SUN Fuqiang ,GUO Hongxuan ,and LIU Jingcheng

1.School of Reliability and Systems Engineering,Beihang University,Beijing 100191,China;2.Science and Technology on Reliability and Environmental Engineering Laboratory,Beihang University,Beijing 100191,China;3.Wuhan Maritime Communication Research Institute,Wuhan 430079,China

Abstract:Fiber optical gyroscope (FOG) is a highly reliable navigation element,and the degradation trajectories of its two accuracy indexes are monotonic and non-monotonic respectively.In this paper,a flexible accelerated degradation testing (ADT) model is used for analyzing the bivariate dependent degradation process of FOG.The time-varying copulas are employed to consider the dynamic dependency structure between two marginal degradation processes as the Wiener process and the inverse Gaussian process.The statistical inference is implemented by utilizing an inference function for the margins (IFM) approach.It is demonstrated that the proposed method is powerful in modeling the joint distribution with various margins.

Keywords:fiber optical gyroscope,bivariate accelerated degradation testing,Wiener process,inverse Gaussian process,time-varying copulas,dependence.

1.Introduction

Degradation is fundamental to all things in nature [1],e.g.,the fatigue and corrosion of metal materials,the wear of mechanical parts,and the parametric drift of semiconductor devices.As a result,degradation testing,especially the accelerated degradation testing (ADT),has been intensively applied to collect information about product performance over time for highly reliable products [2].And then,the lifetime and reliability characteristics of products can be evaluated through analysis of the degradation data and the degradation-threshold failure mechanism [3].In this paper,we study the degradation process of fibre optical gyroscopes (FOGs),where providing extremely precise rotational rate information is the product’s performance requirement.

The FOG is a critical navigational element that senses acceleration and angular motion,providing data outputs for altitude control and guidance using the Sagnac effect[4].FOGs have been widely used in spacecraft,satellites,vehicles,and other fields for their advantages such as high performance,small volume,light weight,and low power consumption [5].With the development of the FOG,how to estimate its lifetime and reliability becomes an urgent problem.However,due to its high reliability and long lifetime,it is unacceptable in terms of time and cost to adopt the traditional failure-time data analysis and testing methods for reliability analysis of the FOG.Generally,the accuracy stability of FOGs will decrease slowly in the long-term operation process due to the irreversible loss of components.In particular,temperature has a great influence on the light circuit part of the FOG,which may cause the photoelectric characteristic parameters of optical elements,such as the optical power,the spectral characteristic parameter,and the electro-optic action coefficient,to change with time.The variation of photoelectric characteristics will affect the output indexes of FOGs,such as the bias stability and the scale factor,and then lead to the degradation of FOGs’ accuracy stability.Therefore,the ADT of FOGs is conducted to investigate the effects of temperature on FOG lifetime.

The essence of the ADT is to find a suitable mathematical model to describe the degradation paths of the specimens tested at different levels,which is called the degradation model.The existing degradation models can be categorized into two broad classes,namely stochastic process models and general path models [3].The stochastic process models have been comprehensively studied and applied because of their time-dependent structures.Typical stochastic models include the Wiener process,the Gamma process,and the inverse Gaussian process.The Gamma process and the inverse Gaussian process are suitable for modeling a degradation process which is always positive and strictly increasing,while the Wiener process is not necessarily limited to the monotonic degradation paths,which makes it more convenient and popular for degradation modeling.Ye et al3] extensively reviewed the related work and variants for the cases involving covariates,random effects,and measurement errors for the Wiener process.Zhang et al6] systematically reviewed the recent development of Wiener process-based methods for degradation data analysis and remaining useful life (RUL)estimation.In many engineering applications,nonlinearity is quite natural due to the complexity of the structures and failure mechanisms of the products.Nonetheless,the linear drift Wiener process cannot be directly used to describe a nonlinear degradation process.To overcome this obstacle,some transformation methods for degradation data have been used,e.g.,time-scale transformation [7−9]and log-transformation [10−12].

