Remaining Useful Life Estimation Based on a ...

Proceedings of the ASME 2017 12th International Manufacturing Science and Engineering Conference MSEC2017 June 4-8, 2017, Los Angeles, CA, USA

MSEC2017-2765

REMAINING USEFUL LIFE ESTIMATION BASED ON A SEGMENTAL HIDDEN MARKOV MODEL WITH CONTINUOUS OBSERVATIONS Zhen Chen Department of Industrial Engineering & Management, Shanghai Jiao Tong University Shanghai, China Contact Author

Tangbin Xia Department of Industrial Engineering & Management, Shanghai Jiao Tong University Shanghai, China H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology Atlanta, GA, USA

Ershun Pan Department of Industrial Engineering & Management, Shanghai Jiao Tong University Shanghai, China

KEYWORDS Remaining useful life; estimation; segmental hidden Markov model; continuous observations.

INTRODUCTION Remaining useful life (RUL) estimation plays an important role in condition monitoring and maintenance strategy making of manufacturing systems. The performance of a manufacturing system is typically affected by its degradation process. There are always some quality characteristics of systems (voltage, crack length, wear, etc.), whose values degrade over time. Failures occur when their degradation paths pass a prefixed critical threshold (Chen et al. [1]). Collecting and analyzing the observations of quality characteristics can inform the engineers about the states of the system and set the ground for RUL estimation and preventive maintenance policy planning. Therefore, an accurate estimation of the RUL could provide sufficient time for the policymakers of production to schedule maintenance actions and guarantee system efficiency. The methods of RUL estimation have been widely studied. Many of them only give the estimate of RUL, but no state information about degradation states. With the extensive application of condition-based maintenance in production systems, estimation methods which not only predict the RUL but also detect the degradation states are urgently required for their potential importance. To meet the requirements above, Hidden Markov model (HMM) can be an appropriate approach. A HMM consists of finite numbers of discrete hidden states with continuous or discrete observations. HMMs have been successfully applied in

ABSTRACT In this paper, a segmental hidden Markov model (SHMM) with continuous observations, is developed to tackle the problem of remaining useful life (RUL) estimation. The proposed approach has the advantage of predicting the RUL and detecting the degradation states simultaneously. As the observation space is discretized into N segments corresponding to N hidden states, the explicit relationship between actual degradation paths and the hidden states can be depicted. The continuous observations are fitted by Gaussian, Gamma and Lognormal distribution, respectively. To select a more suitable distribution, model validation metrics are employed for evaluating the goodness-of-fit of the available models to the observed data. The unknown parameters of the SHMM can be estimated by the maximum likelihood method with the complete data. Then a recursive method is used for RUL estimation. Finally, an illustrate case is analyzed to demonstrate the accuracy and efficiency of the proposed method. The result also suggests that SHMM with observation probability distribution which is closer to the real data behavior may be more suitable for the prediction of RUL.

