastic property.

3.Reliability function

From the concept of FHT, a failure of a degrading system occurs when the degradationX(t) first hits a predefined failure thresholdD.Therefore, the lifetimeTof the system can be defined as

The system reliability at timetis defined as the probability that the lifetime is longer thant.Therefore, it can be given by

In order to deduce the explicit form of reliability function, the two-phase characteristics of the degradation model should be fully considered.Based on the law of total probability, the reliability function can be divided into two parts, which can be expressed as follows:

As mentioned in Section 2,tωis the phase-transition time defined by (2), andis its PDF.

Then, the two items are discussed separately.Considering the first item of (7), we have

whereTω=inf{s:X2(s)≥D−ω}.

For the second item of (7),

Therefore, (7) can be rewritten as

Equation (10) illustrates a general form of reliability function for the proposed modelM0.Specific forms for each item of the reliability function depend on the forms ofX1(t) andX2(t).

where Λ=Λ(u) and Φ(·) are the CDF of standard normal distribution.

For Gamma process

where Λ=Λ(u), Γ(α,β)=is the upper incomplete Gamma function, and ψ(·)=Γ′(·)/Γ(·) is the digamma function.

As important characteristics, the percentile lifetimes are elemental in reliability and safety analysis, wherer-percentile lifetimetrmeans that the system reliability equals tor(i.e.,R(tr)=r).Specially, ifr=0.5,tr=t0.5is the median life of the system.

4.Statistical inference

4.1 Phase model determination

When modeling a phase-transition degradation procedure,suitable stochastic processes (Wiener process or Gamma process) for each phase need to be determined first.Then the parametric forms of Λ1(t) and Λ2(t) should be determined.The types of stochastic processes and parametric forms of transformed time scales can be determined based on mathematical properties, prior knowledge or degradation physics.

Basically, choice of the degradation model depends on the monotonicity of the degradation in each individual phase.If the degradation data shows a non-monotonous characteristic, Wiener process is applicable.Otherwise,Gamma process is more suitable for a monotonous degradation phase.

Furthermore, the parametric forms of Λ1(t) and Λ2(t)can be specified by fitting the average over the testing samples for each phase.Possible choices for the transformed time scales include the linear function, the power law function, and the exponential law function.

4.2 MLE for model parameters

Supposing thatmunits are tested, and the degradation measurements of all test units achieve the phase-transition threshold.Therefore, the test dataset can be separated into two parts by the threshold.Then, the model parameters for the two phases can be estimated by maximum likelihood estimation (MLE) separately.

where λi=, and Σiis variance matrix with the (p,q)th element given by

Taking the first derivatives of the log-LF with respect to µ and σ2, equating them to zero, and solving the corresponding equations, we can obtain the ML estimates of µand σ2, each being conditional on Λ.

If a phase is modeled by Gamma process given by (4),let ΘG=be the vector of all unknown parameters.The LF for unitiwith independent Gamma densities can be expressed as

Then the log-LF of ΘGcan be expressed as

4.3 Initial guesses

To obtain the optimized maximum likelihood estimates from (19) and (21), optimization algorithms have beenwidely applied (for example, the “fminsearch” function in Matlab).A reasonable starting point for unknown parameters has to be identified.

As we know, if parameters of the transformed time scale Λ are determined, (19) for Wiener process can be maximized through one-dimensional optimization [1].Besides the rough estimates α,β for Gamma process can be solved by computing the first partial derivatives of (21)with fixing Λ[38].To this end, the rough estimates of the parameters in Λ are focused in the first step, and one can determine them by minimizing the mean square error

where ν is the rough estimate of µ for Wiener process or α/β for Gamma process.

For Wiener process, the start point of Λ can be obtained.For Gamma process, one can further take the first derivative of (21) with respect to α and β, then equate these two partial derivatives to 0 respectively and solve the equations.As stated above, the starting points for maximizing the profile log-LF can be determined.

5.Simulation study

5.1 Reliability functions verification

To verify the reasonability of the analytical reliability function for the proposed method, we calculate the reliability curves for two general situations by analytical solutions and compare them with the Monte-Carlo simulation results.

The two general situations are Wiener-Gamma process and Gamma-Wiener process, and both linear and nonlinear phases are involved.Without loss of generality, the nonlinear function is preset as widely applied power-law function.Hence, the two cases can be seen as the most general circumstances for the proposed method incorporating both phase-varying temporal stochastics and phasevarying nonlinearity.

The simulation procedure corresponding to the phasetransition degradation model is given as follows:

Step 1Set the initial state for the simulation, including initial degradation stateX(0)=0, initial timet=0 and model parameters.The replication number isM=10 000 and discretized time interval is 0.1, andtj=0.1j.Fori=1, assignj=0, and simulate the degradation paths by Steps 2−3.