For the monotonic degradation scenarios,the Gamma process has received many applications,such as in wear and fatigue crack damage.However,the Gamma process would fail to fit some degradation data in practice [13].Another alternative is the inverse Gaussian process.An important advantage of the inverse Gaussian process over the Gamma process is the closed-form of its first-time passage distribution [14].Furthermore,this process is flexible for incorporating random effects and covariates in the degradation model [15].The inverse Gaussian process was first introduced as a flexible degradation model in [13].Ye et al15] extended the method in [13] by letting the drift parameter [13] and the volatility parameter be random variables respectively to incorporate random effects in the inverse Gaussian process.Peng et al16]proposed a Bayesian inverse Gaussian process to model the degradation processes with time-varying degradation rates.Similarly,Zhou et al17] used the inverse Gaussian process to model the stochastic growth of corrosion defects for the pressurized pipelines.Li et al18] adopted the inverse Gaussian process to implement Bayesian optimum design of the step stress accelerated degradation testing.Liu et al19] presented a Bayesian model averaging method to combine the inverse Gaussian process and the Gamma process for monotonic degradation modeling.

It is also worth pointing out that the previous studies only considered the cases involving one single degradation measure.However,multiple accuracy indexes of the FOG,including bias stability and scale factor,would degrade slowly during operation,and any one of the indexes that reaches its corresponding threshold may cause the product to fail.Since all accuracy indexes of a FOG share several common factors (e.g.,inside structures,the use experience,environmental/operational conditions,and maintenance history),it is unavoidable that they have dependence among multiple degradation parameters.Hence,s-independent assumptions cannot match the engineering practice.Chao et al20] and Mao et al21]adopted the covariance matrix to illustrate the dependence among multiple accuracy indexes of FOGs and then separately used multivariate normal distribution and Wiener process to evaluate the storage reliability of FOGs.However,the covariance matrix method may not be suitable for all conditions because there is a latent premise that all the margins should be normally distributed [22].

Copulas provide an alternative way to model the multiple dependent degradation processes by combining marginal distributions using a joint dependence structure.Sari et al23] used a copula function to describe the correlation between two performance characteristics of the light emitting diode and combined it with a generalized linear regression model for bivariate degradation modeling.Similarly,Pan et al24],Peng et al25],Rodriguez-Picon et al14],and Fang et al26] also utilized the copula method to model the dependency between bivariate degradation measures.Sun et al27] proposed a news-dependent nonlinear ADT model for products with multiple performance parameters using the general Wiener process and copulas.Previous studies are limited to using copulas with constant parameters,which are not appropriate on many occasions because the dependence among degradation processes is usually time-varying.Patton [28] proposed several time-varying copulas and utilized the evolution equations to describe the time-varying characteristics of dependent parameters.Wang et al29] developed ans-dependent competing risk model for systems subject to multiple degradation processes and random shocks using time-varying copulas. Similarly,Sun et al30] and Pan et al31] used the bivariate timevarying copula model to conduct degradation modeling and reliability evaluation of a microwave assembly and a nitrile butadiene rubber O-rings,respectively.However,to the best of our knowledge,the existing bivariate dependent degradation models are all based on the same marginal degradation model,which assumes that the two degradation measures have the same degradation geometric laws.No one has addressed the bivariate dependent degradation scenario with different marginal geometric laws.

To estimate the lifetime and the reliability of FOGs,a high-temperature constant-stress ADT (CSADT) with three stress levels is conducted and the nonlinear accelerated degradation data of its two performance parameters are obtained,including the bias stability and the scale factor.However,the two performance parameters have different degradation patterns,that is to say,the degradation path of the scale factor is strictly and monotonically increasing,whereas the bias stability is non-monotonic and fluctuating with time.Therefore,a novel bivariate time-varying dependent ADT model considering the different degradation characteristics is proposed in this study.Compared with existing works,this paper mainly contributes to the following aspects:

(i) The proposed method employs the time-varying copulas to depict the dynamic dependence structure between two degradation measures with different geometric laws.Here,one degradation measure is monotonic and the other is a non-monotonic degradation path.The model is more general and flexible.

