Research Article Generalized Accelerated Failure Time...
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Research ArticleGeneralized Accelerated Failure Time Frailty Model for SystemsSubject to Imperfect Preventive Maintenance
Huilin Yin,1 Xiaohan Yang,2 and Rui Peng3
1 Department of Electrical & Information Engineering, CDHK, Tongji University, Shanghai 200092, China2Department of Mathematics, Tongji University, Siping Road 1239, Shanghai 200092, China3 School of Electrical & Information Engineering, Tongji University, Shanghai 200092, China
Correspondence should be addressed to Xiaohan Yang; [email protected]
Received 14 August 2014; Revised 13 October 2014; Accepted 13 October 2014
Academic Editor: Gang Li
Copyright Β© 2015 Huilin Yin et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Imperfect preventive maintenance (PM) activities are very common in industrial systems. For condition-based maintenance(CBM), it is necessary to model the failure likelihood of systems subject to imperfect PM activities. In this paper, the modelsin the field of survival analysis are introduced into CBM. Namely, the generalized accelerated failure time (AFT) frailty modelis investigated to model the failure likelihood of industrial systems. Further, on the basis of the traditional maximum likelihood(ML) estimation and expectation maximization (EM) algorithm, the hybrid ML-EM algorithm is investigated for the estimationof parameters. The hybrid iterative estimation procedure is analyzed in detail. In the evaluation experiment, the generated dataof a typical degradation model are verified to be appropriate for the real industrial processes with imperfect PM activities. Theestimates of the model parameters are calculated using the training data. Then, the performance of the model is analyzed throughthe prediction of remaining useful life (RUL) using the testing data. Finally, comparison between the results of the proposed modeland the existing model verifies the effectiveness of the generalized AFT frailty model.
1. Introduction
Condition-basedmaintenance (CBM) has continuously beenan important issue in the area of maintenance strategy. Asdegradation processes before failure of many systems canbe measured, CBM is more effective as corrective main-tenance and time-based preventive maintenance in someaspects, such as catastrophic failure reduction and availabilitymaximization [1, 2]. CBM program includes three steps:data acquisition, signal processing, andmaintenance decisionsupport [1]. Maintenance decision support is categorized intodiagnostics and prognostics. Prognostics are often character-ized by estimating the remaining useful life (RUL) of systemsusing available condition monitoring information. The RULestimation ensures enough time to perform the necessarymaintenance actions prior to failure [3, 4].
The proportional hazards model (PHM) [5, 6], whichis a popular model in survival analysis, can be appliedin reliability evaluation and maintenance optimization. It
has shown its effectiveness for RUL prediction and CBMscheduling of industrial systems [7β9]. PHM is an effectiveCBMmethod due to its strength in handling the influence ofvariable operational conditions. The accelerated failure time(AFT) model [10, 11] is an important alternative to the PHMin survival analysis and has the advantage of being moreintuitively interpretable than the PHM. However, the AFTmethod has been rarely applied in reliability-related fields.It is shown by reviewing and comparing the major math-ematical models of survival analysis and reliability theorythat both fields address the same mathematical problems [6].Because of the unified mathematical models of both fields,the methods for survival analysis might be used for reliabilityanalysis and related fields such as CBM. In this paper, weintroduce the AFT model in reliability fields and investigatethe AFT-based model for CBM.
Imperfect preventivemaintenance (PM) restores a systemto a better state but not βas good as new.β Imperfect PMactivities are considerably common in industrial systems and
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015, Article ID 908742, 8 pageshttp://dx.doi.org/10.1155/2015/908742
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there are many studies on the model of imperfect PM [12β17]. The extended PHM (EPHM) [16, 17] has been proved tobe a superiorly effective method to predict RUL of systemssubject to imperfect PM. Inspired by the EPHM, this paperinvestigates a new alternative model, the generalized AFTfrailty model.
The rest of the paper is organized as follows. Section 2introduces the generalizedAFT frailtymodel and compares itwith the existing model. Section 3 proposes the hybrid max-imum likelihood- (ML-) expectation maximization (EM)algorithm for the calculation of the parameters in the model.Section 4 proves the effectiveness of the proposed model bythe simulation experiment of RUL prediction for systemssubject to imperfect PM. Finally, Section 5 concludes thepaper.
