A degradation path-dependent approach for remaining useful life estimation with an exact and closed-form solution

Stochastics and Statistics

A degradation path-dependent approach for remaining useful life estimation with an exact and closed-form solution

Xiao-Sheng Si

^a,b

, Wenbin Wang

^c,⇑

, Mao-Yin Chen

^b

, Chang-Hua Hu

^a

, Dong-Hua Zhou

^b,⇑

aDepartment of Automation, Xi’an Institute of High-Tech, Xi’an, Shaanxi 710025, PR China

bDepartment of Automation, TNLIST, Tsinghua University, Beijing 100084, PR China

cDongling School of Economics and Management, University of Science and Technology, Beijing, PR China

a r t i c l e i n f o

Article history:

Received 26 January 2012 Accepted 22 October 2012 Available online 3 November 2012

Keywords:

Replacement Remaining useful life First passage time Expectation maximization Prognostics

a b s t r a c t

Remaining useful life (RUL) estimation is regarded as one of the most central components in prognostics and health management (PHM). Accurate RUL estimation can enable failure prevention in a more control- lable manner in that effective maintenance can be executed in appropriate time to correct impending faults. In this paper we consider the problem of estimating the RUL from observed degradation data for a general system. A degradation path-dependent approach for RUL estimation is presented through the combination of Bayesian updating and expectation maximization (EM) algorithm. The use of both Bayesian updating and EM algorithm to update the model parameters and RUL distribution at the time obtaining a newly observed data is a novel contribution of this paper, which makes the estimated RUL depend on the observed degradation data history. As two specific cases, a linear degradation model and an exponential-based degradation model are considered to illustrate the implementation of our presented approach. A major contribution under these two special cases is that our approach can obtain an exact and closed-form RUL distribution respectively, and the moment of the obtained RUL distribution from our presented approach exists. This contrasts sharply with the approximated results obtained in the literature for the same cases. To our knowledge, the RUL estimation approach presented in this paper for the two special cases is the only one that can provide an exact and closed-form RUL distribution utilizing the monitoring history. Finally, numerical examples for RUL estimation and a practical case study for condition-based replacement decision making with comparison to a previously reported approach are provided to substantiate the superiority of the proposed model.

Ó2012 Elsevier B.V. All rights reserved.

1. Introduction

Prognostics and health management (PHM) is an efficient and systematic approach for evaluating the reliability of a system in its actual operating conditions, predicting failure progression, and mitigating operating risks via management actions (Pecht, 2008). In PHM, prognostics can yield an advance warning of impending failure in a system, thereby helping in making mainte- nance decisions and executing preventive actions. The past decade has witnessed a constant research interest on various aspects of PHM due primarily to the fact that PHM has been extensively ap- plied in a variety of fields including electronics, smart grid, nuclear plant, power industry, aerospace and military application, fleet- industrial maintenance, and public health management (Smith et al., 1997; Wang and Zhang, 2005; Nikhil and Pecht, 2006; Lall et al., 2006; Wang, 2007; Mazhar et al., 2007; Tsui et al., 2011).

In each of these applications and documents, one critical quan- tity during prognostics for a system is the prognostic distance within which management decisions and repair actions can be planned effectively prior to failure occurrence to extend system life (Derman et al., 1984; Wang and Christer, 2000; Si et al., 2011). This prognostic distance is closely associated with the definition of the remaining useful life (RUL) which is the length of the time from the present to the end of useful life. In fact, RUL estimation is always a key part in any PHM program and management can make use of RUL information in condition-based maintenance (CBM) to pro- duce economic benefits in engineering, maintenance, logistics, and operations. Therefore, over the past few decades, significant advances have been made in developing RUL estimation ap- proaches (Si et al., 2011).

Stochasticity is one of the main characteristics in system oper- ations, that contributes to the uncertainty in estimating the RUL of the system. Therefore, one fundamental issue in RUL estimation is to find the probability density function (PDF) of the RUL. However this also leads to the main difficulty of RUL estimation since how to make full use of condition monitoring (CM) information to infer a

0377-2217/$ - see front matterÓ2012 Elsevier B.V. All rights reserved.

http://dx.doi.org/10.1016/j.ejor.2012.10.030

⇑ Corresponding authors. Tel.: +86 010 62794461; fax: +86 010 62786911.

E-mail addresses:wangwb@ustb.edu.cn(W. Wang),zdh@mail.tsinghua.edu.cn (D.-H. Zhou).

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RUL distribution is a not-well-solved problem. So far, RUL estima- tion has been regarded as one of the most central components in PHM (Pecht, 2008; Camci and Chinnam, 2010). Thus, our primary interest of this paper is to utilize CM information to adaptively estimate the RUL of the system and then apply it to support deci- sion-related applications.

The current RUL estimation approaches can be generally classi- fied as physics of failure, data driven and fusion. Physics of failure approaches rely on the physics of underlying failure mechanisms.

Data driven approaches achieve RUL estimation via data fitting mainly including machine learning and statistics based ap- proaches. The fusion approaches are the combination of the phys- ics of failure and data driven approaches. However, for complex or large-scale engineering systems, it is typically difficult to obtain the physical failure mechanisms in advance or cost-expensive and time-consuming to capture the physics of failure. In contrast, data-driven approaches attempt to derive models directly from collected CM and life data, and thus are more appealing and have gained much attention in recent years (Si et al., 2011).