(ii) A real application on the accelerated degradation data of a FOG is provided to illustrate the performance and benefits of the proposed model and the statistical estimation method.The results verify the rationality and effectiveness of the proposed method.

The rest of the paper is organized as follows.In Section 2,the failure mechanisms and the accelerated degradation testing of the FOG are described.Section 3 elaborates on the bivariate dependent ADT modeling method,including the univariate ADT models based on the Wiener process and the inverse Gaussian process,the bivariate time-varying dependent ADT model,and statistical inference.In Section 3,a bivariate dependent ADT model of the Wiener process and the inverse Gaussian process is utilized considering time-varying copula functions.The ADT data modeling and reliability prediction of the FOG are given in Section 4.In Section 5,the conclusions are provided.

2.Degradation mechanism and accelerated degradation testing of FOG

2.1 Structure and principle of FOG

FOG is a kind of optical gyroscope that senses the orientation change and measures the angular velocity for the principle of the Sagnac effect [4],and it usually consists of two parts:the electro-circuit part and the light circuit part.The internal structure of the currently widely used interferometric closed-loop FOG is shown inFig.1.The light circuit part is the key factor that affects the reliability of the FOG,including a light source,a coupler,a Y wave-guide (integrated optical component),a fiber coil,and a detector.The electro-circuit part is composed of a light source driving circuit and a signal disposing circuit.

Fig.1 Configuration of a FOG

The operating principle of the FOG is shown inFig.2.When the FOG rotates relative to the inertial space,the light emitted by the light source passes through the coupler and the integrated optical component to form two beams of light transmitted in opposite directions and enters the fiber coil.Due to the Sagnac effect,if the fiber coil rotates around the center,the two beams of light propagating in the opposite direction will produce a phase difference:

Fig.2 Operating principle of the FOG

whereLis the length of the optical fiber,Ddenotes the diameter of the fiber coil,is the average wavelength of a light source,c refers to the speed of light in vacuum,andΩis the angular velocity.

The phase difference ΔΦRwould be transferred into the intensity change of the output light signal through the interference between two beams of light returned from the fiber coil.This light intensity change would be detected by a photo-detector and converted into a change in the electrical signal. Further,the signal disposing circuit could measure the electrical signal to determine the phase difference,which means the angular velocity-sensitive to the FOG can be determined. Therefore,the real-time measurement of the rotational movement of the FOG carrier is achieved.

2.2 Effect of temperature on FOG accuracy indexes

The accuracy indexes of a FOG usually include bias stability,scale factor,etc [4].

(i) Bias stability

The output of the FOG with the input angular velocity of zero is called zero bias.In this scenario,the long-term stable output is a stationary random process.The bias stability is the random error of zero bias,which can be expressed by the equivalent input angular velocity corresponding to the standard deviation of the FOG’s output at the fixed sampling period and the sampling interval.The bias stability is a comprehensive index related to the noise,zero drift,output quantization errors,etc.

(ii) Scale factor

The scale factor is the ratio of the FOG’s output change to the corresponding input change.

The temperature has a great influence on the precision maintaining ability of a fiber optic gyroscope.These effects are mainly caused by the temperature acting on the light circuit part of the FOG.

i) For the light source,long-term exposure to high temperature will result in a decrease in the signal-to-noise ratio of the FOG signal.And it would also cause the physical change of the fiber materials,which in turn leads to a change in the spectral width of the light source.Both reasons may lead to the degradation of bias stability.At the same time,the long-term effect of high temperature may cause the average wavelength of the light source to change,which will make the scale factor stability deteriorate.

ii) For the detector,its response speed will decrease under the long-term exposure to high temperature,and the coupling displacement inside will cause the signal-tonoise ratio of the FOG signal to decrease,which may make the bias stability deteriorate.

