data-driven prognostic method based on bayesian...
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J Intell Manuf (2016) 27:1037–1048DOI 10.1007/s10845-014-0933-4
Data-driven prognostic method based on Bayesian approachesfor direct remaining useful life prediction
A. Mosallam · K. Medjaher · N. Zerhouni
Received: 26 February 2014 / Accepted: 31 May 2014 / Published online: 13 June 2014© Springer Science+Business Media New York 2014
Abstract Reliability of prognostics and health manage-ment systems relies upon accurate understanding of criti-cal components’ degradation process to predict the remain-ing useful life (RUL). Traditionally, degradation process isrepresented in the form of physical or expert models. Suchmodels require extensive experimentation and verificationthat are not always feasible. Another approach that buildsup knowledge about the system degradation over the timefrom component sensor data is known as data driven. Datadrivenmodels, however, require that sufficient historical datahave been collected. In this paper, a two phases data drivenmethod for RUL prediction is presented. In the offline phase,the proposed method builds on finding variables that containinformation about the degradation behavior using unsuper-vised variable selection method. Different health indicators(HIs) are constructed from the selected variables, which rep-resent the degradation as a function of time, and saved inthe offline database as reference models. In the online phase,the method finds the most similar offline HI, to the online HI,using k-nearest neighbors classifier to use it as a RUL predic-tor. The method finally estimates the degradation state usingdiscrete Bayesian filter. The method is verified using batteryand turbofan engine degradation simulation data acquiredfrom NASA data repository. The results show the effective-ness of the method in predicting the RUL for both applica-tions.
A. Mosallam · K. Medjaher (B) · N. ZerhouniFEMTO-ST Institute, AS2M Department, University ofFranche-Comté/CNRS/ENSMM/UTBM, 24 rue Alain Savary,25000 Besançon, Francee-mail: [email protected]
A. Mosallame-mail: [email protected]
N. Zerhounie-mail: [email protected]
Keywords Degradation modeling · Online estimation ·Discrete Bayes filter · Uncertainty representation ·Data-driven PHM
Introduction
The large volume of data gathered continuously from dif-ferent systems has created challenges to interpret such datain order to anticipate the breakdowns. Most large industrieshave specialized engineerswhomare skilled in the use of hightechnology maintenance equipment and have earned specialcertification in the field of maintenance. Nevertheless, it isstill hard to take immediate decisions and predict the systemfailure. The need of computer systems that constantly recordand analyze data to predict the RUL of critical components isparticularly important for facilitatingmaintenance decisions.
In general, maintenance involves performing routineactions to obtain optimal availability of industrial systems(Montgomery andBanjevic 2012).Maintenance routines canbe broadly categorized into two main types, namely, correc-tive and preventive maintenance (Kothamasu et al. 2006).In corrective maintenance, the interventions are performedonlywhen the critical component is fullyworn out and failureoccurred. Preventive maintenance can be further divided intotwo main approaches, namely, time-based maintenance andcondition based maintenance (CBM). In time-based main-tenance, the interventions are placed according to periodicintervals regardless of the assets’ health condition and thusthe service life of the critical components is not fully utilized(Soh et al. 2012). Condition basedmaintenance usesmachinerun-time data to assess the critical component’s state andschedule required maintenance actions prior to breakdown(Peng et al. 2010). Furthermore, predictive maintenance uti-lizes the current health status of a given critical component
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to predict its future condition and plan maintenance actions.Prognostics and health management (PHM) (Jardine et al.2006) is a processwhich links degradationmodeling researchto predictive maintenance policies.
Prognostics and health management consists of four mainmodules: fault detection, fault diagnostics, fault prognosticsand decision making (Medjaher et al. 2012). Fault detectioncan be defined as the process of recognizing that a prob-lem has occurred regardless of the root cause (Dong et al.2012). Fault diagnostics is the process of identifying thefaults and their causes (Choi et al. 2009). Fault prognosticscan be defined as the prediction of when a failure might takeplace (Tobon-Mejia et al. 2012). Finally, decision makingstep uses all the information gathered about the monitoredsystem status to choose the optimalmaintenance actions (Iyeret al. 2006). Among other routines, prognostics have recentlyattracted significant research interest due to the need of mod-els for accurate RUL prediction for different applications.