In conventional data-based approaches, estimating the RUL is achieved by evaluating the conditional lifetime distribution given that a system has survived up to a specific time, e.g. TtjT>t, whereT denotes the lifetime (Maguluri and Zhang, 1994; Alam and Suzuki, 2009). The obtained RUL distributions from these approaches are generally based on the life characteristics of a pop- ulation of identical systems and lifetime data are required. How- ever, such data are scarce in reality or even non-existent at all for systems which are costly or time-consuming to collect the life data (Ma and Krings, 2011). With the advances in CM technologies, degradation data can be obtained from routine CM as feasible and low-cost alternatives to estimate the RUL. These data are usually correlated with the underlying physical degradation process. If they are properly modelled, degradation data can be used to predict unexpected failures and accurately estimate the lifetime of gradually degraded systems (Escobar and Meeker, 2006; Joseph and Yu, 2006). In general, degradation data based methods for RUL estimation can be classified into the models based on indirectly ob- served degradation processes and the models based on directly ob- served degradation processes (Si et al., 2011). The former models considered that the degradation state was hidden and assumed that the available CM data were stochastically related to the under- lying degradation state. In this case, lifetime data must be available to establish the relationship between the CM data and failure. The latter models utilized the observed degradation data directly to describe the underlying degradation state of the system. In this paper, we mainly focus on the directly observed degradation processes.

One common definition of RUL in the directly observed case is re- lated to the concept of the first passage time (FPT) of the degradation process crossing a pre-defined threshold level. The use of the FPT concept as the definition of failure or a terminating event has a long history of application in diverse fields, including medicine, environ- mental science, engineering, business, economics and sociology (Whitmore, 1986; Lee et al., 2004; Lee and Whitmore, 2006; Balka et al., 2009). It is also acknowledged as a mainstream definition of failure in reliability literature based on degradation data (Singpur- walla, 1995; Park and Padgett, 2006; Aalen et al., 2008; Peng and Tseng, 2009; Pandey et al., 2009; Park and Bae, 2010; Li and Ryan, 2011). Thus, in this paper, we pay particular attention to a type of degradation-data-based models and derive the RUL distribution based on the concept of the FPT. Since degradation data are part of CM data, throughout this paper, we use terms ‘CM information’

and ‘degradation data’ inter-changeably.

In most of degradation-data-based models for RUL estimation, an exact and closed-form of the RUL distribution in the FPT sense is only available for some special cases. Frequently, a stepwise

approximation or numerical simulation has to be used for finding an approximated RUL (Yuan and Pandey, 2009; Park and Bae, 2010; Wang and Zhang, 2005, 2008; Xu et al., 2008; Carr and Wang, 2011). In addition, most of these models either do not use the in situ degradation data during lifetime inference or only use information contained at the current observation point. However, the degradation data over the path collected up to date could con- tain more useful information to make the RUL estimation more accurate.

The type of models we specifically consider in this work follows the idea inGebraeel et al. (2005)where two exponential-like deg- radation models were proposed. In their models, stochastic parameters were updated via a Bayesian approach to incorporate real-time CM information. FollowingGebraeel et al. (2005), many variants and applications have been reported in prognostics, main- tenance and inventory management (Li and Ryan, 2011; You et al., 2010; Gebraeel, 2006; Elwany and Gebraeel, 2008, 2009). However, in these papers, they estimated the RUL distribution as the distri- bution of the time that it takes the trajectory of the degradation signal to cross the failure threshold based on an approximated method. In reality, this is not the FPT since the signal may have already crossed the failure threshold, signifying failure, prior to predicting the RUL. In extreme cases where the degradation fluctu- ations are large, this approximation could be significantly crude from the FPT concept. Even when the Brownian motion (BM) used as an error term, the availability of the explicit distribution of the FPT from the BM with a drift, i.e. the inverse Gaussian distribution, was not utilized in their models.Elwany and Gebraeel (2009)used the FPT to approximate the mean RUL but the distribution of the RUL was still evaluated by their approximate approach. As such, the results of RUL estimation inGebraeel et al. (2005)and the fol- lowed works in applications are approximations as opposed to the FPT concept. Furthermore, in above works, the obtained RUL distributions belong to a family of Bernstein distributions. Conse- quently, the moments of the RUL do not exist. But in maintenance practice, the expectation of the RUL is required to be existent sometimes (Derman et al., 1984; Shechter et al., 2008). Also, the stochastic coefficients inGebraeel et al. (2005)and other following works had some prior distributions but no elaborated method is presented to select the hyperparameters of the prior distributions.

Typically, several systems’ historical degradation data of the same type are required to determine the deterministic coefficient and the unknown parameters in the prior distributions of the stochas- tic coefficients. But the scarcity of such historical degradation data of multiple systems is a commonly encountered case in practice, particularly for newly armed systems. As shown in Section5of this paper, an inappropriate selection of these parameters can result in an incorrect estimate of the RUL.

Driven by the above survey over the related works, the purpose of this paper is to develop a degradation path-dependent approach for RUL estimation that allows the estimated RUL distribution to be dependent on a system’s degradation data history and to be adap- tively updated, at the moment that a newly observed data is avail- able. In particular, our goal is to shed light on three fundamental issues: (i) RUL estimation for an individual fielded device without the need of offline data of other similar systems, (ii) parameter estimation/updating of the degradation model from the observed degradation data, and (iii) an exact yet closed-form expression of the RUL distribution given (i) and (ii).

In response to the above issues, the dependency of RUL estima- tion with a system’s past degradation path is presented through the combination of Bayesian updating and expectation maximiza- tion (EM) algorithm. This is a novel contribution of the paper and is not fully explored in the conventional RUL modeling paradigms. As such, the deterministic coefficient and the unknown hyperparam- eters in the prior distributions of the stochastic coefficients can be

updated when the new degradation observation is available. Two specific cases of our general approach, a linear model and an expo- nential-based model which were considered by Gebraeel et al.