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temporal pattern recognition, such as speech and handwriting recognition. In recent years, the HMMs used in condition monitoring and RUL estimation has been receiving increasing attention. Wang and Wang [2] established a HMM with continuous observations for tool condition monitoring and RUL prediction. Le et al. [3] developed multi-branch HMMs for RUL estimation of systems under multiple deterioration. Wang et al. [4] proposed a HMM-based fault detection approach for a multimode process. Ghasemi et al [5] used a HMM to evaluate the reliability function and the mean RUL for equipment with unobservable states. A state in a standard HMM only corresponds to a single observation. Generally speaking, more observations mean more information and a more accurate model of degradation processes can be obtained. Training HMMs using the Baum-Welch algorithm, however is essentially a black-box approach, which does not provide explicit relationship between the degradation paths and the hidden states in the trained HMMs. Therefore, the failure state based on the critical threshold of degradation path cannot be determined directly and precisely. In consideration of the shortcomings of HMM, a segmental hidden Markov model (SHMM) may be a superior option for RUL estimation. The hidden states in SHMMs can generate a segment of observations and thereby can process more data. The relationship between actual degradation and the hidden states can be also created in SHMMs. Dong and He [6] built a segmental hidden semi-Markov model-based diagnostics and prognostics framework and methodology for pumps. Geramifard et al. [7] researched a physically segmental HMM approach for continuous tool condition monitoring including diagnostics and prognostics. However, both of them had no explicit derivations of remaining life time. In addition, the observations in these studies were considered as continuous and assumed to be subject to Gaussian distributions. As real data may be not Gaussian distributed, there are two ways to address this problem (Zhang et al. [8]). One is to employ the Gaussian Mixture Model (GMM) to describe the real data. Theoretically, GMM can fit any distribution (Greenspan et al. [9]). As the number of Gaussian distribution in the GMM increases, both the number of unknown parameters to be estimated and the corresponding computational load will also increase rapidly. The other one is to use a non-Gaussian distribution besides the Gaussian distribution to approximate the real distribution. Both methods are trying to make the model mimic the observed data better, while the latter will not increase the computational load. Sun et al. [10] developed HMMs with log-normal, Gamma and generalized extreme value distributions to solve the problem that the processes may exhibit a non-Gaussian distribution in reality for PM2.5 concentration prediction. Their results show that HMMs with non-Gaussian distributions are able to predict PM2.5 exceedance days correctly and reduces alarms dramatically. In this research, a temporal probabilistic approach, named segmental hidden Markov model (SHMM), is proposed to tackle the problem of remaining useful life estimation. The

continuous observations are fitted by Gaussian, Gamma and Lognormal distribution, respectively. Since the continuous observation space is discretized into N segments with realvalued degradation, the SHMMs can directly depict the correspondence between actual degradation and the hidden states. Furthermore, the provided correspondence can make the degradation data complete, and then the parameters of the SHMMs are estimated by the maximum likelihood method. Then, to select a more suitable distribution, model validation metrics are used as a measure for evaluating the goodness-of-fit of the available models to the observed data. The RUL defined as the remaining time to first reach the final state, is deduced recursively. The effectiveness of the proposed method will be demonstrated by a real case study of fatigue-crack-growth data from Lu and Meeker [11]. Furthermore, these data are used to illustrate the advantages and of the SHMM methods by comparing with the standard HMM. We note that segmental Hidden Markov model with continuous observations has not been applied to model the system degradation and estimate the remaining useful life. Besides, we can describe the degradation process better with the selection of the three presented distributions in this novel SHMM method. The rest of this paper is organized as follows. In Section 2, the standard hidden Markov model is introduced. Then the segmentation of continuous observation space and several distributions for observation fitting are presented. To estimate the unknown parameters of the SHMMs, the maximum likelihood method is proposed based on degradation data. Moreover, a probability-based test statistic is developed. Section 3 deduces the remaining useful life estimation. An illustrate example is provided in Section 4. Finally, some conclusions are drawn in Section 5. MODEL DESCRIPTION Unlike a standard HMM with continuous observations, the segmental HMM proposed in this research discretize the continuous observation space into N segments with real-valued degradation. Then three distributions, namely, Gaussian distribution, Gamma distribution and Log-normal distribution, are used to describe values observed from certain state. Throughout this paper, SHMMs with Gaussian, Gamma and Lognormal distribution are abbreviated as SHMM-Gaussian, SHMM-Gamma and SHMM-LN for convenience, respectively. Since the hidden state can be confirmed based on the correspondence between the actual degradation values and the states in the SHMMs, the maximum likelihood method (MLE) is adopted for parameter estimation. To select a more suitable model, model validation metrics are presented as a measure for evaluating the goodness-of-fit of the available models to the observed data. Hidden Markov Model HMM is a doubly stochastic process [9], which has a hidden layer and a set of discrete or continuous observations. Formally, a HMM can be characterized by the following elements:

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

A finite set of hidden states: S  S1 , S2 ,



aij  P  qt 1  S j qt  Si  , 1  i, j  N ,



An observation sequence, O   o1 ,

i

i

1.