2. Generalized AFT Frailty Model
2.1. Models. First, we introduce the Cox PHM, the AFTmodel, the generalized AFT model, and the PHM frailtymodel in sequence. Then, we propose the generalized AFTfrailty model.
The Cox PHM [5] is a popular model in survival analysis,and it is an effective method for RUL prediction and CBMscheduling of industrial systems. Let π(π‘ | X) be the hazard/failure function at time π‘ given the covariates X; the PHM isexpressed as
π (π‘ | X) = π0 (π‘) exp (π½X) , (1)
where π0(π‘) is the baseline hazard/failure function, which
considers the age of the system at the time of inspection. Xis the vector of the covariates and π½ is the regression coef-ficient vector. In a CBM program, the systemsβ degradationindicators are usually chosen to be the covariates.
The AFT model [10, 11, 18] is an important alternative tothe PHM. Suppose π
ππis the failure time of system π in cluster
(or group) π. There is a censoring random variable, which wedenote as πΆ
ππ. We only observe πβ
ππ= min(π
ππ, πΆππ) and the
linear regression (i.e., AFT) model is
log (πππ) = π½X
ππ+ πππ, (2)
whereXππis the vector of observed covariates, π½ is the regres-
sion coefficient vector, and πππis the random error.This linear
form of the AFTmodel deals with the regression relationshipof the covariates and the failure times logarithms, which hasthe similar form to the generic linear regression model. Inthe PHM, the baseline failure function and the covariates inPHM are independent, which limits the modeling of sometypes of the survival data in medical research. Collett [19]has introduced the influence of the covariates in the baselinefailure function and proposed the following generalized AFTmodel:
π (π‘ | Xππ) = exp (βπ½X
ππ) π0(π‘ exp (βπ½X
ππ)) . (3)
In this model, the whole part of the covariates exp(βπ½Xππ)
has been introduced into the baseline failure function.
The effectiveness of this model has been verified by themodeling of the survival data in medical field [19].
PHM and AFT are suitable for the mutually independentfailure time data. In reality, correlated or clustered failuretime data are very common in the fields of survival analysisand reliability. For example, systems operate in the sameenvironment with the same temperature and humidity. Theshared environment of the subjects leads to the dependenceamong the observed failure time. Frailty is a good tool torepresent the random effect shared by subjects in the samecluster (or group) and it induces dependence among thecorrelated or clustered failure time data. The PHM frailtymodel [20] is
π (π‘ | Xππ, ππ) = πππ0(π‘) exp (π½X
ππ) , (4)
where ππis the frailty term and models the random effect
that is shared by the systems in the πth cluster (group). Frailtyhas been also introduced in the AFT model to represent thepossible correlation among failure times [12]. Consideringboth the generalized AFT model (3) and the PHM frailtymodel (4), the following generalized AFT frailty model [21]is proposed:
π (π‘ | Xππ, ππ) = ππexp (βπ½X
ππ) π0(π‘ exp (βπ½X
ππ)) . (5)
2.2. Comparison between the Generalized AFT Frailty Modeland the Existing Model. The EPHM has been proposedby You et al. [16, 17] to model the failure likelihood forsystems subject to imperfect PM activities. The EPHM hasthe following form:
βπ(π‘+
π
β
π=1
ππ,X(π‘ +
π
β
π=1
ππ))
= π΄πβ0(π‘+ πππ¦π) exp(π½X(π‘ +
π
β
π=1
ππ)) ,
(6)
where βπis the hazard/failure function of the system after the
πth PM activity and before the (π + 1)th PM activity, ππis the
πth PM interval, and π‘ is the random local time between theπth and the (π + 1)th PM activity.π΄
π= π0β π1β β β ππ, where π
π
is the hazard rate increase factor (HRIF) due to the πth PMactivity, and π
πis the age reduction factor (ARF) due to the πth
PMactivity,π¦π= ππ+ππβ1ππβ1+ππβ1ππβ2ππβ2+β β β +(β
πβ1
π=1ππ)π1.
The HRIF modifies the increase rate of the hazard/failurerate of the system after an imperfect PM activity, and theARF measures the extent to which the PM activity brings thesystem to a younger age.