iii) For the Y-waveguide integrated optics component and the coupler,the long-term effects of high temperature will cause the signal-to-noise ratio to decrease and the extinction ratio to change,which would result in the change of bias stability of the FOG.Meanwhile,the light spectral loss caused by high temperature will deteriorate the scale factor stability of the FOG.

iv) For the fiber coil,the long-term exposure to high temperature will lead to the decrement of the FOG’s signal-to-noise ratio,non-reciprocity errors,and the physical and chemical changes in the cured adhesive.These factors may cause stress variations in the distributed optical path,which would lead to the degradation of bias stability.Meanwhile,the spectral loss of the fiber coil,and the physical and chemical changes in the cured adhesive may also cause the changes in the diameter and the length of the fiber coil,which may affect the stability of the scale factor.

In summary,the optical components may produce loss deterioration,material changes,physical or chemical changes,etc.under the long-term effects of high temperature,which may lead to the degradation of bias stability and scale factor stability.

2.3 CSADT of FOGs

Previous degradation mechanism analysis shows that high temperature is the key factor leading to the degradation of FOG accuracy indexes.Due to its high reliability and long lifetime,the degradation processes of these accuracy indexes are usually slow and long-term.As mentioned earlier,ADT attempts to obtain a product’s degradation information more quickly by exposing the test units to the harsher-than-normal conditions.

To quickly evaluate the lifetime and reliability of the FOG,a high-temperature CSADT is conducted as shown inFig.3.The test specimens are divided into three groups by the CSADT,and each group’s testing is conducted under a certain accelerated temperature stress level until the testing ends.The bias stability and the scale factor of each FOG specimen in the CSADT are measured,and the sampling interval Δtis 480 h.The censoring time at each accelerated level is 3840 h.The other settings of the test are shown inTable 1,whereS1,S2,andS3are the accelerated stress levels,and the normal operating conditionS0is 25 °C.The accelerated degradation data of all FOG accuracy indexes obtained by the CSADT are shown inFig.4.

Fig.3 CSADT of FOGs

Fig.4 Accuracy indexes of FOGs

Table 1 Testing parameters of FOGs’ CSADT

During the measurement process,no FOGs are damaged.Thus,we suppose every datum of the degradation tests is valid.As can be seen fromFig.4,the degradation paths of the bias stability and the scale factor are both nonlinear.The difference of the two accuracy indexes’ degradation processes is that the degradation of the scale factor is a strictly increasing monotonic process,while the degradation process of the bias stability has no such characteristic.Therefore,different degradation models should be utilized to better trace the degradation processes of the two accuracy indexes.Furthermore,in order to achieve more accurate and practical evaluation results of the FOG’s CSADT,it would be more appropriate to consider the time-varying dependence between these two degradation processes.

3.Bivariate dependent ADT modeling

The stress-acceleration model and the degradation model are the major issues in the ADT modeling,where the stress-acceleration model describes the relationship between the degradation rate and the stress levels,and the degradation model expresses how the degradation process evolves. In this section,the nonlinear univariate CSADT models of FOG accuracy indexes based on the Wiener process and the inverse Gaussian process are introduced respectively.

3.1 Assumptions

To analyze the bivariate CSADT data of the FOGs,the following assumptions are made:

(i) The FOG specimens in the CSADT are independent.

(ii) The bias stability and the scale factor of all FOGs are measured at the same time at each accelerated stress level.

(iii) A FOG is considered to fail if one of the accuracy indexes reaches its corresponding failure threshold for the first time.

(iv) The degradation processes of bias stability can be depicted by the Wiener process,and the inverse Gaussian process is employed to describe the degradation of the scale factor.

(v) The degradation processes of the bias stability and the scale factor are not independent,and the dynamic dependency between them can be characterized by a timevarying copula function.