, oT  ,

(3) (4)





b j  ot   P ot qt  S j , 1  j  N .

S5

D4

S4

D3

D1

S3 S2 S1

0

Time

FIGURE 1. AN ILLUSTRATIVE EXAMPLE FOR THE CORRESPONDENCE BETWEEN THE SEGMENTATION AND DISCRETIZATION VALUES OF ACTUAL DEGRADATION AND THE HIDDEN STATES IN SHMM.

(5)

As the degradation processes are often irreversible and the number of observations is more than that of possible states, the samples can only stay in the same state or go to the next degradation state at each time step and their failure rates increase as time passes. Therefore, the left-right SHMM is appropriate for degradation modeling. Then, the transition matrix A can be given by (the final state is an attraction point):

In general, more data mean more information. Hence substantial degradation data are expected to be collected in a reasonable duration. Unlike a state in a standard HMM only corresponding to a single observation, a hidden state in a segmental HMM can generate a segment of observations and thereby can process more data. To make full use of the degradation data and obtaining a precise remaining useful life estimation, the SHMM approach is more preferred. In addition, the degradation data from engineering practice acquired by the estimation are typically continuous. Therefore, using the HMM in the SHMM method with continuous observations, the observation space should be discretized into N segments for N hidden states. Segmentation and Discretization A uniform discretization method is adopted to discretize the continuous observation space into N segments with real-valued , DN 1 , in which Di is the endpoint of degradation,  D1 ,

0 0  p1 1  p1 0  p 1  p 0 2 2   (7) A N  N    0 0 p3 0 .   0   0 0 0 1  An illustration for left-right SHMM is described in Figure 2, where υi , 1  i  N ,



N

υ  T is the number of time

i 1 i

steps of observations belonging to the ith state. The initial states of all samples are assumed as S1 . This means the initial state probability distribution π  1, 0 ,

the ith segment and DN 1 is the critical threshold of failure. The initial degradation values of samples are assumed as zero. Thus, N hidden states correspond to these N ordinal degradation stages and the Nth state is failure state. As shown in Figure 1, after segmenting and discretizing the degradation paths, segments are labeled sequentially from D1 to DN 1 . The

S1 ,

S6

D5

D2

Obviously, a complete HMM requires the specifications of A, B and π . For convenience, the whole element set can be denoted by a triplet λ   A, B, π  . (6)

explicit relationships between the states

S7

D6

(2)

where T is the number of time steps of the observations, and should be infinite if the observation space is continuous. An observation probability distribution, B  b j   , where

, DN 1 are

Actual degradation path Discretized degradation value

An initial state probability distribution, π  πi  , where

π

D1 ,

thereby built. For example, if the observation of actual degradation is between Di 1 and Di , the state is specified as Si .

a  1 and qt represents the hidden state at time. j ij πi  P q1  Si  , 1  i  N ,



SHMM and the actual degradation values

(1)

where N is the number of the hidden states and can be determined by a model selection technique such as cross validation. A state transition probability distribution, A  aij  , where



, SN  ,

Degradation Path



, 0 . Different samples

may degrade with different rates and thereby arrive the following states at different times. In addition, a sample is considered to have failed once it reaches the final state S N for the first time.