Step 4Calculate the empirical reliability curve for the degradation model based onMsamples of failure timeT.

The main results are summarized in Fig.2, where the analytical and simulation results are shown in solid lines and dotted lines, respectively.From Fig.2, one can see that the analytical results are consistent with the simulation results.Simulated failure times are the realizations of the failure time distribution, so when the sample size is enough, the simulated curves approach the true reliability curves.Therefore, above comparisons empirically validate the reliability function.

Fig.2 Comparative reliability curves using analytical and simulation methods

5.2 Model misspecification analysis

To show the necessity of adopting the proposed degradation model in degradation modeling, our proposed model and three relative models are adopted for comparison.A Wiener (linear)-Gamma process (nonline(ar) is focused),i.e.,X1(t)=µ1t+σ1B(t) andThe process can be seen as a general case ofM0.

Three relative models are used for comparison.First, a Wiener-Gamma process with linear time scale is focused,i.e.,X1(t)=µ1t+σ1B(t) andX2(t−tω)∼Ga(α2(t−tω),β2).The process can be described by modelM1, which is the linear case of proposed modelM0.Second, modelM2incorporatingX1(t)=µ1t+σ1B(t) andX2(t−tω)=µ2.(t−tω)+σ2B(t−tω) without considering the phase-varying stochastic property is also used for comparison.In addition, although nonlinearity has hardly been considered in multi-phase degradation modeling, it has been well studied in single-phase research.Therefore, the nonlinear Wiener process expressed by (3) with power-law time scale, i.e.,X(t)=is used for comparison further,denoted asM3.

Degradation data ofmproducts are simulated with the following parameters: µ1=1,=1,α2=4,β2=2,b2=1.5.The transition and failure thresholds are predefined as ω=10 andD=50, respectively.Each product is measured until its degradation reaches 30 with the measure frequency of ∆t=1.

First, to demonstrate the effectiveness of parameter estimation method under correct the modelM0, the biases of MSEs for the parameters are estimated based on 10 000 Monte-Carlo replications under different sample sizem=5, 10, 15, 20.For Wiener process, because µ and σ2can represent the change of expectation and variance for degradation process independently, corresponding statistics for them are given directly.But for Gamma process,because the expectation and variance are given byit can be found that α/β and α/β2represent the change rates of expectation and variance.Therefore, the biases and MSEs of α/β and α/β2are calculated to show the estimation accuracy for the mean and dispersion of Gamma process.

The results are listed in Table 1.From the table, it can be observed that the parameter estimation method can give accurate estimates of degradation features, including mean, dispersity, and nonlinearity.Whenmincreases, the performance of the estimators becomes better.Evidently,the parameter estimation method is effective.

Table 1 Biases and MSEs of the ML estimates based on M0

Second, modelsM0,M1,M2, andM3are all used to fit the simulated degradation data.For comparison, the average log-LF results, and average Akaike information criterion (AIC) values are calculated to show the fitting goodness, where AIC can be defined as AIC=−2max(log−LF)+2pandpis the number of unknown parameters in the corresponding model.In addition, the MSEs of medium lifet0.5and 90-percentile lifet0.9are calculated to compare the prediction accuracy.The comparison is based on 10 000 Monte-Carlo replications under different sample sizem= 5, 10, 15, 20.

Given that the degradation threshold isD=50, the comparison results are listed in Table 2.Clearly, under all circumstances with different sample size, it can be observed that modelM0always outperforms other models in both terms of the fitting goodness and prediction accuracy.

Table 2 Comparison results of M0, M1, M2, and M3

It is clear thatM3gives worst fitting results and unacceptable prediction biases.Even though nonlinearity is considered inM3, the omit of phase-transition characteristics leads to unreliable evaluation results.The comparison between single-phase modelM3and other two-phase models demonstrates the importance of considering phase transition.

As forM1andM2,M2assumes that the stochastic property is unchanged in the two phases, then the Gamma process for Phase 2 is misspecified as Wiener process (which is the same as the first phase).In addition, even thoughM1can correctly describe the monotonicity of different degradation phases, the phase-varying nonlinearity is ignored.However, phase-varying features are not completely considered in both two-phase models, resulting in unreliable results.

To sum up, the comparison study demonstrates the importance of considering phase transition, stochastic property and nonlinearity simultaneously.

