A Simulation-based Remaining Useful Life Prediction Method Considering the Influence of Maintenance Activities Zhao-Qiang Wang1*, Chang-Hua Hu1, Wenbin Wang2, Xiao-Sheng Si1 1.Department of Automation, High-Tech Institute of Xi'an, Xi'an, Shaanxi 710025, P. R. China. 2.Dongling School of Economics and Management, University of Science and Technology Beijing, Beijing 100083, P. R. China. Email:
[email protected] The researches on RUL prediction have gained momentum during these years, especially along with the rapid advances of sensor technology and computer science [4-6]. Among the existing studies for RUL prediction in the literature, stochastic models have been well acknowledged as the mainstream in both industrial and academic communities [3-5]. This is because the degradation process of a practical system is typically stochastic and uncertain over time, especially when the environment under which the system operates is dynamic. As one of the most commonly used stochastic models, Wiener processes have been extensively investigated in literature over the past decades. For example, Whitmore and Schenkelberg [7] used Wiener process to model the degradation of the selfregulating heating cables, and the lifetime of a new cable was then estimated based on the inverse Gaussian (IG) distribution which was the first hitting time (FHT) distribution of Wiener process. Tseng et al. [8] also proposed a Wiener-process-based method to determine the lifetime and the burn-in parameters for contact image scanners.
Abstract—As the key of the prevalent prognostics and health management, remaining useful life prediction has attracted considerable attentions during the past decades. However, almost all of the existing remaining useful life prediction methods were implemented under the premise that the deteriorating systems were not maintained over the whole life cycle. For the deteriorating systems experiencing maintenance activities during their life profiles, this paper presents a simulation-based remaining useful life prediction method taking the influence of maintenance activities into account. Specifically, the Wiener process with jumps is employed to model the degradation path of a deteriorating system, where the jump parts are used to characterize the influence of maintenance activities on the system degradation. The parameters in the degradation model are estimated by the maximum likelihood estimation method. To acquire the remaining useful life distributions of the deteriorating system, we design a simulation-based algorithm on the basis of the Markov Chain Monte Carlo method. Accordingly, the interested statistics associated with the remaining useful life can be obtained numerically. Finally, a numerical example is provided to show the implementation of the newly proposed remaining useful life prediction method. Keywords-Remaining useful life; maintenance activities; simulation
I.
degradation
It should be noted that most of the early studies in this area dedicated to estimating the lifetime for a population of systems with the same kind and the real-time CM data of an individual system were not used. To predict the RUL of a specific system according to its real-time health condition, Gebraeel et al. [9] incorporated the real-time observed CM data into the RUL predictions based on Wiener process with a linear or exponential (can be linearized) drift via a Bayesian mechanism. Following the work in [9], numerous extensions and applications were put forward in literature, such as [10-13]. However, the Brownian motion (BM) term in the degradation model in [9-13] was treated as the error term. This led to the obtained RUL distributions therein were only approximations belonging to the Bernstein distributions, whose moments did not exist, rather than exact RUL distributions [3]. Still under a dynamic update framework, Wang et al. [14] realized the realtime updating of the exact RUL distributions of Wiener process, namely the IG distributions, by resorting to Kalman filtering method. Based on the results in [9] and [14], Si et al. [3] modeled the unit-to-unit heterogeneity in RUL predicting by incorporating the stochasticity of the drift coefficient of Wiener process into RUL predictions. More recently, Si et al. [15] presented a general degradation model based on Wiener process for RUL prediction, where three sources of variability (i.e. temporal variability, unit-to-unit variability, and
modeling;
INTRODUCTION
The past decade has witnessed the rapid development of prognostics and health management (PHM) due to its ability in enhancing system performance and reducing unexpected failures while cutting down the operational cost [1]. PHM includes the following two parts: prognostics and health management, where the former is recognized to play a dominant role since it always acts as the prerequisite or basis in PHM implementation [1-3]. Specifically, prognostics is often related to find the probability density function (PDF) or mean of a practical system’s remaining useful life (RUL) based on the observed condition monitoring (CM) data [4]. Once the interested system’ RUL is available, appropriate health management activities, such as maintenance scheduling, inventory controlling, logistic supporting, etc., can be planned in advance to guarantee the safe and reliable operation of the system.
The research reported here is partially supported by NSFC under grant numbers: 61025014, 71231001, 61174030, 61203007, and 61374126.