Comparing (5) and (6), we observe that the HRIFπ΄πand
the frailty term ππact multiplicatively on the hazard/failure
rate function and play the same role in the models. Whenthe system starts operation after imperfect PM activities, thesystem has already aged somewhat. The ARF measures theextent to which the imperfect PM affects the effective age.Thegeneralized AFT frailty model describes the hybrid influenceof imperfect PM on both the hazard/failure rate and theeffective age, which is shown in Figure 1.
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(a) After perfect PM(b) Influence on effective age
(c) Influence on failure rate(d) Hybrid influence
(a)(b)
(c)(d)
t
h(t)
First PM
Figure 1: Hybrid influence of imperfect PM on both failure rate andeffective age.
To some extent, the generalized AFT frailty model isunifiedwith the EPHM. Sowe introduce the generalizedAFTfrailty model (5) into the CBM field to model the failurelikelihood of the systems subject to imperfect PM.
3. Parameter Estimation Algorithm
After the introduction of the generalized AFT frailty model,we investigate its parameter estimation algorithm. The EMalgorithm has been a popular method for computing MLestimates from incomplete data. Due to the unknown frailtyterms, the EM algorithm has been used to estimate param-eters in the PHM frailty model [22] and AFT frailty model[12]. Vu and Kuniman [23] have proposed a hybrid ML-EMmethod for the calculation of ML estimates in PHM frailtymodels and have verified that the hybrid method is morecomputationally efficient than the standard EM method andfaster than the standard direct ML method. In this paper, weestimate the parameters in the generalized AFT frailty modelwith a hybrid ML-EM algorithm.
If the failure time is smaller than the censoring randomvariable, the indicator function has a value of one; otherwise,it has a value of zero:
πΌππ= {
1 if πππβ€ πΆππ
0 otherwise
(π = 1, 2, . . . , π; π = 1, 2, . . . , ππ) .
(7)
The likelihood function of right censoring data is
πΏ =
π
β
π=1
ππ
β
π=1
π (π‘ππ)πΌπππ (π‘ππ) , (8)
where π (β ) is the survival/reliability function.The frailty terms are random variables, andwe supposeπ
π
obeys the distribution π(ππ, π), with the parameter π, which
can be a vector. Thus there are two unknown vectors π and π½in model (5).
The parameter π is computed with the ML estimationmethod. The ML estimation is based on the marginal like-lihood function, which is written as
πΏmarginal =π
β
π=1
β«
β
0
ππ
β
π=1
πππ(π‘ππ)πΌπππ ππ(π‘ππ) π (π
π, π) ππ
π
=
π
β
π=1
β«
β
0
ππ
β
π=1
ππ
πΌππ exp (βπ½πΌππXππ) ππΌππ
0(π‘πππβπ½Xππ)
Γ exp (βππΞ0(π‘πππβπ½Xππ))
Γ π (ππ, π) ππ
π,
(9)
where Ξ0is the cumulative failure function. For the sake of
convenient calculation, we take the logarithm of the aboveequation so that the multiplications can be converted intoadditions. The logarithmic form is
πΏπ=
π
β
π=1
ππ
β
π=1
[ β π½πΌππXππ+ πΌππlog (π
0(π‘πππβπ½Xππ))
+ log(β«β
0
ππ·π
πexp (βπ
πΞπ) π (π
π, π) ππ
π)] ,
(10)
whereπ·π= βππ
π=1πΌππ, Ξπ= βππ
π=1Ξ0(π‘πππβπ½Xππ).
Given the initial values of π½ and Ξ0, we can calculate the
ML estimate of π. Given the estimate of π, we can estimateπ½ by an EM algorithm. Namely, the estimations of π and π½interact with each other as both preconditions and results. Soweupdate both parameterswith an iterative process until theyconverge.
The parameter π½ is estimated with the EM algorithm.We start the EM algorithm with the full likelihood function,which is written as
πΏ full (π,π½, π0,π) =π
β
π=1
ππ
β
π=1
πππ(π‘ππ)πΌπππ ππ(π‘ππ) π (π
π, π)
=
π
β
π=1
ππ
β
π=1
ππΌππ
πexp (βπ½πΌ
ππXππ) π0(π‘πππβπ½Xππ)πΌππ
Γ exp (βππΞ0(π‘πππβπ½Xππ))
Γ π (ππ, π) .