3.2 Univariate ADT model

3.2.1 Stress-acceleration model

To determine a suitable stress-acceleration model is a premise for using the degradation data under the accelerated stress levels to assess the FOG’s reliability at the operational conditions.Usually,a stress-acceleration model can be obtained from either physical knowledge about the product or empirical observations [32].The Arrhenius model is often used for describing the phenomenon that the degradation rate increases with high temperature.Hence,the following Arrhenius model can be used for FOG’s temperature ADT modeling:

whereSis the absolute temperature;k=8.617 1×10−5eV/°C is the Boltzmann constant;Eais the activation energy with unit eV;andAis a constant.

For convenience,the linear normalization method is usually applied to standardize the stress levels.Let the standardized function of thejth stress levelSjbe φ(S j),which can be written as follows [33]:

whereS0

where α0=lnA−Ea/kS0,and α1=Ea/k·(1/S0−1/SH).

Thus,the degradation rate under the normal stress level can be expressed ash(S0)=exp(α0).

3.2.2 Wiener process ADT model

The Wiener process is a widely used method to describe the non-monotone increasing degradation paths [3].A well-adopted expression of the Wiener process {X(t),t≥0} for degradation modeling is given as follows:

whereX(t) is the degradation value at timet,and for simplicity,we assumeX(0)=0;μis the drift coefficient reflecting the degradation rate;σis the diffusion coefficient,and is often considered as a constant under various stress levels in the ADT analysis;B(·) is the standard Brownian motion;andΛ(t) is a time-scale transformation function depicting the nonlinearity of degradation processes.It is usually assumed that Λ(t)=tr,whererdenotes the time transformation coefficient and it is positive,andΛ(0)=0[8].

The Wiener process has independent increments,which means that the degradation increment Δx(t) follows the normal distribution with meanμΔΛ(t) and varianceσ2ΔΛ(t),where ΔΛ(t)=Λ(t+Δt)−Λ(t)=(t+Δt)r−tr,and Δtis a non-overlapped interval.Then the PDF of Δx(t) can be expressed as

When employing the Wiener process for ADT modeling,the drift coefficientμis often regarded as the function of stress levels.Combined with the stress-acceleration model (4),we have

wherea0anda1are the parameters of the stress-acceleration model under the Wiener process,φ(S) is the standardized stress function.Thus,the degradation rate under the normal stress level can be expressed asμ(S0)=exp(a0).

It is well-known that the degradation failure distribution can be calculated based on its first passage time(FPT),i.e.,the time when the degradation pathX(t) exceeds the failure thresholdω1for the first time.Folks et al34] proved that the FPT of the Wiener process follows an inverse Gaussian distribution.Hence,the probability density function (PDF) of the FPT distribution under the Wiener process [35] can be written as

The associated reliability function of the product using the Wiener process model under normal operational conditionS0can be expressed as

Based on the independent increments property of the Wiener process,the unknown parametersz1=(σ,r,a0,a1)of the Wiener process ADT model (5) and (7) could be estimated using the maximum likelihood estimate (MLE)method as follows.

Considering the case under CSADT,the observations,X(tpij) of theith specimen under thepth accelerated stress level at the corresponding timetpijare obtained,wherep=1,2,···,P,i=1,2,···,Np,j=0,1,2,···,mp.Pdenotes the number of accelerated stress levels in CSADT,mpandNpare the number of measurements and the sample size under thepth accelerated stress level respectively,andNis the total number of the specimens in CSADT,i.e.,ΣNp=N.Thus,the increment of the degradation can be expressed as Δx(tpij)=X(tpij)−X(tpi(j−1)),whereX(tpi0) is the observation value of the initial value.From the Wiener process,we know that Δx(tpij) follows the normal distribution,that is to say,∆x(tpij)∼N(µ∆Λ(tpij),σ2∆Λ(tpi j)),where ∆Λ(tpij)=Then,according to (6) and(7),the full log-likelihood function of CSADT can be expressed as

wherez1=(σ,r,a0,a1) is the unknown parameters vector,σis the diffusion coefficient of the Wiener process,rdenotes the time transformation coefficient,anda0anda1are the stress-acceleration model parameters under the Wiener process.