978-1-4799-7958-5/14/$31.00 ©2014 IEEE
284
that the maintenance activities are executed periodically with a fixed period T0 , and ⎣⎢ ⎦⎥ rounds the element to the nearest integer towards zero.
measurement variability) were simultaneously taken into account. However, all the above-mentioned Wiener-process-based degradation modeling and the according RUL prediction method did not consider the influence of maintenance activities. In other words, they all assumed that, once the maintenance activity was executed to the deteriorating system, the degradation process will be renewed for RUL prediction. To the best of the authors’ knowledge, this is a common limitation in current RUL prediction studies. The difficulty of considering the influence of maintenance activities lies in: 1) how to characterize the effects of maintenance activities on degradation modeling; 2) how to derive the RUL distributions of the deteriorating system. In view of these problems, this paper attempts to explore the degradation modeling and RUL prediction method in a simulation manner. Toward this end, the system’s degradation is modeled by a linear-drifted Wiener process with negative jumps, and the jumps are used to characterize the influence of maintenance activities. Given the complexity of the degradation model, we establish a simulated algorithm for RUL prediction by Markov Chain Monte Carlo (MCMC) method. Both the empirical distributions and the interested statistics (such as mean and standard deviation) of the RUL can be obtained accordingly.
Based on the FHT concept, the RUL Lt (with its realization lt ) of a deteriorating system at time t associated with degradation model (1) can be defined as Lt = inf{ X (t + lt ) ≥ ω | lt ≥ 0, ∀t ′ ∈ [0, t ], x(t ′) < ω} , where ω is the pre-set failure threshold level. To realize the RUL prediction defined in (2), the unknown parameters therein should be estimated at first. We will proceed to the parameter estimation procedure in the forthcoming section. III.
⎢t ⎥ ⎢ ⎥ ⎣ T0 ⎦ i =0
-related part ∑ Yi , the unknown parameters Θ cannot be estimated from x1:N directly. To facilitate the parameter estimating process, a transformation should be executed on the data in advance. Specifically, from (1), we have ⎢ t ⎥ ⎢ ⎥ T
X (t )+∑ i⎣=00 ⎦ Yi = x(0) + θ t + σ B (t ) .
DEGRADATION MODELING AND RUL PREDICTION
⎢t ⎥ ⎢ ⎥ T
(3)
Denote the left hand side of (3) as Z (t ) , we have
Let X (t ) (with its realization x(t ) ) denote the system degradation at time t , which can be characterized by
X (t ) = x(0) + θ t − ∑ i⎣=00 ⎦ Yi + σ B(t ) ,
PARAMETER ESTIMATION
The unknown parameters in degradation model (1) include Θ = ( μθ , σ θ , σ )′ . These parameters can be estimated by the maximum likelihood estimation (MLE) method. Denote the observed CM data from time t1 to time t N as x1: N = {x1 , x2 , , xN }′ . Due to the existence of the maintenance
The remainder of this paper is organized as follows. Section II describes the degradation modeling and RUL prediction problem. Section III focuses on estimating the unknown parameters. Section IV formulates a set of simulated algorithms for RUL prediction. Section V provides a numerical example for illustration, and Section VI concludes the paper with some discussions. II.
(2)
⎢ t ⎥ ⎢ ⎥ T
Z (t ) = X (t )+ ∑ i⎣=00 ⎦ Yi .
(4)
where Z (t ) (with its realization z (t ) ) can be viewed as the intermediate variable which is essentially unobservable.
(1)
where x(0) = 0 is the initial degradation at time t = 0 ;
Based on (3) and (4), we can easily obtain the following counterpart relationship:
θ ~ N ( μθ , σ θ2 ) is the drift coefficient, and the stochasticity is
used to characterize the unit-to-unit heterogeneity among different systems in a population; σ is the diffusion coefficient; B(i) is the standard BM characterizing system dynamics. We use θ t + σ B (t ) to represent the normal degradation. These specifications are common in Wienerprocess-based degradation modeling practices [3, 9-15].
Z (t ) = x(0) + θ t + σ B(t ) .
(5)
The estimate for Θ can be realized by (5), however, (4) should be used first to calculate Z (t ) on the basis of x1:N . The transformed data from (4) at each time tn ( n = 1, , N ) are denoted as z1:N = {z1 , z2 , , z N }′ , which can be viewed as a realization of the intermediate stochastic vector Z1: N = {Z (t1 ), Z (t2 ), , Z (t N )}′ . Let t1: N = {t1 , t2 , , t N }′ denote the according CM times for x1:N (or z1:N ). From (5), the following lemma can be provided.