(11)
The logarithmic form of πΏ full is denoted as πΏπΉ and thecorresponding full log-likelihood is
πΏπΉ(π,π½, π
0,π) = πΏ
1(π,π) + πΏ
2(π½, π0,π) , (12)
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where
πΏ1(π,π) =
π
β
π=1
[
[
log (π (ππ, π)) +
ππ
β
π=1
πΌππlogππ]
]
, (13)
πΏ2(π½, π0,π) =
π
β
π=1
ππ
β
π=1
[βπ½πΌππXππ+ πΌππlog (π
0(π‘πππβπ½Xππ))
β ππΞ0(π‘πππβπ½Xππ) ] .
(14)
TheπΈ-step calculates the expectation of the full likelihoodfunction with respect to π, in order to eliminate its effect onthe likelihood function.The corresponding equations for theexpectation calculation are
πΈ (πΏ1) =
π
β
π=1
[
[
πΈ (log (π (ππ, π))) +
ππ
β
π=1
πΌπππΈ (logπ
π)]
]
, (15)
πΈ (πΏ2) =
π
β
π=1
ππ
β
π=1
[βπ½πΌππXππ+ πΌππlog (π
0(π‘πππβπ½Xππ))
βπΈ (ππ) Ξ0(π‘πππβπ½Xππ) ] .
(16)
From (15) and (16), we observe that the calculation of πΈ(πΏ1)
depends only on the parameter π and is independent of π½.The calculation of πΈ(πΏ
2) contains π½. So we estimate π½ only by
analyzing (16).The π-step maximizes πΈ(πΏ
2) and calculates the ML
estimate of π½. Due to both the unknown parts π0(π‘πππβπ½Xππ)
and Ξ0(π‘πππβπ½Xππ) in πΈ(πΏ
2), we cannot handle πΈ(πΏ
2) directly.
With kernel smoothing processes [24], we convert thesemiparametric form of (16) into the parametric form. Thelikelihood function is written as
πΈ (πΏπ
2(π½))
=
π
β
π=1
ππ
β
π=1
[πΌππlog( 1
πππππ
π
β
π=1
ππ
β
π=1
πΌπππΎ(
πππβ πππ
ππ
))
β πΌππlog( 1
πππ
π
β
π=1
ππ
β
π=1
β«
(πππβπππ)/ππ
ββ
πΈ (ππ)πΎ (π ) ππ )] ,
(17)
where πππ= log π‘
ππβπ½Xππ,πΎ(π ) is the kernel function, and π
πis
the window width of the kernel function. By maximizing thelikelihood function πΈ(πΏπ
2(π½)), we obtain the ML estimate of
π½.
Based on the estimate οΏ½ΜοΏ½, we can calculate the failure den-sity function, the reliability function, and the correspondingcumulative failure function:
π (π‘) = (ππππ‘)β1
π
β
π=1
ππ
β
π=1
πΌπππΎ(
πππ(οΏ½ΜοΏ½) β log π‘ππ
) , (18)
οΏ½ΜοΏ½ (π‘) = 1 β β«
π‘
0
π (π₯) ππ₯, (19)
ΞΜ0 (π‘)
= β log οΏ½ΜοΏ½ (π‘)
= β«
log π‘
ββ
[[
[
(πππ)β1βπ
π=1βππ
π=1πΌπππΎ((πππ(οΏ½ΜοΏ½) β π ) /π
π)
πβ1βπ
π=1βππ
π=1β«(πππ(οΏ½ΜοΏ½)βπ )/ππ
ββπΎ (π’) ππ’
]]
]
ππ .