RUL prediction of critical components is a non-trivial taskfor many reasons. Sensor signals, for instance, are usuallyobscured by noise and thus it is very challenging to processand to extract informative representation of the RUL (Kam-ran et al. 2013).Another problem is the prediction uncertaintydue to the variation of the end of life time that can differ fortwo components made by the samemanufacturer and operat-ing under the same conditions. Therefore, proposed modelsshould include such uncertainties and represent them in aprobabilistic form (Saha and Goebel 2008; Pal et al. 2011).
RUL prediction models can be realized using two differ-ent methods, namely, physics based and data-driven methods(Heng et al. 2009). Physics based methods build physicalmodels of the desired critical components by the means ofstate-space models (Isermann 2006) and dynamic ordinaryor partial differential equations (Vachtsevanos et al. 2006).These models require extensive experimentation and modelverification (Luo et al. 2003). However, these models arevery reliable at least until the system is upgraded or changed(Chaari et al. 2009). Data-driven methods can be used whenthe first principles of the system operation are complex suchthat developing an accurate physics of failure model is notfeasible (Zhang et al. 2013; He et al. 2012). Such methodsemploy pattern recognition and machine learning techniquesto characterize the desired critical components’ degradationbehavior (Schwabacher 2005). One way to do data-drivenRUL prediction is by first estimating the current health statusof the desired component and when the degradation exceedsthe alarm threshold, the algorithms start predicting the RUL(Zhang 2003; Gorjian et al. 2009; Benkedjouh et al. 2013).Different regression models have been proposed in the liter-ature to deal with data-driven RUL prediction problem suchas the auto regressive model and the multivariate adaptiveregression splines (Box and Jenkins 1976; Lewis 1992; Tsay2000; Wu et al. 2007; Yan et al. 2004; Lee et al. 2006).
A drawback of using regression methods is that when avail-able component degradation history is incomplete the extrap-olation may lead to large errors (Wang et al. 2008). Therehave been more interests lately on various types of neuralnetworks and neural-fuzzy systems (Gebraeel et al. 2004;Satish and Sarma 2005; Huang et al. 2007; Lei et al. 2007;Vassilopoulos et al. 2007; Tian 2012; Kamran et al. 2013;Brezak et al. 2012; Gajate et al. 2012; Purushothaman 2010;Yeo et al. 2000). However, these methods generate blackbox models and it is difficult to select the structure of thenetwork (Ramasso et al. 2013). Similarity-based methodsare shown to be very effective in performing RUL predic-tion. A similarity-based method based on linear regression toconstruct offline degradation models is proposed in Wang etal. (2008). The method measures the similarity between testinstance and offline models and the selected offline instanceis used for RUL prediction. The RUL probability density ofthe test instance is estimated from the multiple local predic-tions using the kernel density estimation method. The mainproblemwith thismethod is themanual selection of the infor-mative sensor data. Another similarity-basedmethod that uti-lizes k-nearest neighbors (k-NN) and belief function theoryto estimate the health and from that deduce the RUL of turbo-fan engines is proposed in Ramasso et al. (2013). The authorsmanually annotate the health status of the offline data sets andthen themethod predicts the RULwhen the degradation levelreaches a predefined alarm threshold.
Alternatively, instead of learning the degradation from thedata and predict the RUL; direct RUL prediction algorithmslearn the relation between the sensor data and the end of lifeto predict the RUL. To do this, health indicators are extractedfrom the raw monitoring signals, which may have originatedfrom single sensor or from a number of sensors aggregatedto represent the degradation evolution over time. Althoughthis type of RUL prediction is relatively easy to implement,there are few published examples in the literature (Sikorskaet al. 2011).
In this work, direct RUL prediction method is presented.The aim of this work is to model the relation between sensordata and end of life to predict the RUL without the need forpredefined alarm threshold. The method builds on extract-ing health indicators from the training data, which are usedas reference models. For new data, the method finds in thedatabase the most similar signal to be used as a RUL predic-tor. The method then estimates the new signal’s health statususing a Bayesian approach.