(2005), are used to illustrate the implementation of our presented approach. A major contribution of this paper under these two special cases is that our approach can obtain an exact yet closed- form RUL distribution respectively, and we show that the moment of the obtained RUL distribution from our approach exists. This contrasts sharply with the approximated results obtained in the literature for the same cases. To our knowledge, the RUL estimation approach presented in this paper for the two special cases isthe only onethat can provide an exact but closed-form RUL distribution utilizing the monitoring history.

Furthermore, in our approach the parameter updates in each iteration of the EM algorithm have explicit formulas. This implies that each iteration of the EM algorithm can be performed with a single computation, which leads to an extremely fast and simple estimation procedure. This computation advantage plus the exact yet closed-form RUL distribution are particularly attractive for practical applications.

We have performed extensive numerical studies to substantiate the superiority of the proposed approach in comparison with previously reported models. Since the presented method allows real-time updating the RUL distribution as new observations from CM are available, such updating mechanism enables the estimates less sensitive to the selection of parameters in the prior distribu- tions, i.e. our estimation method is robust with prior or initial parameters, as revealed by the experimental results. This is an- other important character since it can make the engineering imple- mentation rather reliable. We also provide a practical case study to test the performance of the developed approach in condition-based replacement decision making. The use of our estimated RUL in CBM decision making allows us to generate new insights on the ef- fect of the estimated RUL from CM data upon decision making and to explore in more detail of how estimated parameters influence the RUL estimation and further the replacement decision.

The remainder parts are organized as follows. Section 2 first constructs a general stochastic process-based degradation model and then presents a degradation path-dependent approach for adaptive RUL estimation via real-time CM data. Sections3 and 4 consider a linear model and an exponential model for illustrating the working mechanism of the proposed approach, respectively.

Section5provides several simulations and a case study to illustrate the application and usefulness of the developed approach. Section 6concludes the paper. All proofs associated with this paper are provided in theSupplementary materialdue to the limited space.

2. A degradation path-dependent approach for adaptive RUL estimation

In the following, a general parametric degradation model is developed first and then we present a degradation path-dependent approach that utilizes online CM sensory information to adaptively compute RUL distribution.

2.1. A general description of stochastic process-based degradation models

As discussed previously, a degradation process is stochastic in nature due to inherent randomness in manufacturing and opera- tions. Therefore, it is natural to model a degradation process as a stochastic process (Singpurwalla, 1995; Aalen et al., 2008; Pandey et al., 2009). In this paper, the degradation model is represented as a stochastic process {X(t), tP0} where X(t) is the degradation signal att. As mentioned earlier, a degradation signal is a charac-

teristic pattern from the sensory information that captures the physical transitions associated with the degradation process. Some examples of degradation signals have been extensively illustrated in the literature, such asMeeker and Escobar (1998) and Elwany et al. (2011).

Usually, a degradation model consists of deterministic and sto- chastic parts. The deterministic part represents a constant physical phenomenon common to all systems of a given population. While the stochastic part captures the variation of the degradation pro- cess of an individual system, particularly represented by a proba- bility distribution. The stochastic part of the noise and random effects associated with the degradation signals are usually repre- sented by a random term

e

(t), which is modelled as a stochastic process in this paper.

With the above considerations in place, and without loss of generality, we assume that the degradationX(t) at timetcan be represented by the following general expression:

XðtÞ ¼hðt;

e

ðtÞ;h;

u

Þ; ð1Þ

whereX(t) is driven by a functionh() with stochastic process

e

(t), characterizing the dynamics/uncertainty of the degradation process withhand

u

as the parameters. The functional formh() depends on the type of the system under consideration and represents a rela- tionship between the operating time and the degradation signal.

This functional form may follow a linear, polynomial, exponential, or any other trend. Considering that each system possibly experi- ences different sources of variations during its operation, for a deg- radation model to be realistic, we treathas a random-effect vector representing unit-to-unit variability, and

u

as a fixed effect vector that is common to all systems. For simplicity, we assume that h and

e

(t) are s-independent. The ideas of random effects and the independent assumption betweenhand

e

(t) have been widely used in degradation modeling literature (seeMeeker and Escobar, 1998;

Gebraeel et al., 2005; Peng and Tseng, 2009; Park and Bae, 2010).

2.2. A degradation path-dependent approach for adaptive RUL estimation via real-time CM data

We have established the degradation model using a general stochastic process. We now illustrate how to estimate the RUL based on the established model. DefineX_1:k= {x₁,x₂,. . .,x_k} as the observed degradation at CM timest1,t2,. . .,tk, which could be irreg- ularly spaced. It is noted that the degradation modeling paradigm for RUL estimation in most of conventional models is based on one assumption that the estimated PDF of RUL depends only on the currently observed degradation data,xk. As such, it is highly de- sired to construct a model which can be conditional on all the data up totk, that is,X1:k. Consequently, using the FPT of the degrada- tion process {X(t),tP0} crossing the thresholdwand conditional on the observation historyX1:k, we define RULLkat timetkas:

Lk¼infflk:XðlkþtkÞPwjX1:kg ð2Þ with PDFfLkjX1:kðlkjX1:kÞand cumulative distribution function (CDF) F_LkjX1:kðl_kjX_1:kÞ. It can be observed from Eq.(2)that the defined RUL is degradation path-dependent.