(20)
With the calculated estimates οΏ½ΜοΏ½ and ΞΜ0, we calculate the
ML estimate π by maximizing πΏπ
in (10). Then, with theestimated value οΏ½ΜοΏ½, we follow the EM process to calculate οΏ½ΜοΏ½and ΞΜ
0again. Thus, we repeat this iterative process until the
parameters convergence.We summarize the above-analyzed process as the follow-
ing estimation procedure:
Step 1: obtain initial estimate οΏ½ΜοΏ½ from the independent AFTmodel and estimate the cumulative failure functionΞΜ0(π‘πππβπ½Xππ);
Step 2 (ML estimation): given the current estimates οΏ½ΜοΏ½ andΞΜ0, update the estimate οΏ½ΜοΏ½ bymaximizing themarginal
log-likelihood πΏπin (10);
Step 3: prepare the full log-likelihood function for the EMalgorithm;
Step 4 (πΈ-step): calculate the expectation of the full log-likelihood function with respect to π in order toeliminate its effect on the likelihood function;
Step 5 (π-step): given the current estimate οΏ½ΜοΏ½ (from Step 2),update the estimate οΏ½ΜοΏ½ by maximizing the partial log-likelihood function πΈ(πΏπ
2(π½)) in (17) and calculate the
corresponding cumulative failure function ΞΜ0with
(20);Step 6: repeat Steps 2, 3, and 4 until both parameters οΏ½ΜοΏ½ and οΏ½ΜοΏ½
converge.
This iterative ML-EM procedure is illustrated in Figure 2.
4. Evaluation Experiment
We generated the data on the basis of a typical degradationmodel to simulate the industrial processes of the systemssubject to imperfect PM activities. With the simulated datafor training, we calculated the parameter estimates of the gen-eralizedAFT frailtymodel. In order to test the effectiveness ofthe model, we calculated the RUL prediction with the testing
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Mathematical Problems in Engineering 5
Initial value
E-step
M-step
End
Convergence?No
Marginal likelihood function equation (9)
Full likelihood function equation (11)
Expectation of partial full log-likelihood function equation (15) and (16)
With kernel smoothing process handled expectation equation (17)
Full log-likelihood functionequation (12), (13) and (14)
Marginal log-likelihood functionequation (10) ML estimation
Preparation for EM process
Yes
ML-EM process
, ΞΜ0
, ΞΜ0
, ΞΜ0
equation (18), (19) and (20)Calculation ΞΜ0
ΞΜ0
οΏ½ΜοΏ½
οΏ½ΜοΏ½
οΏ½ΜοΏ½
οΏ½ΜοΏ½
οΏ½ΜοΏ½
Maximization equation (17) with respect to π½
οΏ½ΜοΏ½, οΏ½ΜοΏ½
οΏ½ΜοΏ½
οΏ½ΜοΏ½
Maximization equation (10) with respect to π
Figure 2: Parameters estimation procedure.
data and compare the results with those of the EPHM. By thecomparison, it was proved that the generalized AFT frailtymodel is effective for the systems subject to imperfect PMactivities.
4.1. Process-Data Generation. The functional form of thetypical degradation process is as follows:
π(π‘) = πΏ + ππ‘ + π (π‘) , (21)
where π(π‘) denotes the degradation indicator, and thestochastic parameters are assumed to follow the normaldistribution; namely, πΏ βΌ π(π
0, π2
0) and π βΌ π(π
1, π2
1). The
error term π(π‘) is included in themodel to capture system andenvironmental noise, signal transients, measurement errors,and variations due to monitoring system. π(π‘) is assumed tofollow a Brownian motion process π(π‘) βΌ π(0, π2π‘). Equation(21) is used to generate the degradation indicator over time,and a highermagnitude of degradation indicator correspondsto a worse system state.
On the basis of (21), the degradation process with imper-fect PM activities was simulated. The simulation mechanismand experiment design are introduced by You andMeng [17].It was demonstrated that the simulated degradation processes
were close to real data in practice. In our simulation exper-iment, we also considered the single degradation indicatorπ(π‘) as a covariate for consistency with the simulation workof You and Meng [17]. It was assumed that the PM durationwas ignored as zero, and the πth PM activities occur at timeππ. After PM, the degradation process was simulated as a new
process plus a residual degradation due to the PM activitywith a random imperfect effect. For example, after the firstPM, sample π has the following degradation process:
ππ (π‘) = ππ1ππ (π1) + (ππ1 + ππ1 (π‘ β π1) + ππ1 (π‘ β π1)) .