⎢t ⎥ ⎢ ⎥ ⎣ T0 ⎦ i =0
Below let’s pay attention to the part ∑ Yi in (1), which is used to characterize the accumulated influence of maintenance activities. Yi denotes the effect of the i th maintenance activity upon degradation, and Yi is assumed to follow a known positive random distribution with Yi ~ f ( y; γ ) and Y0 = 0 . It needs to be mentioned that γ can be directly estimated based on the maintenance effect data in practice, and it is assumed to be known in this paper. We further assume
Lemma 1 [16]. The stochastic vector Z1: N = {Z (t1 ), Z (t2 ), , Z (t N )}′ associated with (5) follow a multi-variable Gaussian distribution with mean and covariance:
285
μN = μθ t1:N ,
(6)
Σ N = σθ2 t1:N t1:N′ + ΩN ,
(7)
⎡t1 ⎢t where ΩN = σ 2 ∇ N , and ∇ N = ⎢ 1 ⎢ ⎢ ⎣t1
t1
2) Otherwise, X (k Δt ) = x((k − 1)Δt ) + θΔt + σδ Δt .
Remark 1: The condition that tk = k Δt can be divisible by T0 implies that the system is maintained at time tk since the
t1 ⎤ t2 ⎥ ⎥ . ⎥ ⎥ tN ⎦ N × N
t2 t2
maintenance activities are carried out at time mT0 ( m ∈ N + ). In such a case, the system degradation at time tk should take into account the influence of the maintenance (i.e. (9)). Otherwise, the system degrades normally as shown in (10).
The proof of Lemma 1 can be found in [16]. Based on Lemma 1 and the transformation data z1: N , the log-likelihood function of Θ can be formulated as
(Θ | z1:N ) = −
N 1 1 ln(2π ) − ln ∑ N − ( z1:N − μ N )′ ∑ −N1 ( z1: N − μ N ) 2 2 2 . (8)
ˆ = ( μˆ , σˆ , σˆ )′ can be obtained by The estimators Θ θ θ maximizing (8). Up to now, the unknown parameters in degradation model (1) have been estimated. In the following, we will design two algorithms for data generation and RUL prediction, respectively. IV.
(10)
Based on (9) and (10), we will provide the two successive algorithms (i.e. Algorithm 1 and Algorithm 2) for data generation and RUL prediction, respectively. For uniformity, we denote all of the generated CM data from Algorithm 1 as x1: N = {x1 , x2 , , xN }′ . In such a case, t fht = t N . These collected data are treated as the observed CM data, which can be used to estimate the unknown parameters as shown in section III. The estimators based on x1:N are denoted as ˆ = ( μˆ , σˆ , σˆ )′ . Based on the x and Θ ˆ , the RUL can be Θ
SIMULATED ALGORITHMS FOR RUL PREDICTION
In this section, we first design a simulation algorithm to generate the data to be used, and then we design another algorithm for RUL prediction based on the MCMC method. Considering the fact that the samples (i.e. CM data) are usually observed at discrete points, we first discretize the degradation model in (1) using the Euler approximation method [17] in the following way.
θ
θ
1:N
predicted from time t fht to the FHT of a specified threshold level ω . From this, we can provide a new algorithm (i.e. Algorithm 2) for RUL prediction by employing the MCMC method.
1) If tk = k Δt ( k ∈ N + ) can be divisible by T0 , namely the remainder of k Δt T0 is equal to zero, then X (k Δt ) = x((k − 1)Δt ) + θΔt − f ( y; γ ) + σδ Δt ,
Remark 2: In (9), to guarantee X (k Δt ) ≥ 0 , which is a practical requirement, we assume that the maintenance amount f ( y; γ ) in (9) at time tk is not greater than τ X (k Δt ) with a known ratio τ ∈ (0,1) . To be specific, if the sampler from f ( y; γ ) is not larger than τ X (k Δt ) in (9), then the CM data can be sampled from Eq. (9). Otherwise, f ( y; γ ) is chosen to be equal to τ X (k Δt ) in Eq. (9), and the new CM data can then be sampled accordingly.
1 2 M Based on the simulated TRUL = {tRUL , tRUL , , tRUL } in Algorithm 2, the histogram of TRUL and the according empirical RUL distributions can be obtained. And, the interested statistics (such as mean, standard deviation) of the RUL can also be calculated.