Now we need to focus on how to estimatefLkjX1:kðlkjX1:kÞin an adaptive way, namely, when the newly observed data is available, the PDF of the RUL can be updated in order to make the estimated RUL depend onX1:k. In order to computefLkjX1:kðlkjX1:kÞ, considering the stochastic nature ofh, we can formulatefLkjX1:kðlkjX1:kÞby the law of total probability as follows:

fLkjX1:kðlkjX1:kÞ ¼ Z

fLkjh;X1:kðlkjh;X1:kÞpðhjX1:kÞdh: ð3Þ From Eq.(3),fLkjh;X1:kðlkjh;X1:kÞandp(hjX1:k) must be known, and the unknown parameter

u

is needed to be estimated fromX1:k, for the

sake of calculating fLkjX1:kðlkjX1:kÞ. There are four steps which are shown below to accomplish the task.

2.2.1. Step 1. Determine prior information forh

As for the stochastic parameter vector,h, it can be specified as a prior distribution asp(h) in a Bayesian framework. The prior distri- bution ofp(h) contains hyperparametera. Once the observed data and the sampling distributionp(X1:kjh) are available, the posterior distribution ofh,p(hjX1:k), can be computed by the Bayesian rule.

During the course of selecting the prior distribution, p(h), one convenient way is to makep(h) belong to the conjugate family of the sampling distributions,p(X1:kjh), which can lead to a tractable posterior distribution ofh. In our illustration cases, we use such method to make the posterior distribution ofhtractable.

2.2.2. Step 2. Update the posterior distribution forh

Once a new observation attkis available, the posterior distribu- tion ofhcan be updated via the Bayesian rule as follows:

pðhjX1:kÞ ¼pðX1:kjhÞ pðhÞ

pðX1:kÞ /pðX1:kjhÞ pðhÞ: ð4Þ

If analyticalp(hjX1:k) is not available, Gibbs sampling or Metrop- olis–Hastings algorithm can be used to simulate all distributions (Gelfand and Simth, 1990; Tierney, 1994). In this paper, we do not consider the general case involving such as Gibbs sampling, but construct a conjugate prior distribution for the special cases considered in this paper instead.

From Eq.(4)and using the chain rule in probability, we have

pðhjX1:kÞ ¼pðX1:k;hÞ

pðX1:kÞ ¼pðxkjX1:k1;hÞ pðhjX1:k1ÞpðX1:k1Þ pðX1:kÞ

¼pðxkjX1:k1;hÞ pðhjX1:k1Þ pðxkjX1:k1Þ

/pðxkjX1:k1;hÞ pðhjX1:k1Þ: ð5Þ Eq.(5)shows the recursive relationship between the prior attk1

and the newly observed information att_k.

2.2.3. Step 3. Estimate the unknown parameters via the EM algorithm Now we return to estimate unknown parameters

u

in Eq.(1) and parametersain prior distributionp(h). For simplicity, we de- note unknown parameter vector consisting of

u

andaasH= [

u

, a]. In order to estimateH, we calculate the maximum likelihood estimation (MLE) ofHonce new observationxkis available. In this case, the log-likelihood function forX1:kcan be written as:

‘_kðHÞ ¼log½pðX1:kjHÞ; ð6Þ

wherep(X1:kjH) is the joint PDF of the degradation dataX1:k. Then the MLEHbkofHconditional onX1:kcan be obtained by:

b

Hk¼arg max

H ‘kðHÞ: ð7Þ

Due to the random effect and unobservability ofh, Eq.(7)will be too difficult to maximize with respect toH. However, the EM algo- rithm (Dempster et al., 1977) provides a possible way for resolving this difficulty. The essential idea in the EM algorithm is to manip- ulate the relationship betweenp(X1:kjH) andp(X1:k,hjH) via the Bayesian rule so that estimatingHcan be achieved by two steps:

E-step andM-step.

E-step

Calculate‘HjHb^ðiÞ_k

¼E

hjX_1:k;b^HðiÞ_kflogpðX1:k;hjHÞg; ð8Þ whereHb^ðiÞ_k denotes the estimated parameters in theith step condi- tional onX1:k.

M-step

CalculateHb^ðiþ1Þ_k ¼arg max

H ‘HjHb^ðiÞ_k

: ð9Þ

The above steps are iterated multiple times to produce a se- quencenHb^ð0Þ_k ;Hb^ð1Þ_k ;Hb^ð2Þ_k ;. . .o

of increasingly good approximations Hb_ktoHbk. The iterations are usually terminated using a standard criterion such as the difference betweenHb^ðiÞ_k andHb^ðiþ1Þ_k falling be- low a pre-defined threshold. The properties of the convergence of the log-likelihood function and parameter estimates are discussed inDempster et al. (1977) and Wu (1983).

2.2.4. Step 4. Update the RUL distribution conditional on the observed information

After obtaining the requiredfL_kjh;X_1:kðlkjh;X1:kÞ; pðhjX1:kÞ, and esti- matedHconditional onX1:kin previous three steps in place, the updated PDF and CDF of the RUL at timetkcan be formulated via the law of total probability as follows:

fL_kjX_1:kðlkjX1:kÞ ¼ Z þ1

1

fL_kjh;X_1:kðlkjh;X1:kÞpðhjX1:kÞdh

¼EhjX1:kfLkjh;X1:kðlkjh;X1:kÞ

; ð10Þ

FLkjX1:kðlkjX1:kÞ ¼ Z þ1

1

FLkjh;X1:kðlkjh;X1:kÞpðhjX1:kÞdh

¼EhjX_1:kFL_kjh;X_1:kðlkjh;X1:kÞ

: ð11Þ

Clearly,fLkjX1:kðlkjX1:kÞcontains the whole history of observation tot_k, which is introduced by two updating procedures, updatingh via the Bayesian rule and then updatingHvia the EM algorithm.