The log-likelihood function can be expressed as

Then,the unknown parametersz1=(σ,r,a0,a1) of the Wiener process ADT model can be estimated by maximizing the log-likelihood function.

3.2.3 Inverse Gaussian process ADT model

The inverse Gaussian process can be utilized for modeling the strictly increasing degradation processes. If a stochastic process {Y(t),(t>0)} has the following properties,we call it the inverse Gaussian process [36]:

(i)Y(0)=0 with probability 1;

(ii)Y(t) has independent increments,that is,Y(t2) −Y(t1) andY(t4) −Y(t3) are independent for 0≤t1

(iii) The increments follow an inverse Gaussian distribution,i.e.,Δy(t)=Y(t+Δt)−Y(t)～IG(νΔΩ(t),λ(ΔΩ(t))2),whereν>0 is the drift coefficient,λ>0 is the diffusion coefficient;Ω(t) is the time-scale transformation function which represents the nonlinearity of the degradation process.ΔΩ(t)=Ω(t+Δt) −Ω(t),Ω(0)=0.

Suppose thatΩ(t)=tγ,γdenotes the time transformation coefficient and it is positive,then ΔΩ(t)=(t+Δt)γ−tγ[37].

When employing the inverse Gaussian process for ADT analysis,the diffusion coefficientλis usually assumed to be a constant for all the stress levels,that is,λ1=λ2=···=λp.And,the drift coefficientνdenotes the rate of degradation and is often regarded as the function of stress levels [15].According to the stress-acceleration model(4),we have

whereb0andb1are the parameters of the stress-acceleration model under the inverse Gaussian process,and φ(S)is the standardized stress function.Similarly,the degradation rate under the normal stress level can be expressed asv(S0)=exp(b0).

Since the degradation increments follow an inverse Gaussian distribution with meanνΔΩ(t) and varianceν3ΔΩ(t)/λ,i.e.,Δy(t)～IG(νΔΩ(t),λ(ΔΩ(t)2),the PDF of Δy(t) can be defined by

Similarly,a product is assumed to fail when inverse Gaussian processY(t) firstly reaches a given thresholdω2.The cumulative distribution function (CDF) of FPT for inverse Gaussian processY(t) is

Then,the reliability function of the product derived from the inverse Gaussian process can be expressed as

The unknown parametersz2=(λ,γ,b0,b1) of the inverse Gaussian process ADT model could be also estimated by using the MLE method,wherez2is the unknown parameters vector,λis the diffusion coefficient of the inverse Gaussian process,γdenotes the time transformation coefficient,b0andb1are the stress-acceleration model parameters under the inverse Gaussian process.

Suppose the CSADT observations,Y(tpij) of theith specimen under thepth accelerated stress level at the corresponding timetpijare obtained,wherep=1,2,···,P,i=1,2,···,Np,j=0,1,2,···,mp.Thus,the increments of the degradation can be expressed as Δy(tpij)=Y(tpij)−Y(tpi(j−1)),whereY(tpi0) is the observation value of the initial value.Then,according to (13),the full log-likelihood function of CSADT can be expressed as

The log-likelihood function can be expressed as

Then,the unknown parametersz1=(σ,r,a0,a1) of the inverse Gaussian process ADT model can be estimated by maximizing the log-likelihood function.

3.3 Bivariate ADT modeling with time-varying copulas

After determining the marginal distributions of bias stability and scale factor,the time-varying copulas theory is employed for capturing the dependence structure of bivariate degradation data and identifying the joint probability distribution of the FOG.

3.3.1 Time-varying copulas

IfH(x,y) denotes the joint distribution of the marginsF(x)andG(y),then there exists a copula functionCsuch that:

Then,a bivariate copula could be defined as follows.

Definition 1A two-dimensional copula (or briefly,a copula) is a functionCfromI2toI=[0,1] with the following properties [38] :

(i) Boundary conditions:for everyxinI,C(x,0)=C(0,x)=0,andC(x,1)=C(1,x)=x.