The assumptions taken in this work can be summarized asfollows:
1. The method can only be applied to critical components,which are already identified by the system expert.
2. Historical data should contain degradation evolution ofthe critical component over time.
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Fig. 1 The method’s generalscheme
Online signalsHealth
indicator construction
K-NN
Online estimation
Offline signalsHealth
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Data baseVariable selection
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3. Historical data should contain sufficient number of train-ing instances to build representativemodels of the desiredcritical component’s behavior.
4. The predicted RUL values will span between the valuesavailable in the offline data sets.
This paper is structured as follows. Section 2 presents theproposed method. The experimental set-up and the simula-tion results are depicted in Sect. 3. Finally, Sect. 4 concludesthe paper.
Data-driven prognostic method based on Bayesianapproaches for direct remaining useful life prediction
Measurements observed from monitored components areusually noisy multidimensional time series signals. Thus, itis essential in the offline phase to first extract information thatrepresents the degradation evolution over time. The relationbetween the extracted information and the end of life shouldbe modeled to predict the RUL. To do this, the proposedmethod selects interesting sensor signals and builds healthindicators that are used as offlinemodels. In the online phase,the method estimates the current status from the unseenonline data, using only the sensors selected on the offlinephase, and predicts the RUL by measuring the similarity tothe offline data. The method is summarized in Fig. 1 andexplained hereafter.
Offline phase
In order to build offline reference models, representative fea-tures should be extracted from the training data. Those fea-tures are later labeled with the end of life time and savedin the database. To do that, a trend construction method is
applied (Mosallam et al. 2013). The method builds on twomain steps, namely, variable selection and health indicatorconstruction.
Variable selection
Not all signals from the monitored component are infor-mative. Signals that have non-random relationships containinformation about the system degradation. To select such sig-nals, an unsupervised variable selection algorithm based oninformation theory is applied (Mosallam et al. 2011). Thealgorithm first calculates pairwise symmetrical uncertainty(SU ), as depicted in Fig. 2a, for all the input signals:
SU (X,Y ) = 2 × I (X,Y )
H(X) + H(Y )(1)
where, I (X,Y ) is the mutual information between two ran-dom variables X and Y ; H(X) and H(Y ) are informationentropy values of the random variables X and Y , respec-tively. Then, the algorithm groups the variables based on theSU distance using hierarchical clustering shown in Fig. 2b.The algorithmfinally ranks the resulting clusters according tothe quality of the included signals in representing interestingrelationships using normalized self-organizing map distor-tion measure. A cluster gets low rank if it contains randomsignals. On the other hand, a cluster gets high rank if it con-tains signals that exhibit nonrandom relationship and thosesignals will be used for later processing.
Health indicator construction
The following task, after selecting the interesting variablesfrom the initial monitoring raw signals, is to extract smoothmonotonic signals, which are correlated with the compo-nent’s endof life. Thesemonotonic signals are later processed
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Fig. 2 Variable selection step for 11 sensor signals from NASA battery B0005. a SU similarity matrix. b Tree representation of variables relations
to extract representative features over the time, which can beused as health indicators and are saved in the database asreference models. To do this, three processing steps, namelyvariable compression, trend extraction and feature extraction,are applied to the selected variables.Variable compression: The goal of this step is to com-press the n signals selected in the previous step onto one-dimensional space. From each cycle, the selected variablesare compressed using standard principal component analy-sis (PCA) method. The first principle component retains themaximum variance while reducing the dimensionality to onedimension. Therefore, only the first principle component isused to represent the health status evolution with respect totime.Trend extraction: The compressed variables are then fur-ther processed at each cycle to get monotonic trends that canrepresent the variation of end of life using empirical modedecomposition algorithm (EMD) (Huang et al. 1998). EMDis a method employed to decompose a signal into successiveintrinsic mode functions (IMF) and a residual signal rn(t),which should be a constant or monotonic signal that can berepresented as:
rn(t) = X (t) −n∑
i=1
im fi (t) (2)