In the subsequent sections, we will give two specific models based on above-presented framework to illustrate the implemen- tation process of our presented degradation path-dependent approach. Let us first consider a linear degradation model for RUL estimation.

3. Linear model

The linear degradation model is typically used for modelling degradation processes where the degradation rate is approxi- mately a constant, seeChrister and Wang (1992), Gebraeel et al.

(2005), and Elwany and Gebraeel (2008, 2009). In this paper, we consider a linear degradation model based on a Wiener process as follows:

XðtÞ ¼

u

þhtþ

r

BðtÞ; ð12Þ

where

u

is the initial degradation,hand

r

are the drift and diffusion parameters, andB(t) denotes the standard BM, which represents the stochastic dynamics of the degradation process, as denoted by

e

(t) in Eq.(1). In this model, we assume thathis the stochastic coeffi- cient while

u

and

r

are deterministic. Without loss of generality, we further assumet0= 0 and x0= 0 and thus

u

= 0 in this case.

Now, we illustrate the step-by-step implementation of our approach presented in Section2.

3.1. Step 1

In Eq.(12), stochastic parameterhis generally assumed to fol- low a prior distribution,p(h). Here we assume thathis normally distributed with mean

l

0and variance

r

²₀. Then according to the properties of the standard BM, for givenh, the sampling distribu- tion ofX1:k= {x1,x2,. . .,xk} is multi-variable normal, distributed by the following expression:

pðX1:kjhÞ ¼ 1 Qk

j¼1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2

pr

²ðtjtj1Þ

p exp X^k

j¼1

ðxjxj1hðtjtj1ÞÞ² 2

r

²ðtjtj1Þ

" #

:

ð13Þ In Bayesian framework, in order to calculate posteriorp(hjX1:k), it is assumed that the prior distribution ofhfollowsN

l

0;

r

²0

. Note that such prior distribution actually falls into the conjugate family of sampling distribution p(X1:kjh). Consequently, the posterior estimate ofh conditional onX1:kis still normal, that is, hjX1:k Nð

l

_h;k;

r

²_h;kÞ. Other prior distributions can also be used, but evaluat- ing the posterior may involve numerical techniques such as Gibbs sampler.

3.2. Step 2

GivenhN

l

0;

r

²0

; pðhjX1:kÞcan be calculated from Eq.(4)as:

pðhjX1:kÞ /pðX1:kjhÞ pðhÞ /exp X^k

j¼1

ðxjxj1hðtjtj1ÞÞ² 2

r

²ðtjtj1Þ

" #

exp ðh

l

₀Þ² 2

r

²₀

" #

/exp ðh

l

_h;kÞ² 2

r

²_h;k

" #

: ð14Þ

Due to the property of the normal distribution ofhjX1:k, we can obtain,

pðhjX1:kÞ ¼ 1

r

h;k

ffiffiffiffiffiffiffi 2

p

p exp ðh

l

_h;kÞ² 2

r

²_h;k

" #

; ð15Þ

with

l

_h;k¼

l

₀

r

²þxk

r

²₀ tk

r

²₀þ

r

²;

r

²_h;k¼

r

²

r

²₀ tk

r

²₀þ

r

²; ð16Þ where we can learn that the posterior estimate ofhcan be easily up- dated once new observation is available.

Remark 1. It is noted that, if we write the mean drift as

l

h;k¼xk=tk, the posterior estimate ofhin Eq.(16)can be rewritten as

l

h;k¼w1

l

h;kþw2

l

0, where w1¼tk

r

²0 tk

r

²0þ

r

² and w2¼

r

²=t_k

r

²₀þ

r

², and

r

²_h;k¼

r

²

r

²₀ t_k

r

²₀þ

r

²<

r

²₀. It is easily verified that

r

²_h;k decreases monotonically approaching

r

²/tk as tk?1, namely, the uncertainty about the true value of h decreases.

l

h,kis a linearly weighted combination of

l

_h;k and

l

0

with weighted coefficients w1 and w2. Thus

l

h,k always lies somewhere between

l

h;k and

l

0. It approaches

l

h;k as tk?1.

Therefore, if we have the adaptive adjustment mechanism for the prior parameters

l

0;

r

²₀, and fixed parameter

r

², the initial guess of the truehwill be hopefully improved so as to make it closer to the new mean drift with less uncertainty. This motivates us to develop such adjustment mechanism in Step 3.

Now let us first focus on how to calculatefLkjh;X1:kðlkjh;X1:kÞattk

and then go the further details about the parameters updating for

l

0,

r

²₀, and

r

². Note that we consider the case of the directly observed degradation process (e.g. at the current CM pointtk, the current degradation statexkis observed). Therefore, for in service RUL estimation attk, givenhandxk, we can translate the original degradation process as:

XðtÞ ¼xkþhðttkÞ þ

r

ðBðtÞ BðtkÞÞ; fortPtk: ð17Þ Further translating this model with time scale over residual timelk, i.e. RUL, as:

XðlkþtkÞ ¼xkþhlkþ

r

ðBðlkþtkÞ BðtkÞÞ: ð18Þ In order to calculatefLkjh;X1:kðlkjh;X1:kÞ, we first show that the fol- lowing holds in general.