(22)
After the πth PM the corresponding degradation signal is
ππ (π‘) = πππππ (π1 + β β β + ππ) + ππ (π‘ β π1 β β β β β ππ)
= πππππ(π1+ β β β + π
π)
+ (πππ+ πππ(π‘ β π
1β β β β β π
π)
+ πππ(π‘ β π
1β β β β β π
π)) ,
(23)
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Table 1: Updated degradation model parameters.
Number ofPM activities π0π π1π π0π π1π ππ
π = 0 1.00 4.00 1.00 1.00 0.15π = 1 1.53 4.77 0.99 0.98 0.15π = 2 1.99 5.48 1.04 1.02 0.16π = 3 2.45 5.89 1.56 1.54 0.16
0 50 100 150 190200 250 270 300β100
0100200300400500600700
Time
Mag
nitu
de o
f the
deg
rada
tion
indi
cato
r
Figure 3: Degradation processes with a maximum of three imper-fect PM activities.
where π is the coefficient quantifying the amount of residualdegradation, which is defined as
πππβΌ π (1 β exp (β0.5π) , 0.005) ; (24)
πΏππ, πππ, and π
ππare the model parameters of sample π after
it receives its πth PM activity. The typical method for theestimate of the degradation model parameters after eachPM performs a Bayesian update, which combines the priordistribution parameters and the real-time information toobtain the posterior distribution parameters [25].
We simulate the degradation processes of 400 systems,which are subject to a maximum of three imperfect PMactivities. Some systems fail before they receive any PM. Somesystems fail after the first PM, second PM, or third PM.Some systems operate in a normal state up to the end of thesimulation time.The updated degradation model parametersof one sample are shown in Table 1.
Twenty degradation processes are illustrated in Figure 3.The horizontal axis represents time. The vertical axis repre-sents the magnitude of the degradation indicator.
The three moments of PM activities are π1= 100, π
2=
190, and π3= 270, respectively. Each section represents
the degradation process before the corresponding PM.Whenthe degradation indicator exceeds the threshold (defined asπ· = 500), the process is considered to have failed. Amongthe twenty illustrated degradation processes, four processesfail before receiving any PM activity, three processes fail afterreceiving the first PM activity, and eleven processes fail afterreceiving the second PM activity.
Next, we demonstrate the effectiveness of the simulateddata for imperfect PM. We take the local lifetime as the timeinterval between the PM moment and the time of failure ineach section. In Figure 4, π‘
0, π‘1, and π‘
2are the times of failure
in the three sections, respectively.
Time
D
0
Mag
nitu
de o
f deg
rada
tion
indi
cato
r
t0
t0E
t1
t1E
t2
t2E
T1 T2 T3
Figure 4: Schematic drawing of local lifetimes.
Table 2: Average local lifetime in each section.
Section Number offailed signalsAverage
local lifetimeBefore 1st PM 33 96.5Between 1st and 2nd PM 49 85.3Between 2nd and 3rd PM 80 72.9
The corresponding local lifetimes are π‘0πΈ= π‘0β 0, π‘1πΈ=
π‘1βπ1, and π‘
2πΈ= π‘2βπ2. Concerning all of the 400 simulated
processes, the number of failed processes and the averagelocal lifetime in each section are shown in Table 2.
From Table 2, we observe that the average local lifetimedecreases with the increases in the times of imperfect PMactivities. These simulated data are appropriate for processeswith imperfect PM activities, which restore the system tobetter but not βas good as newβ states.
4.2. RUL Prediction and Results Comparison. We used theRUL prediction, an example to test the effectiveness of thegeneralized AFT frailty model. Among the 400 simulateddegradation processes, we used 300 samples for trainingand 100 samples for testing. Parameters of the model wereestimated with training samples, and RUL predictions werecalculated with testing samples. During the parameters esti-mation procedure, we used the typicalWeibull distribution asthe baseline failure function. We chose the gamma distribu-tion with mean 1 and variance π as the frailty distribution forthe following two reasons. Firstly, the gamma distribution isthemost popular choice for the frailty distribution [12, 18, 23].The posterior distribution of the gamma frailty is still thegamma distribution. Secondly, the gamma distribution withmean 1 and variance π is appropriate for the characterizationof the frailty terms in our model. Frailty acts multiplicativelyon the failure rate function and represents the change inthe failure rate after PM relative to that before PM. Thismultiplication factor is theoretically greater than or equal to1 but sometimes smaller than 1 in reality [17].