where, X (t) is the input signal, im fi is the IMF and n is themaximum number of IMFs. The generated residual can rep-resent the relation between the generated trends and end oflife time. For example, Fig. 3a shows an acceleration sig-nal acquired from a degraded bearing that was worn outafter around 9h and Fig. 3b shows a non-degraded bear-ing were the experiment stopped at the same time of thedegraded bearing (Nectoux et al. 2012). EMD was applied
to both of the two signals and the resulting residuals areshown in Fig. 3c. The experiments show that the residualof the degraded component was a monotonic signal whilethe non-degraded component generated almost a constantresidual.Feature extraction:So far, trends are extracted from the com-pressed variables. These trends should be used to build anoffline model, which can be used to classify new online data.In order to make the classification task more efficient, dis-criminant features should be extracted from acquired trends.Different approaches have been proposed for extracting fea-tures such as mean, variance, multi-exponential function,curve fitting, discrete wavelet transform and discrete Fouriertransform (Marco et al. 2009). However, selecting appropri-ate features is mainly problem specific. Recalling Fig. 3c,the slope of the trend can be a discriminant characteristicof the trend. A trend with more RUL tends to have smallerslope and vice versa. The y-intercept of the curve fit showsthe beginning value of the extracted trend, which also can bea discriminant feature. Another discriminant feature for thisproblem is the mean of the extracted trend. Every data valuein the trend contributes to the mean value, and the changeof the data over time will affect the mean value. Finally, thevariance of the extracted trend describes the spread of a trendwith respect to end of life time, which is also an importantfeature to extract.
In this work, a feature vector F = [a, b, x̄, s2] is extractedfrom each trend at each time, where, a and b are the slope andthe y-intercept of a linear curve fit of the input trend respec-tively, x̄ and s2 are the mean and the variance of the inputtrend, respectively. The feature vector is extracted from eachtrend starting from time 0 until current time t . Figure 4 showsan example of the feature extraction process from three differ-ent trends extracted at three constitutive cycles, namely, cycle40, cycle 100 and cycle 167. The method extracts the feature
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Fig. 3 Residual variation according to the health status. a Degraded bearing. b Non-degraded bearing. c Residual of both bearings
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Fig. 4 Feature extraction from input residual at cycles 40, 100 and 167. a Slope and y-intercept values. b Mean and variance values
vector, fromeach trendbuilt in previous step, labels the vectorwith the cycle number and end of life value and saves it in theoffline database. This process is repeated recursively, until themethod reaches the end of life, to generate a representation ofthe degradation as a function of time. The resulting function
or health indicator, as depicted in Fig. 5, is then used to rep-resent the corresponding critical component according to itsend of life time. Each group of health indicators with similarend of life time is considered as a class and saved in the offlinedatabase.
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Fig. 6 Finding the end of life for the online signal at time = 50 cycles using k-NN classifier. a Selecting the most similar group. b End of life forthe online signal
Online phase
In this phase, new sensor data are collected online from thecritical component(s) from only the sensors that are selectedin the offline phase. The processes applied in the offlinephase, such as extractingmonotonic trends and feature vectorF , are applied to the online signals. The generated vector Fis then fed to a k-NN classifier to find the most similar offlinesignal (or case). The end of life value of the offline signal isthen considered to be the RUL of the test signal. The onlineestimator recursively estimates the trends value until it stopsat end of life time.
Classification using k-nearest neighbours
In order to build the predictive model, a k-NN classifier isapplied in this work. The extracted feature vector F at time tis passed to the k-NN to find the most similar offline group inthe database at the same time as shown in Fig. 6a. The clas-
sification decision is based on largest posterior probabilityof the tested sample at time t , therefore, a probability valuewill be assigned to the prediction output:
p(EOLk |Ft ) = p(Ft |EOLk) × p(EOLk)
p(Ft )(3)
where, Ft is the online feature vector, EOLk is the class orend of life value for group k, p(Ft |EOLk) is the probabilityof observing Ft given EOLk , also known as the likelihood,p(EOLk) is class prior and p(Ft ) is the marginal likelihood.The end of life with the highest posterior probability will beused as the end of life for the new signal as depicted in Fig.6b.