Theorem 1. Given tk, for any tP0, the stochastic process, {W(t), tP0}, with W(t) = B(t + tk)B(tk) is still a standard BM, where {B(t), tP0}is a standard BM.

Based onTheorem 1andEq.(18),the estimated RUL at t_kcan be calculated as the FPT of the following processfX⁰ðlkÞ;lkP0gcrossing threshold w,

X⁰ðlkÞ ¼xkþhlkþ

r

ðBðlkþtkÞ BðtkÞÞ

¼xkþhlkþ

r

WðlkÞ; forlkP0; ð19Þ with

WðlkÞ ¼BðlkþtkÞ BðtkÞ: ð20Þ

Therefore,fX⁰ðlkÞ;lkP0gis still a BM with a drift parthlkand initial value X⁰(0) = xk. We further show that the following holds.

Theorem 2. OnceX1:kis available at tk, the followings hold, FL_kjh;X_1:kðlkjh;X1:kÞ ¼FL_kjh;Xðt_kÞ¼x_kðlkjh;xkÞ and fL_kjh;X_1:kðlkjh;X1:kÞ

¼fLkjh;XðtkÞ¼xkðlkjh;xkÞ:

ð21Þ Note the FPT of BM with a drift follows an inverse Gaussian distribution.

FromTheorem 2andEq.(21),it is direct to obtain the PDF and CDF of the RUL at tk, associated withEq.(19),as follows:

fL_kjh;X_1:kðlkjh;X1:kÞ ¼fL_kjh;Xðt_kÞ¼x_kðlkjh;xkÞ

¼ wxk

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2

p

l³_k

r

²

q exp ðwxkhlkÞ² 2

r

²lk

!

; lk>0;ð22Þ

FLkjh;X1:kðlkjh;X1:kÞ ¼FLkjh;XðtkÞ¼xkðlkjh;xkÞ

¼1U ^wxkhlk

r

^ffiffiffiffilk

p

!

þexp 2hðwxkÞ

r

²

U ^{ðwxkÞ hlk}

r

^ffiffiffiffilk

p

!

: ð23Þ

In the next step, we illustrate how to estimate the unknown parameters H¼ ½

r

²;a ¼

r

²;

l

₀;

r

²₀. In order to incorporate the updating nature of H, we use Hk¼^h

r

²_k;

l

0;k;

r

²_0;kⁱ to denote the parameter needed to be estimated based onX1:kand the estimated parameters are denoted byHbk¼^h

r

^²_k;

l

^0;k;

r

^²_0;kⁱ.

3.3. Step 3

In order to estimateHk, from Eq.(8), we first evaluate the com- plete log-likelihood function lnp(X1:k,hjHk), which is

lnpðX1:k;hjHkÞ ¼lnpðX1:kjh;HkÞ þlnpðhjHkÞ

¼ kþ1

2 ln 2

p

1 2

X^k

j¼1

lnðtjtj1Þ k 2ln

r

²_k

X^k

j¼1

ðxjxj1hðtjtj1ÞÞ² 2

r

²_kðtjtj1Þ 1

2ln

r

²_0;k

ðh

l

_0;kÞ²

2

r

²_0;k ^{: ð24Þ}

Given Hb^ðiÞ_k ¼^h

r

^^2ðiÞ_k ;

l

^^ðiÞ_0;k;

r

^^2ðiÞ_0;kⁱ as the estimate in the i th step based onX1:k, the expectation‘ðHkjHb^ðiÞ_kÞof lnp(X1:k,hjHk), can be computed as follows:

‘ðHkjHb^ðiÞ_kÞ ¼E

hjX_1:k;b^HðiÞ_kflnpðX1:k;hjHkÞg

¼ kþ1 2 ln 2p1

2 X^k

j¼1

lnðtjtj1Þ k 2lnr²k

X^k

j¼1

ðxjxj1Þ²2lh;kðtjtj1Þðxjxj1Þ þ ðtjtj1Þ² l²h;kþr²h;k

2r²kðtjtj1Þ 1

2lnr²0;kl²h;kþr²h;k2lh;kl0;kþl²0;k

2r²_0;k ^{: ð25Þ}

Let^@‘ðHkjbH^ðiÞ_kÞ

@Hk ¼0, we obtainHb^ðiþ1Þ_k as follows,

r

^^2ðiþ1Þ_k

¼1 k

X^k

j¼1

ðxjxj1Þ²2

l

h;kðtjtj1Þðxjxj1Þ þ ðtjtj1Þ²

l

²_h;kþ

r

²_h;k

ðtjtj1Þ ;

ð26Þ

l

^^ðiþ1Þ_0;k ¼

l

_h;k;

r

^^2ðiþ1Þ_0;k ¼

r

²_h;k:

ð27Þ

Theorem 3. Hb^ðiþ1Þ_k obtained by Eqs. (26) and (27) is uniquely determined and located at the maximum of‘HkjHb^ðiÞ_k

.

Remark 2. It is observed fromTheorem 3that theM-step in our approach can be solved analytically and obtains the unique maxi- mum point. In other words, parameter updates in each iteration of the EM algorithm have explicit formulas. This implies that each iteration of the EM algorithm can be performed with a single com- putation, which leads to an extremely fast and simple estimation procedure. This computation advantage is particularly attractive for practical applications.

3.4. Step 4

We note that the estimated RUL by Eqs.(22) and (23)only uses the current degradation data, but not the system’s degradation his- tory beforetk. As discussed previously, ideally the future FPT de- pends on the path that the degradation has involved to date. In this step, we attempt to achieve such desired feature, i.e. to obtain fLkjX1:kðlkjX1:kÞ.