(ii) It is 2-increasing onI2,i.e.,for everyx1,x2,y1,y2inIsuch thatx1≤x2andy1≤y2,

wheref(x) andg(y) are the PDFs of the marginal distributionF(x) andG(y) respectively.

The copulas method has been extensively applied as a statistical tool for understanding the dependence between several random variables because of its attractive properties[39].First,copulas allow us to model separately the marginal distributions and the joint dependence structure.Second,they are invariant under increasing and continuous transformations.Third,copulas do not require the marginals to be elliptically distributed,unlike correlation.

Generally,the parameter of the copula function,θ,is a constant,i.e.,the strength of dependence is assumed to be unchanged throughout the degradation process.It is not appropriate in many scenarios,because the degradation process is time-dependent and the degradation dependency is also time-variant.There are many ways of capturing possible time variation in the copula.Patton [28] assumed that the functional form of the copula remains fixed over the sample whereas the parameters vary according to some dynamic evolution equation.The following auto-regressive moving average (ARMA) process is employed to illustrate the parameterθtof a time-varying copula,namely,the dynamic evolution process is ARMA(1,10) process:

whereΘ(·) is a modified logistic transformation function to ensure thatθtis always within the domain of definition.is used to describe the dynamic dependence between the random variables.c0,c1,andc2are the parameters of the dynamic evolution equation.

Table 2lists several common time-varying copulas and their logistic transformation functions.The Akaike information criterion (AIC) could be used to compare the goodness-of-fit for the candidate time-varying copulas,

Table 2 Some common time-varying copulas

whereqdenotes the number of unknown parameters in the model;and lnLis the maximum value of the log-likelihood function.

3.3.2 Joint reliability

As mentioned earlier,the FOG has two performance parameters,i.e.,the bias stability and the scale factor.The degradation processes of bias stability and scale factor are assumed to satisfy the Wiener processX(t) and the inverse Gaussian processY(t),respectively.And the FOG failure is defined to be the event thatX(t) orY(t) firstly crosses its pre-specified thresholdω1orω2.The corresponding FPT of {X(t),Y(t)} is denoted as (T1,T2).The lifetime of the FOG isT=min(T1,T2),and the marginal reliabilities of the bias and the scale factor areRW(t) andRIG(t).Then the FOG reliability under normal conditionS0can be expressed as

If bias stability and scale factor are independent with each other,then the joint reliability can be written as

Considering the dependence between the bias stability and the scale factor,time-varying copulas can be utilized to calculate the joint distribution for the marginal distributionsFW(t) andFIG(t).

Thus,the reliability of the FOG under normal operating conditionS0can be performed as

Furthermore,Nelsen et al40] improved the Frechet-Hoeffding bounds for a bivariate copula distribution of continuous random variables with given margins and measures of association and derived the further narrowed pointwise infimum and supremum.Ifanddenote the pointwise infimum and supremum of a copula withρin [−1,1],then for any (u,v) inI2,

Hence,the best-possible bounds forC(u,v) in terms of Spearman’sρcan be expressed as

Therefore,if Spearman’sρis known,the upper and lower bounds for the FOG’s reliability (26) can be represented as

3.3.3 Parameters estimation method

As stated before,z1=(σ,r,a0,a1) andz2=(γ,τ,b0,b1) are the unknown parameters sets of two margins.Then,the joint PDF of FOG based on equations (25) and (20) can be written as

wherec(·) is the density function ofC,fWandfIGare the PDFs of the marginal distribution.Thus,the log-likelihood function can be expressed as

Then,the parameters (z1,z2,θt) can be calculated by maximizing (32).However,it is difficult to obtain the optimal solution in the high dimension scenario using the MLE method directly.Joe [41] proposed the IFM method as a computationally attractive alternative.The IFM method separately estimates the parameters of marginal distributions and the parameters of the copula in two stages.In the first stage,the marginal distribution parametersz1andz2are estimated by using MLE according to Section 3.2.2 and Section 3.2.3. In the second stage,based on the marginal distribution parameters estimated in the first stage,the time-varying copula parameter θtis computed as follows:

The time-varying copulas contain more parameters than constant copulas and the log-likelihood function may take a non-convex form.Therefore,we adopt the Matlab toolbox developed by Patton [28] to estimate the timevarying copula parametersθt.