, S N  in the

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p1

Initial state Observations

S1 o1 ,

, ov1

1  p1

S2 ov1 1 ,

state can be confirmed based on the correspondence between the actual degradation values and the states in this research, there is no necessary to use the conventional algorithm. Here, the maximum likelihood method (MLE) is adopted. Given the complete data which consist of information on degradation and state, the parameters of the SHMM can be directly estimated. The set of unknown parameters is λ   A, B, π  . Assume that

1

p2

1  p2

1 pN1

, ov1  v2

Failure state

SN

ov1 

 vN 1

,

, oT

Figure 2. A LEFT-RIGHT SHMM.

there are M samples available. The complete is denoted by

Continuous Observations Since the degradation data are continuous, the distributions of observation are usual specified using a parametric model family. Generally, the observations are assumed to follow a Gaussian distribution or can be characterized by a mixture of Gaussian distributions. However, some real data does not always distribute normally. To address this problem, we use two non-Gaussian distributions besides the Gaussian distribution to approximate the real observation probability here. Then a model validation metric is developed to select a suitable distribution from the three distributions for the real data. This can make the applied model mimic the real situations better. According to the descriptions above, Gaussian distribution, Gamma distribution and Log-normal distribution are used to describe values observed from certain state in this research.  Under the assumption of Gaussian distribution, the probability density function (pdf) of the observations is given by (8) P  ot qt  Si =Gau  ot ; μi , σi  ,



M

, X M λ   P  X k λ .

(14)

k 1

In Eq. (10), the joint probability distribution for the kth sample given the parameter λ , can be computed as P  X k λ   P O k , qk λ  T T . (15)  P  q1k   P qtk qtk1  P otk qtk , λ t 2









t 1

, oTk  and q k   q1k ,

where O k  o1k ,



, qTk  . Due to the

assumption that the initial states of all samples are S1 , the

 

probability P q1k  P  S1  . Considering the transition matrix in Eq. (7), the joint probability density function can be rewritten as follows N 1

P  X k λ  =  pi  i

  o  μi 2  Gau  ot ; μi , σ i  = exp   t  . (9) 2 σ i 2π  2  σ i   Under the assumption of Gamma distribution, the pdf is given by (10) P  ot qt  Si =Gam  ot ; αi , βi  ,

i 1

v k 1

T

1  pi    P  otk t 1



qtk , λ .

(16)

Accordingly, the complete-data log-likelihood for the SHMM of M samples is given by M

 λ X    ln  P  X λ   . k

(17)

k 1

On differentiation of Eq. (17) with respect to pi and equating the result to zero, we obtain M pˆ i  1  M . (18)  k 1 vik

where Gam  ot ; αi , βi  , ot  0 , is the Gamma density (Coit and Jin [13]) corresponding to the state Si  o ot αi 1 exp  t βi Γ  αi   βi

 (11) .  Under the assumption of Log-normal distribution, the pdf is given by (12) P  ot qt  Si =L  ot ; ηi , εi  , αi

Then the estimates of the initial state probabilities are given by i 1  pˆ1 ,  k πˆi  P  q1   1  pˆ1 , i  2 . (19) 0, i3  Estimation for the unknown parameters of the observation probability distribution depends on the choice of the distribution. As in the Subsection 2.3, Gaussian distribution, Gamma distribution and Log-normal distribution are used to describe the observations. Let lik , 1  i  N , denote the starting time step of the ith state. Furthermore, the relationships between lik and vik are

Where L  ot ; ηi , εi  is the Log-normal density (Siano [14]) corresponding to the state Si

L  ot ; ηi , εi =



P  X 1,

1



, in which the elements

k

SHMM can be formulated as

corresponding to the state Si

Gam  ot ; αi , βi  

k

are independent and identically distributed, and consist of the observation sequences and the hidden state sequences, i.e., O k , q k . Hence, the joint probability distribution for the

where Gau  ot ; μi , σ i  , ot  0 , is the Gaussian density



X 

  ln ot  ηi 2  exp    . (13) 2 εi ot 2π 2  εi    1

Parameter estimation The Baum-Welch algorithm is the common method for parameter estimation of HMMs. However, since the hidden

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LN model, without a term not involving ηi and εi , is given by 2 k M N li 1 1  ln otk  ηi      . (27) 2πεi ot + 3  λ X      ln 2  2  εi  k 1 i 1 t  li j    The first partial derivatives of Eq. (27) are taken with respect to ηi and εi , and then set to zero. The estimates of the unknown parameters are calculated as follows