In order to calculatefL_kjX_1:kðl_kjX_1:kÞ, we first present following two results.

Lemma 1. If YN(0, 1) and k,

c

,2R, then E_Y [U(k+

c

Y)] can be formulated as:

EY½Uðkþ

c

YÞ ¼U k=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

c

²þ1

q

:

Proof. We have

EY½Uðkþ

c

YÞ ¼E½EðIfZ6kþcYgjYÞjY ¼PrðZ6kþ

c

YÞ

¼PrðZk

c

Y60Þ ¼U k=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

c

²þ1

q

: ð28Þ

In the derivation process,Udenotes the standard normal CDF, I{Z6k+cY}is the indicator function,Zis standard normal and indepen- dent ofYandZYN(k,

c

²+ 1). This completes the proof. h Theorem 4. If ZN(

l

,

r

²), and w, A, B, D2R, C2R⁺, then the fol- lowing holds:

ð1Þ EZ½expðAZÞUðCþDZÞ ¼exp A

l

þA² 2

r

²

!

U ^CþD

l

þAD

r

²

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þD²

r

²

p

!

; ð29Þ

ð2Þ EZ½ðAZÞ expððBZÞ²=2CÞ

¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C

r

²þC r

A

r

²Bþ

l

C

r

²þC

exp ðB

l

Þ² 2ð

r

²þCÞ

!

: ð30Þ

The updated RUL distribution at time tkcan be summarized in the following theorem by usingLemma 1andTheorem 4.

Theorem 5.The PDF and CDF of the RUL conditional on the observa- tions up to t_kcan be written as:

fLkjX1:kðlkjX1:kÞ ¼ wxk

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2

p

l³_kð

r

²_h;klkþ

r

²Þ q

expððwxk

l

_h;klkÞ².2lkð

r

²_h;klkþ

r

²ÞÞ

; lk>0;

ð31Þ

FL_kjX_1:kðlkjX1:kÞ ¼1U ðwxk

l

_h;klkÞ=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

r

²_h;kl²_kþ

r

²lk

q

þexp 2

l

_h;kðwxkÞ

r

^{2 þ}

2

r

²_h;kðwxkÞ²

r

⁴

!

U ²

r

²_h;kðwxkÞlkþ

r

²ð

l

_h;klkþwxkÞ

r

² ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

r

²_h;kl²_kþ

r

²lk

q 0

B@

1 CA:ð32Þ

where the degradation history is introduced by parameters updating.

So far, we have accomplished our presented degradation path- dependent approach for RUL estimation in the linear case. In the following, we give some remarks through comparing our approach with previously reported approaches.

Remark 3. Gebraeel et al. (2005)and the related works directly used Pr(Lk6lkjX1:k) = Pr(X(lk+tk)PwjX1:k) to calculate the RUL distribution, which ignored the possible hitting events within (tk, tk+lk). This implies that their results are approximate in the sense of the FPT. However, our obtained results in Eqs.(31) and (32)are exact but with explicit form.

In order to further compare the obtained results with the results byGebraeel et al. (2005), we first introduce the concept of stochastic comparison between two random variables.

Definition 1Ross, 2007. Given random variablesnandf,nis sto- chastically greater thanfif

Prðn>

v

ÞPPrðf>

v

Þ; for all real

v

^{; denoted bynP}stf: ð33Þ Denote the estimated RUL by Gebraeel’s approach asL⁰_k, then we have the following conclusion.

Theorem 6. Conditional on the degradation history to t_k, i.e.X_1:k, and adopting the same parameter estimation procedure, we have L⁰_kPstLk. FromTheorem 6, the result following the approach developed byGebraeel et al. (2005)overestimates the RUL and then can lead to under-maintenance or delayed-maintenance.

Remark 4.The moment of the RUL distribution obtained by Gebraeel et al. (2005) and Elwany and Gebraeel (2009), does not exist since their obtained RUL distributions belong to the family of Bernstein distributions, known without moments, but this is not the case for our result. For example, the mean of RUL can be easily formulated by:

EðLkjX1:kÞ ¼E½EðLkjh;X1:kÞjX1:k ¼E wxk

h jX1:k

¼wxk

r

²_h;k ^exp

l

²_h;k

2

r

²_h;k

! Zl_h;k 0

exp u² 2

r

²_h;k

! du

¼ ffiffiffi2 p ðwxkÞ

r

h;k

D

l

_h;k

ffiffiffi2 p

r

h;k

!

: ð34Þ

whereDðzÞ ¼expðz²ÞRz

0expðu²Þduis the Dawson integral, which is known to exist. Particularly, if we assume Pr(h< 0) = 0, which im- plies

l

h,k

r

h,k. Then using the approximation property of Dawson integral for largez,D(z) 1/2z, thenE(LkjX1:k) = (wxk)/

l

h,k.

The PDF and CDF in Eqs.(31) and (32)enable the construction of a replacement decision model that incorporates the probability of failure before a particular instant conditioned on the degrada- tion history to date. At each monitoring point throughout the life of a system, an optimal replacement time can be scheduled using the renewal-reward theory and the long run expected cost per unit time. When the RUL distribution is used in condition based replacement, the following is usually minimized to decide the optimal replacement time (Wang and Zhang, 2005),

CðTR;kÞ ¼ cpþ ðcfcpÞPrðLk<TR;ktkjX1:kÞ tkþ ðTR;ktkÞð1PrðLk<TR;ktkjX1:kÞÞ þRT_R;kt_k

l_k¼0 lkfL_kjX_1:kðlkjX1:kÞdlk

;

ð35Þ whereTR,kis the decision variable representing the planned replace- ment time determined at the kth CM point, cp is the cost of a preventive replacement, andcfis the replacement cost with the fail- ure. It is well-known that ifcpPcf, no preventive replacement is optimal. In this case,T_R,kwill approach positive infinity, i.e.T_R,k?+- 1. Then the last term in the denominator of Eq.(35)will beE(Lk-

jX1:k). Therefore, the non-existence ofE(LkjX1:k) may lead to the non-existence of Eq.(35). However, our result can avoid this prob- lem and makes Eq.(35)hold in general.