(9)
where Δt is the discretization step, and δ ~ N (0,1) .
Algorithm 1: Data generation algorithm. step 1: Initial setting: Initializing the distribution of θ with N ( μ0,θ , σ 0,θ ) , the diffusion coefficient σ , the parameter vector γ , 2
maintenance period T0 , discretization step Δt , initial time t 0 , initial state x0 , ratio τ , and upperbound: ωstop . step 2: Sampling for θ : Sampling one θ from N ( μ0,θ , σ 0,θ ) . 2
0
+
step 3: Data generation: Based on the initial setting and sampler in step 1 and step 2, sampling xn ( n ∈ N ) from (9) and (10) at each CM time t n . step 4: Termination judgment. If xn ≥ ωstop is satisfied at time t n , then set the termination time t fht = t n and the simulation process is 0
terminated. Otherwise, set n = n + 1 , and return to step 3 for a new sampler.
Algorithm 2: RUL prediction algorithm.
286
step 1: Parameters setting: The estimated distribution of θ with N ( μˆθ , σˆ θ ) , the diffusion coefficient σˆ , the parameter vector γ , 2
maintenance period T0 , discretization step Δt , initial time t fht , initial state xin (equal to ωstop ), ratio
τ
, the number of sampling paths M ,
and threshold level: ω . step 2: Sampling for θ : Sampling one θ from N ( μˆθ , σˆ θ ) . 2
+
step 3: Data generation: Based on the parameters setting and the sampler in step 1 and step 2, sample the j th ( j = N ) CM data in the
i th ( i = 1, 2,
, M ) sampling path xi , j from (9) and (10) at each CM time ti , j = t fht + j Δt .
step 4: Termination judgment. If xi , j ≥ ω is satisfied at time ti , j , then set the RUL of the i th sampling path from the initial time t fht i
as t RUL = ti , j − t fht . Otherwise, set j = j + 1 , and return to step 3. 1
2
step 5: Repetition. Repeating step 3 and step 4 until all of the M RULs are simulated, i.e. TRUL = {t RUL , t RUL ,
V.
coefficient, respectively. As mentioned above, these parameters can be estimated from x1:N by the collaboration of (8) and (11).
EXPERIMENTAL STUDY
A. Preliminary A simulated example will be given in this section for illustration based on the above designed algorithms for data generation and RUL predictions. As a comparison, Wiener process with a linear drift without considering the influence of maintenance activities is employed as a benchmark model here, which can be represented as [3, 15, 16] X (t ) = x(0) + θ t + σ B(t ) .
B. Results and Discussions The maintenance effect f ( y; γ ) = 1 ( yσ γ 2π ) e
t
FL (lt ) = 1 − Φ( t
Φ (−
2π lt3 (σˆ 2 + σˆθ2 lt )
w − xt − μˆθ lt
σˆ 2lt + σˆθ2lt2
exp{−
) + exp{
2σˆθ2 (w − xt )lt + σˆ 2 (μˆθ lt + w − xt )
σˆ 2 σˆ 2 lt + σˆθ2 lt2
( w − xt − μˆθ lt )2 2lt (σˆ 2 + σˆθ2 lt )
2μˆθ (w − xt )
σˆ 2
+
},
σˆ 4
with γ = [ μγ , σ γ ]′ . The
simulation process is terminated at t106 = 10.6 with the FHT t fht = t N =10.6 and x106 = 2.0283 . That is, N = 106 suits of CM data (i.e. x1:106 = {x1 , x2 , , x106 }′ ) are collected excluding the initial state x0 . The simulated CM data are plotted in Fig .1. It can be observed from Fig. 1 that the degradation path declines significantly at each maintenance time (i.e. t =mT0 (m ∈ N + ) ). Based on the obtained CM data x1:N , the unknown parameters in (1) and the benchmark model (11) can be estimated by the MLE method. They are shown in Table I.