1   i 1 v kj , i  2 j 1 . (20) l  1, i 1  When the observations are Gaussian distributed, the complete-data log-likelihood for the SHMM-Gaussian model, on ignoring an additive term not involving μi and σ i , is given by k i

M

1

N

 λ X    v k 1 i 1

k i

ln





k N li 1 1

M

2πσ i   

o

k t

 μi 

2  σi 

k 1 i 1 t  li j

2



2

. (21)

k M li 1 1

The maximum likelihood estimates of μi and σ i can be derived explicitly from equating the derivative of Eq. (21) to zero to give lik1 1

M

μˆ i 

o M

k 1

2

,

v

M

N

  o k 1 t  lik

M

 μi 

(22)

 εˆi 

2

.

v

(23)

k i

α ln  βi   ln  Γ  αi   

. (24)  ok      αi  1 ln  otk   t  βi  k 1 i 1 t  lik  On equating the derivative of Eq. (24) with respect to αi and βi to zero, we have k M li 1 1

o k 1 t  lik M

v k 1

k i



k t

,

(25)



2

   ln o 

(28)

k t

k 1 t  lik

 μˆ i 

2

M

 vik

.

(29)

 M li1 1  k    ln ot   k 1 t lik βˆi  exp   Ψ α  i  . M k   k 1 vi   

Hhannan-quinn criterion: HQC  2   ln  ln M   δ

(32)

where is the maximized value of the log-likelihood function of the estimated model, δ is the number of model parameters and M is the number of the observed sequences. When there are several potential available models, the one with the smallest AIC/BIC/HQC among these could be selected as the best fitting model.

βi

k

where Ψ 

v

,

k i

Model validation metric Since there are three models presented above, model validation metrics with exact powers would help us to select a suitable distribution for the observations of degradation processes. Three criteria on the basis of maximum likelihood, namely, the Akaike information criterion (AIC), the Bayesian information criterion (BIC) and the Hannan-quinn criterion (HQC), are the commonly employed model validation metrics. Therefore, we can use them to evaluate the fit of the candidate models. The specific forms of these criteria are expressed as follows:  Akaike information criterion: AIC =  2  2δ (30)  Bayesian information criterion: BIC  2   ln M  δ (31)

k i i k 1 i 1 k M N li 1 1

αˆi 

M

k 1

k t

k 1

 λ X    v

k 1

k t

t  lik

k 1

When the observations follow Gamma distributions, the complete-data log-likelihood for the SHMM-Gamma model, apart from a term not involving αi and βi , is given by 2

  ln o

k M li 1 1

k i

k M li 1 1

 σˆi  

ηˆi 

k t

k 1 t  lik



(26)

REMAINING USEFUL LIFE ESTIMATION

is the digamma function. Obviously, Eq.

From the discussions above, the remaining useful life (RUL) can be estimated by calculating the remaining number of time steps denoted by τ to reach the final state S N . Given all

(26) is a transcendental equation which does not have a closedform solution. Therefore, the one-step Newton-Raphson method proposed by McLachlan and Krishnan [15] can be employed here to find the approximate solutions. When the observations are described by Log-normal distribution, the complete-data log-likelihood for the SHMM-

the observations O   o1 ,

, oT  from time step 1 up to T, the

current state can be obtained, i.e. qT  Si . The corresponding flowchart for the application of the proposed method is given in Figure 3. Our task is to estimate the remaining number of time

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Due the lack of the upper bound for τ , the summation in Eq. (37) may be complex and time-consuming. Instead, approximate solutions can be found as an alternative for the RUL estimation. With the increase of τ , the value of P  RUL  τ qT  Si  would tend to zero. Therefore, we can

steps τ to first arrive the failure state S N from the current time T. Accordingly, the RUL can be expressed as (33) RUL  inf τ  0 : qT  τ  S N O, λ .

define a positive tolerance ξ that is close to zero. If

Collect degradation data

P  RUL  τu qT  Si   ξ , then τ u is selected as the upper

bound for τ in the summation of Eq. (37). In this way, the calculation becomes simple.