Additionally, it can be proved that the above cost function equals to the cost function used inElwany and Gebraeel (2008).

Theorem 7.Let FLkjX1:kðTR;ktkjX1:kÞ ¼PrðLk<TR;ktkjX1:kÞ and F_LkjX1:kðl_kjX_1:kÞ ¼1F_LkjX1:kðl_kjX_1:kÞ. Then

CðTR;kÞ ¼cpþ ðcfcpÞFLkjX1:kðTR;ktkjX1:kÞ tkþR^TR;kt_k

l_k¼0 FLkjX1:kðlkjX1:kÞdlk

:

Based on above results, we have the following conclusion for the cost functions when using the approximated RUL distribution by Gebraeel’s approach and our exact RUL distribution. Denote the cost function using the RUL distribution obtained from Gebraeel’s approach as C⁰(TR,k).

Theorem 8.Conditional on the degradation history to tk, i.e.X1:k, and adopting the same parameter estimation procedure, if cfPcp, then C(TR,k)PC⁰(TR,k).

Theorem 8implies that, when using approximated RUL distribu- tion in decision making, the operating risk represented by the expected cost per unit time will be underestimated and then the maintenance action may be delayed. This further confirms the statement implied by Theorem 6.

Remark 5. Note that variance parameter

r

²and parameters

l

₀;

r

²₀

in prior distributionp(h) ofGebraeel et al. (2005)and the related works are prior determined from the offline degradation data of multiple other systems. However, once these parameters are deter- mined and they are then fixed even if real-time CM data are avail- able. This makes the RUL estimation non-robust over these parameters. Particularly, if these parameters are not determined

accurately enough, then the estimated RUL may be hardly accurate.

In contrast, our approach can adaptively adjustH¼

r

²;

l

₀;

r

²₀via the EM algorithm in line with real-time data. In this sense, our approach relies less on prior information. To facilitate the imple- mentation of the developed approach, the main steps are summa- rized in the following table.

4. Exponential model

The exponential-like degradation model is another typical mod- el representing a degradation process where the cumulative dam- age has a particular effect on the rate of degradation, but the degradation path can be linearized by log-transformation. So far, it has long been thought to be a good approximation for nonlinear degradation processes such as corrosion, bearing degradation, deterioration of LED lighting, see Tseng et al. (2003), Park and Padgett (2005), Gebraeel et al. (2005), Elwany et al. (2011), and Chen and Tsui (2012). In this section, we borrow this kind of model but obtain some novel results which were not reported before. In general, an exponential degradation model can be represented as:

XðtÞ ¼

u

þh⁰exp b⁰tþ

r

BðtÞ

r

²

2t

; ð36Þ

where

u

is a known constant,

r

is a constant representing the deterministic parameter,h⁰andb⁰are random variables characteriz- ing the unit-to-unit variability, andB(t) is also a standard BM.

For an exponential-like model, it is more convenient to work with its logged format. Thus, we defineS(t) attas follows:

SðtÞ ¼ln½XðtÞ

u

¼lnh⁰þ b⁰

r

²

2

tþ

r

BðtÞ

¼hþbtþ

r

BðtÞ; ð37Þ

whereh= lnh⁰andb=b⁰

r

²/2, and we denoteh= [h,b].

As well as the case study part inGebraeel et al. (2005), Elwany et al. (2011), and Chen and Tsui (2012), it is assumed that

u

= 0 to simplify the analysis. In general, however,

u

could be any known constant. In the following, we mainly focus on the transformed degradation observations fromS(t) and illustrate the implementa- tion process of our approach for RUL estimation in this exponential case.

4.1. Steps 1–2

As the same assumption used for the prior distributions ofh⁰ and b⁰ in Gebraeel et al. (2005), we let lnh⁰ and b⁰ follow Nð

l

₀;

r

²₀Þ and N

l

⁰₁;

r

²₁ respectively, where

l

⁰1=

l

1+

r

²/2, and we further assume thath⁰,b⁰ andB(t) are mutually independent.

Therefore, we have hNð

l

₀;

r

²₀Þ and bNð

l

₁;

r

²₁Þ. As a result, once the new CM data is available, the posterior estimates of h andbcan be evaluated by the Bayesian rule.

LetS1:k= {s1,s2,. . .,sk}, wheresk= lnxk. Then, givenh,b, the sam- pling distribution ofS_1:kis multi-variable normal as

pðS1:kjh;bÞ ¼ 1 Qk

j¼1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2

pr

²ðtjtj1Þ p

exp ðs1hbt1Þ² 2

r

²t1

X^k

j¼2

ðsjsj1bðtjtj1ÞÞ² 2

r

²ðtjtj1Þ

" #

:

ð38Þ Then the joint posterior estimate ofhandbconditional onS1:kis still normal resulted from the fact of the normal distribution assumption of h and b. In other words, h;bjS1:kNð

l

_h;k;

r

²_h;k;

l

_b;k;

r

²_b;k;

q

_kÞ. To be more precise, we have