6.Practical case study

A set of widely cited phase-transition degradation data of LCD is revisited, which was originally studied by Wang et al.[19].The vibration of LCD has an apparent characteristic of two distinctive phases divided by a phase-transition threshold approximated to 30 mm, and the failure threshold of LCD isD=40 mm.Because the first phase is monotonous while the second phase is nonmonotonous, a Gamma-Wiener process with linear time scale was adopted in [19], i.e.,M1:X1(t)∼Ga(α1t,β1) andX2(t−tω)=µ2(t−tω)+σ2B(t−tω).However, the linear assumption may ignore nonlinear property of the degradation.In the current study, a Gamma-Wiener process with power-law time scale is utilized, i.e.,M0:X1(t)∼andBesides, a Wiener-Wiener process with linear time scale corresponding toM2type and nonlinear Wiener processM3with power-law time scale are also adopted for a clear comparison.

For comparison, we summarize the estimation results of the parameters, values of log-LF and AIC in Table 3 to show model fitting goodness.We have also calculated the median lifet0.5and 90-percentile lifet0.9from the fitted models, and results are also listed in Table 3.One can see that, the estimated results ofb1andb2for the proposed modelM0can clearly show the phase-varying nonlinear characteristics.Therefore,M0outperforms other two reference modelsM1andM2in both terms of Log-LF and AIC.In addition, the fitting goodness indicators ofM1andM2are very close, which means that even though the monotonicity of different degradation phases have been correctly described, the ignorance phase-varying nonlinearity will also lead to unsatisfactory model fitting results.On the contrary, single-phase modelM3shows poorest fitting.It demonstrates the necessity to consider phase transition in degradation modeling.

Table 3 Comparisons of four degradation models with LCD data

Different life prediction results may be produced by the models, whereM0gives more conservativet0.5andt0.9than other two linear modelsM1andM2becauseb2>1(which illustrates that the degradation rate gradually increases).Moreover,t0.5andt0.9results are slightly different betweenM1andM2, even though only the first phase is modeled by different stochastic processes.In addition, because of incorrect structure ofM3, the dispersity estimate is very large, leading to quite big distance betweent0.5andt0.9.

To further verify the reasonability ofM0, graphic methods are adopted to assess the fitting goodness.First, the estimated mean degradation paths based on four models are obtained and shown in Fig.3.It can be observed that the estimated mean degradation path ofM0matches the sample average well for both phases, while the linear time scales ofM1andM2and single-phase modelM3cannot well describe the real degradation trend.

Fig.3 Estimated mean degradation paths derived by four models

Second, the Q-Q plot is considered.For the Wiener process described by (3), increment δi,jis distributed asand for the Gamma process described by(4),is approximately normally distributed with meanand varianceaccording to [5].Therefore, the Q-Q plot for both phases based on four models are obtained and depicted in Fig.4.A straighter regulation of blue sample points means a better fitting.For the first phase, the Gamma processes adopted inM0andM1are better than the Wiener process incorporated inM2, indicating that monotonous Gamma process is more reasonable.Furthermore,M0with nonlinear time scale fits better thanM1with a linear one.For the second phase,all the three models are Wiener processes, but the nonlinear one inM0is significantly better than those linear ones inM1andM2.As forM3, because the data from two different phases are analyzed as one phase, the Q-Q line is curved and does not match the red line at all.To sum up,M0is a judicious choice for modeling LCD degradation.

Fig.4 Q-Q plots of four degradation models with LCD data

The reliability curves for four models are plotted in Fig.5 to demonstrate the influence on reliability evaluation caused by model misspecification.M0is apparently more conservative thanM1andM2, because it can consider the nonlinearity characterization.This is coincident with our intuition.Meanwhile,M1andM2are only slightly different, because both are linear and involve a same stochastic property in the second phase.The reliability curve ofM1is a bit on the right of that fromM2, because the first phase ofM1is monotonous whileM2is non-monotonous.Besides,compared with other two-phase models, the reliability curve ofM3is very different from others.This is becauseM3does not catch phase-transition characteristics,resulting in improper description of the degradation process.

Fig.5 Reliability curves derived by four models

7.Conclusions

In this paper, we mainly concentrate on modeling the phase-transition degrading system and analyze its reliability considering phase-varying stochastic property and nonlinearity simultaneously.To achieve such a goal, a generalized phase-transition degradation model is constructed.Wiener process and Gamma process with different time scale functions are adopted to capture the monotonicity and nonlinearity of different degradation phases.Under the concept of FHT, a generalized form of analytical reliability function corresponding to the proposed model is derived.For real applications, the determination of phase model and the initial guesses for parameters optimization are also discussed.Finally, the reasonability and effectiveness of the proposed model are verified by a simulation study and practical case study.

For further study, the following issues can be considered: (i) Stochastic processes for different phases are not limited to Wiener process or Gamma process, other stochastic processes, such as skew Wiener process, IG process can be adopted in practical engineering; (ii) This research concentrates on modeling degrading systems with one phase transition, if the deterioration regulation changes multiple times in the whole lifetime, the degradation model can be expanded to a multi-phase one, and the corresponding reliability analysis is worth further investigating.