(12)
2σˆθ2 (w − xt ) 2
chosen as
σ = 0.2 , μγ = -1.7006 , σ γ = 0.4270 , T0 = 1 , Δt = 0.1 , t0 = 0 , x0 = 0 , τ = 0.3 , ωstop = 2 . Based on Algorithm 1, the
(11)
CDF of RUL Lt (with its realization lt ) at time t associated with degradation model (11) are [3, 15, 16] w − xt
distribution is
− (ln y + μγ )2 (2σ γ2 )
inputs of Algorithm 1 are set as: μ0,θ = 0.4 , σ 0,θ = 0.01 ,
Compared with degradation model (1), the difference between (1) and (11) is that the influence of maintenance activities is not modeled in the benchmark model. Note that the other assumptions associated with (1) also apply to the benchmark model (11). Of course, the parameters estimation procedure will be more simplified for (11). The data transformation step in (4) will be not needed and the parameters in the degradation model can be estimated directly from x1:N via (8). Based on the FHT concept, the PDF and
f L (lt ) =
M , tRUL }.
}
) , (13)
where Φ(i) denotes the CDF of the standard normal distribution; σˆ is the estimated diffusion coefficient, and μˆθ and σˆθ are the estimated mean and standard deviation of drift
287
and 1.1015, respectively. These results can be used to instruct the health management activities in practice.
2.5
Upperbound 2
0.45 0.4
Empirical PDF Kernel smoothing density estimate for empirical PDF The benchmark model
CM data
1.5
0.35 0.3 PDF of RUL
1
0.5
0.25 0.2 0.15
0
0.1 0
1
2
3
4
5 6 Sampling points
7
8
9
10
0.05
Fig. 1. The simulated CM data
TABLE I.
0
0
5
10
15 RUL
20
25
30
Fig. 2. Comparison results of the PDF of the RUL
THE ESTIMATED PARAMETERS IN OUR MODEL AND THE BENCHMARK MODEL 1
μˆθ
σˆθ
σˆ
Our model
0.3341
0.0291
0.2014
0.8
The benchmark model
0.1824
0.0288
0.2363
0.7 CDF of RUL
0.9
We set the threshold of this simulated path as ω = 3 . In the following, we will focus on predicting the RUL from time t fht = 10.6 to the FHT of the threshold ω = 3 . The RUL predicting results of both these two rival modeling methods are both provided for comparison. Specifically, the PDF and CDF of the RUL by the benchmark model can be calculated analytically by substituting the according parameter estimators in Table I into (12) and (13). Below, we will specify how to obtain the empirical RUL distributions by Algorithm 2, namely our modeling method.
0.6 0.5 0.4
Empirical CDF Lower confidence bound for the empirical CDF Upper confidence bound for the empirical CDF The benchmark model
0.3 0.2 0.1 0
0
5
10
15 RUL
20
25
30
Fig. 3. Comparison results of the CDF of the RUL
The inputs of Algorithm 2 are set as follows: μˆθ = 0.3341 , σˆθ = 0.0291 , σˆ = 0.2014 , t fht =10.6 , xin = 2 , M = 10000 , τ = 0.3 , ω = 3 . By invoking Algorithm 2, M suits of RULs 1 2 M (i.e. TRUL = {tRUL , tRUL , , tRUL } ) are collected. Based on TRUL , the empirical PDF and CDF of the RUL can then be obtained. Specifically, the empirical PDF of RUL can be obtained in a histogram form, as shown in Fig. 2. The kernel smoothing density estimate for the empirical PDF is also plotted in Fig.2 using a normal kernel smoother [18]. In Fig. 3, the KaplanMeier estimates [18, 19] for the CDF of TRUL with a confidence level of 95% are plotted. For comparison, the exact PDF and CDF of the RUL by the benchmark model are also plotted in Fig. 2 and Fig. 3, respectively. In addition, the statistics of the RUL can also be calculated based on TRUL , such as the mean and standard deviation of TRUL are 6.7950
Two observations can be obtained from Fig. 2 and Fig. 3. First, the RUL distributions predicted by the benchmark model cannot track the empirical distributions well. Second, the RUL predicted by the benchmark model is much smaller than that of the empirical distribution. The reason may lie in that the maintenance activities can slow the deterioration progression; however these influences are ignored in the benchmark model. From this, we can conclude that the influence of maintenance activities should be considered in RUL prediction. VI.
CONCLUSIONS
In this paper, a simulation-based RUL prediction method, which took the influence of maintenance activities for the deteriorating systems into account, was proposed. The
288
degradation of a deteriorating system was modeled by the Wiener process with negative jumps, where the influence of maintenance activities was incorporated. The parameters in the degradation model were estimated by the MLE method. Two simulation algorithms were designed for data generation and RUL prediction, respectively. Finally, a simulated example was provided as an illustration, and the empirical PDF and CDF of the RUL were calculated. Comparison works were also made with the common Wiener process model in literature. The results showed that the influence of maintenance activities should be considered in RUL prediction. One promising direction in future is to derive the analytical forms of the RUL distributions under the framework proposed in this paper.