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Mathematical Problems in Engineering 7
Table 3: Statistical analysis of the predicted RUL using G-AFT-F and EPHM for the testing samples that fail before receiving any PM activity.
Actual RUL 10 9 8 7 6 5 4 3 2 1G-AFT-F
Mean 9.93 8.96 7.90 7.06 6.05 4.86 3.80 3.07 2.09 1.24SD 0.65 0.76 0.43 0.86 0.75 0.75 0.86 0.43 0.35 0.27
EPHMMean 9.91 9.03 7.95 7.04 6.01 5.03 3.98 3.06 2.12 1.32SD 0.57 0.69 0.97 0.48 0.59 0.76 0.57 0.58 0.63 0.59
βSD = standard deviation.
0 2 4 6 8 100
1
2
3
4
5
6
7
8
9
10
11
Actual RL
Mea
n va
lues
of p
redi
cted
RL
After the first PM
G-AFT-TEPHM
(a)
0
1
2
3
4
5
6
7
8
9
10
11
Mea
n va
lues
of p
redi
cted
RL
0 2 4 6 8 10Actual RL
G-AFT-TEPHM
After the second PM
(b)
Figure 5: Statistical analysis of the predicted RUL using G-AFT-F and EPHM for the testing samples that fail after the first and second PM.
With the estimated parameters οΏ½ΜοΏ½ and οΏ½ΜοΏ½, we calculatedthe RUL prediction of the 100 testing samples followingthe typical procedure [17]. The minimum of the actuallocal lifetimes in three sections were 43.5, 39.6, and 37.3,respectively. Theoretically, all the samples whose RUL is lessthan the minimum of the actual local lifetime could be usedfor further analysis. However, because the prediction of beingclose to failure is the most essential information in a CBMprogram,we only collect the samples whose actual RUL is lessthan 10. One part of the statistical analysis of the predictedRUL based on the generalized AFT frailty model (G-AFT-Ffor short) and the EPHM with respect to the actual RUL aresummarized in Table 3.
The statistical analysis for the samples that fail afterreceiving the first PM activity and the second PM activityis shown in Figure 5. Mean values of the statistical analysisresults are regarded as predicted RUL values, so only meanvalues are illustrated.Thehorizontal axis represents the actual
RUL, and the vertical axis represents the mean values of thepredicted RUL.
From Table 3 and Figure 5 we can see that there is noobvious performance difference between the results usingboth models. The superior performance of the EPHM overother classical models has been proved [17]. Similar tothe EPHM, the generalized AFT frailty model can achievereasonably accurate and reliable RUL prediction. Thus, itis demonstrated that the generalized AFT frailty model isappropriate to model the failure likelihood of the systemssubject to imperfect PM activities.
5. Conclusions and Future Work
Imperfect PM activities are considerably common in indus-trial systems. This paper has introduced the methods ofsurvival analysis into the CBM field and used the statisticaltools to solve practical industrial problems. The generalized
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8 Mathematical Problems in Engineering
AFT frailty model has been investigated to model the failurelikelihood of systems subject to imperfect PM activities.On the basis of the ML estimation and the EM method,the hybrid ML-EM algorithm for parameter estimation hasbeen analyzed. In the evaluation experiment, the data ofthe typical degradation model has been generated. The RULprediction has been taken as a calculation case and theresults of the proposed model have been compared withthose of the existing model. The comparison demonstratesthe effectiveness of the generalized AFT frailty model.
The data in the experiment are generated from thetypical degradation processes and are consistent to practicalindustrial cases. That is why we performed the evaluationonly with simulation data. However, in future research, it willbe necessary to evaluate the performance of the proposedmodel using real data. In addition, it would be interesting tointroduce more methods of survival analysis into reliability-related fields and investigate their applicability.
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper.
Acknowledgment
This research work was supported by the National Natu-ral Science Foundation of China under the Contract nos.60903033 and 11271289.
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