4.Statistical inference and discussion

The proposed bivariate ADT modeling method is implemented to model and analyze the CSADT data of FOG.

First,the Wiener process and the inverse Gaussian process are used to model the univariate CSADT data of the bias stability and the scale factor,respectively.The model parameters are estimated through the MLE methods mentioned in Section 3.2.2 and Section 3.2.3,and the results are presented inTable 3.Suppose that the degradation failure thresholds of the bias stability and the scale factor areω1=0.02 andω2=0.0001,respectively.The marginal reliability under normal operational conditionS0=25 °C for two accuracy indexes of FOG are obtained,as shown inFig.5.For the case when the accuracy indexes are independent of each other,the system reliability of the FOG is also displayed inFig.5.FromFig.5,one can see that the FOG’s reliability is less than the marginal reliabilities of the individual degradation process.

Fig.5 Marginal reliability function for each accuracy index

Table 3 Estimations of the univariate CSADT model parameters

Secondly,the time-varying copulas method is utilized to describe the dependence between two marginal degradation distributions of the bias stability and the scale factor.The copula toolbox developed by Patton [28] is employed to estimate the parameters of the time-varying copulas.And the comparison between the time-varying copulas and some commonly used constant copulas is also conducted.The parameter estimation results for timevarying copulas are shown inTable 4.The parameters for the time-varying copulas and the corresponding constant copulas are illustrated inFig.6,where the solid lines indicate the dynamic evolution processes of time-varying copulas,and the dashed lines display the parameter estimation of the corresponding constant copulas.

Table 4 Parameter estimation and goodness-of-fit candidate copulas

Fig.6 Parameter estimation of candidate time-varying copulas

Then,the reliability of the FOG can be obtained by substituting the marginal distributions of the bias stability and the scale factor into (26),as shown inFig.7.To determine the best-fitted copula function,the AIC is used as the criteria.As can be seen inTable 4,the time-varying Gumbel copula has the minimal AIC value which is−1 002.12.Hence,the time-varying Gumbel copula might be the best choice from the candidates.

Fig.7 System reliability with different time-varying copulas

Finally,the reliability of FOG under the normal stress levelS0can be calculated by using the best fitting timevarying Gumbel copulas.Fig.8displays the joint reliability curves of FOG,and the system reliability under thesindependent assumption is also illustrated inFig.8.

Fig.8 Comparison of reliability functions of FOG and its accuracy indexes

FromFig.8,one can see that thes-independent reliability curve is lower than thes-dependent reliability curve.Making thes-independent assumption may under-estimate the system reliability,which further supports the conclusion that there exists a certain dependence among multiple accuracy indexes of a FOG.Therefore,it is more reasonable to use the proposed method in FOG reliability estimation.The upper and lower bounds of the FOG’s reliability,in terms of Spearman’sρ,can be obtained by using (30),as shown inFig.9.

Fig.9 Upper and lower bounds of the FOG’s reliability

5.Conclusions

Because of the multiple variations and the nonlinearity of the increments,the analysis of the degradation processes is usually difficult.To solve this problem,a time-varying copula-based reliability model is proposed for the reliability estimation of the FOG in this paper.Firstly,as the degradation paths of FOG’s bias stability and scale factor have different features,the Wiener process and the inverse Gaussian process are utilized to model the degradation paths of the two parameters separately.After that,the reliability of each parameter is gained.A time-varying copula method is chosen for depicting the dependence between the two parameters to calculate the reliability of the FOG system through the AIC method.An IFM method is then applied to find the parameters of the proposed model.Finally,a case study on CSADT data of the FOG is implemented to demonstrate the usability and validity of the proposed model.The result reveals that this method can serve as an efficient tool for system reliability evaluation under ans-dependent assumption.