Modeling and discretization

NUMERICAL EXAMPLE In order to evaluate the performance of the proposed approach, a case study is carried out to train the SCHMMs and estimate the RUL. The advantages of the proposed model are demonstrated in comparison with other estimation methods.

Parameter estimation for SHMM

Obtain the current state

RUL estimation

Figure 3. THE FLOWCHART OF THE IMPLEMENTATION STEPS FOR RUL ESTIMATION. From Eq. (33), the RUL is regarded as a discrete variable and its conditional probability on the current state is Riτ  P  RUL  τ qT  Si  .  P  qT  τ  S N , qT  τ 1  S N , , qT 1  S N qT  Si  (34) Under the assumption of Eq. (7), the RUL can be recursively calculated as follows:  When qT  S N 1 , 1  pN 1 , RNτ 1   τ 1  pN 1 RN 1 ,



τ 1 τ2

.

FIGURE 4. THE CRACK-LENGTH MEASUREMENTS VERSUS TIME. We use fatigue-crack-growth data from Lu and Meeker [11]. There are 21 sample paths of degradation. We define a critical crack length of 1.6 inches to be a “failure state”. Figure 4 is a plot of the crack-length measurements versus time (in million cycles).

(35)

When qT  Si , i  N-2 ,

τ 1 0, Riτ   . (36) τ 1 τ 1  pi Ri  1  pi  Ri 1 , τ  2 Then, the RUL can be obtained by deriving the expectation RUL   P  RUL  τ qT  Si  P  qT  Si O, λ   τ . (37) τ

TABLE 1. THE AIC OF THE THREE MODELS UNDER DIFFERENT NUMBERS OF STATES. SHMM-Gaussian SHMM-Gamma SHMM-LN N=3 -198.9 -168.5 410.8 N=4 -215.7 30646.6 304.8 N=5 -199.7 60547.4 239.6 N=6 -183.9 104444.1 192.6 N=7 -168.7 133741.4 164.9 N=8 -160.6 152998.4 150.3 N=9 -148.9 173307.7 131.7

i

For the calculation of P  qT  Si O, λ  , the forward variable  t  i  and the backward variable θt  i  should be defined. These two variables can be computed using the forward-backward algorithm [12]. Hence, P  qT  Si O, λ  can be computed as follows: P  qT  Si O, λ   t  i   θT  i  .

(38)

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Before the RUL estimation, we should select a suitable model among SHMM-Gaussian, SHMM-Gamma and SHMMLN first. Table 1 shows the values of AIC of the three models under different numbers of states. Obviously, the values of AIC of SHMM-Gaussian are always the smallest. Thus, the SHMMGaussian is the best model for the given data. The number of hidden states of the SHMMs can also affect the generalization of the models. Hence, after selecting the SHMM-Gaussian model, the value of N should be fixed. There is no appropriate formula to the determination of the number of states. Here, we adopt cross-validation methods for the optimization. The root mean square error (RMSE) between the estimate of RUL of SHMM-Gaussian and the real RUL is used as the criterion. M

  RUL

SHMM

 RULreal 

2

. (39) M Figure 5 shows a plot of RMSE versus number of states. It can be seen that the RMSE is sensitive to the number of states, achieving a local minimum at N  7 from this process. Then the unknown parameters of the SHMM-Gaussian can be estimated using the proposed MLE method. In order to demonstrate the accuracy of the estimation intuitively, performance of the SHMM is compared with the standard HMM method in the RUL estimation. Both the observation probability distributions of SHMM and HMM here are Gaussian distribution. At each time step, the estimates of RUL are calculated based on the observations from 1 to the current time step by using the estimation approach in the previous section. Figure 6 depicts the estimates of RUL of SHMM and standard HMM along with the actual one. As it can be seen from Figure 5, the SHMM outperforms the standard HMM, which indicates that the establishment of the correspondence between actual degradation and the hidden states can make the modeling more effective and precise. RMSE 

k 1

FIGURE 6. THE ESTIMATE OF RUL OF SHMM AND HMM, ALONG WITH THE TRUE RUL.