[8]
[9]
[10]
[11]
[12]
REFERENCES [1] [2]
[3]
[4]
[5]
[6]
[7]
[13]
M. Pecht, Prognostics and health management of electronics, John Wiley, New Jersey, 2008. B. Sun, S. Zeng, R. Kang and M. G. Pecht. “Benefits and challenges of system prognostics,” IEEE Transaction on Reliability, vol. 61, no. 2, pp. 323-335, 2012. X.S. Si, W. Wang, C.H. Hu, M.Y. Chen and D.H. Zhou, “A Wienerprocess-based degradation model with a recursive filter algorithm for remaining useful life estimation,” Mechanical Systems and Signal Processing, vol. 35, no. 1, pp. 219–237, 2013. X. S. Si, W. Wang, C.H. Hu and D.H. Zhou, “Remaining useful life estimation-A review on the statistical data driven approaches,” European Journal of Operational Research, vol. 213, no. 1, pp. 1-14, 2011. J.Z. Sikorska, M. Hodkiewicz and L. Ma, “Prognostic modelling options for remaining useful life estimation by industry,” Mechanical Systems and Signal Processing, vol. 25, no. 5, pp. 1803-1836, 2011. W. Wang, “Condition-based maintenance modelling,” Complex system maintenance handbook, Springer Series in Reliability Engineering, pp. 111-131, 2008. G.A.Whitmore and F. Schenkelberg, “Modelling accelerated degradation data using Wiener diffusion with a time scale transformation,” Lifetime Data Analysis, vol. 3, no.1, pp. 27-45, 1997.
[14]
[15]
[16]
[17] [18]
[19]
289
S.T. Tseng, J. Tang and I.H. Ku, “Determination of optimal burn-in parameters and residual life for highly reliable products,” Naval Research Logistics, vol. 50, no.1, pp. 1-14, 2003. N.Z. Gebraeel, M.A. Lawley, R. Li and J.K. Ryan, “Residual-life distributions from component degradation signals: A Bayesian approach,” IIE Transactions, vol. 37, no. 6, pp. 543-557, 2005. A.H. Elwany and N.Z. Gebraeel, “Sensor-driven prognostic models for equipment replacement and spare parts inventory,” IIE Transactions, vol. 40, no. 7, pp. 629-639, 2008. N.Z. Gebraeel and J. Pan, “Prognostic degradation models for computing and updating residual life distributions in a time-varying environment,” IEEE Transactions on Reliability, vol. 57, no. 4, pp. 539550, 2008. M.Y. You and G. Meng, “A predictive maintenance scheduling framework utilizing residual life prediction information,” Proceedings of the Institution of Mechanical Engineers, Part E: Journal of Process Mechanical Engineering, vol. 227, no. 3, pp. 185-197, 2013. M.Y. You, F. Liu, W. Wang and G. Meng, “Statistically planned and individually improved predictive maintenance management for continuously monitored degrading systems,” IEEE Transactions on Reliability, vol. 59, no. 4, pp. 744-753, 2010. W. Wang, M. Carr, W. Xu and K. Kobbacy, “A model for residual life prediction based on Brownian motion with an adaptive drift,” Microelectronics Reliability, vol. 51, no. 2, pp. 285-293, 2011. X.S. Si, W. Wang, C.H. Hu and D.H. Zhou, “Estimating remaining useful life with three-source variability in degradation modeling,” IEEE Transactions on Reliability, vol. 63, no. 1, pp. 167-190, 2014. Z.Q. Wang, C.H. Hu, W. Wang and G.J. Dong, “An online Wiener process-based remaining useful life prediction method for the draught fan in steel mills,” Journal of University of Science and Technology Beijing, 2014 (in press). P. Kloeden and E. Platen, “Numerical Solution of Stochastic Differential Equations,” New York: Springer, 1995. Z.S. Ye, M. Xie and L.C. Tang, “Reliability evaluation of hard disk drive failures based on counting processes,” Reliability Engineering & System Safety, vol. 109, pp. 110-118, 2013. E.L. Kaplan and P. Meier, "Nonparametric estimation from incomplete observations," Journal of the American Statistical Association, vol. 53, no. 282, pp. 457-481, 1958.