FIGURE 7. THE RELATIVE ERROR OF RUL ESTIMATION OF SHMM AND HMM. Moreover, the relative errors of the estimation and the actual RUL from the SHMM and HMM are shown in Figure 7. The contrast of these two curves in Figure 7 can imply that the SHMM has the ability to process more data. When the number of cycles is small, which means only few data can be collected, there is no big difference between the SHMM and HMM in the estimation. However, when the number of cycles increase to a certain amount, the relative error of HMM is rising rapidly and the relative error of SHMM still keep steady. Since more data signify more information, the characteristics of degradation can be better caught by the SHMM and thereby a more accurate estimate of RUL can be obtained. CONCLUSIONS This study proposed a segmental hidden Markov modelbased approach to handle the problem of remaining useful life estimation. The observations were fitted by Gaussian, Gamma

FIGURE 5. ILLUSTRATION OF THE CROSS-VALIDATION METHOD FOR DECIDING ON NUMBER OF STATES

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and Lognormal distribution, respectively. Then, model validation metrics are used as a measure for evaluating the goodness-of-fit of the available models to select a more suitable distribution for the given data, Moreover, the correspondence between actual degradation and the hidden states were built. The parameters of the SHMMs were estimated by the maximum likelihood method with the complete data. The RUL defined as the remaining time to first reach the final state were deduced recursively. The numerical example showed accuracy of the SHMM model through comparison and analysis. The ability of processing data was also reflected. There are some possible extensions to further enrich this research. The multidimensional observations are more common in practice. Thus, multivariate distributions should be considered into the model. Additionally, the estimates of RUL that we can get are only are based on the time steps. However, many failures may occur between two sequential steps. A higher accuracy method of the RUL estimation based on the SHMM is worth further investigating.

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ACKNOWLEDGMENTS The authors would like to thank anonymous referees for their remarkable comments. This research is supported by National Natural Science Foundation of China (51475289 and 51505288), and China Postdoctoral Science Foundation funded project (2014M561465). REFERENCES [1] Chen Z, Xia T, Pan E. Optimal multi-level classification and preventive maintenance policy for highly-reliable products. International Journal of Production Research. 2016, Online, DOI: 10.1080/00207543.2016.1232497. [2] Wang M, Wang J. CHMM for tool condition monitoring and remaining useful life prediction. The International Journal of Advanced Manufacturing Technology, 2012, 59(58): 463-471. [3] Le T T, Chatelain F, Bérenguer C. Multi-branch hidden Markov models for remaining useful life estimation of systems under multiple deterioration modes. Proceedings of the Institution of Mechanical Engineers, Part O: Journal of Risk and Reliability, 2016: 1748006X15624584. [4] Wang F, Tan S, Yang Y, Shi H. Hidden Markov ModelBased Fault Detection Approach for a Multimode Process. Industrial & Engineering Chemistry Research, 2016, 55(16): 4613-4621. [5] Ghasemi A, Yacout S, Ouali M S. Evaluating the reliability function and the mean residual life for equipment with unobservable states. IEEE Transactions on Reliability, 2010, 59(1): 45-54. [6] Dong M, He D. A segmental hidden semi-Markov model (HSMM)-based diagnostics and prognostics framework and methodology. Mechanical systems and signal processing, 2007, 21(5): 2248-2266. [7] Geramifard O, Xu J X, Zhou J H, Li X. A physically segmented hidden Markov model approach for continuous

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