Online estimation
To estimate the actual value of the online health indicator atthe predicted end of life value, a recursive discrete Bayesianfilter is applied to the online trends. This filter, decomposes
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the state space into many regions and represents the cumu-lative posterior for each region by probability values, seeAlgorithm 1.
Input : {pk,t−1}, ztOutput: {pk,t }forall the k do
p̄k,t = ∑ip(Xt = xk |Xt−1 = xi )pi,t−1
pk,t = ηp(zt |Xt = xk) p̄k,tend
Algorithm 1: Discrete Bayesian filter.
The input to the algorithm is a discrete probability dis-tribution {pk,t} along with the recent measurement zt . Thefirst line of the Algorithm 1, p̄k,t = ∑
i p(Xt = xk |Xt−1 =xi )pi,t−1 , calculates the prediction for the new state basedon previous state uncertainty and state transition model.The prediction is then updated in the second line, pk,t =ηp(zt |Xt = xk) p̄k,t , so as to incorporate the measurement.Discrete Bayesian filters apply to problems with finite statespace, where the random variable Xt = x1,t ∪ x2,t ∪ · · · xk,t .A straightforward decomposition of Xt is amultidimensionalgrid, where each xk,t is a bin or region. The size of each binis dx = xmax−xmin
n , where xmax is the maximum state value,xmin is the minimum state value and n is the number of bins.Each bin can then be represented as a Gaussian function witha mean value at each state and a common variance:
p(Xt |Xt−1) = ‖dx × N (Xk,t , σ2)‖ (4)
where, p(Xt = xk |Xt−1) is the state transition model, dx isthe bin size andN (Xk,t , σ
2) is the normal distribution at stateXk,t . Moreover, Eq. (4) is normalized to turn this quantityinto a probability distribution. Similarly, the measurementprobability model can be calculated in the same manner asthe transition model. Figure 7 shows the final result of theproposed method.
The estimation algorithm stops once it reaches the pre-dicted end of life. The uncertainty about the prediction andcurrent status are represented in probabilistic forms. Theoverall method is summarized in Algorithm 2.
Themachine degradation information, i.e. predictedRUL,the estimated health status and corresponding uncertainties,produced by this method can be used as an input for main-tenance decision making routine. Decision-making routineconsiders both machine degradation information and systemstructure to assist the plant manager in making a dynamicmaintenance planbasednot only on the optimization of singlecomponent/subsystem plan, but also on the global schedul-ing of whole system for optimized maintenance prioritiza-tion (Xia et al. 2012). Maintenance prioritization is crucialand important to reduce unnecessary maintenance activities,
p(EOLk | Ft)
Fig. 7 Estimation of the health indicator using Bayesian filter
Data: {trainingData, test Data}Result: {Dl , RUL}
1 for ∀trainingData do2 selectedVariables = FindBestGroup(trainingData);3 end4 Offline phase5 for i = 1 : numberO f (trainingData) do6 EOL = lengthO f (trainingData(i));7 for j = 2 : EOL do8 selectedVariables =
get SelectedVariables(trainingData(i));9 i p = selectedVariables(1 : j);
10 f irstComponent = Get FirstComponent (i p);11 residual = Get EMDResidual( f irstComponent);12 f eatures = Get Features(residual);13 H I = append([ f eatures, i]);14 end15 Dl = append([H I, EOL]);16 end17 Online phase18 selectedVariables = get SelectedVariables(test Data);19 f irstComponent =
Get FirstComponent (selectedVariables);20 residual = Get EMDResidual( f irstComponent);21 testingFeatures = Get Features(residual);22 EOL = kN N (test Features, Dl);23 rulEstimation =
discreteBayesianFilter(test Features, EOL);24 RUL = (EOL , rulEstimation);
Algorithm 2: The general algorithm of the proposedmethod
especially when availability of maintenance resources arelimited (Li and Ni 2009).
Applications and results
Two real life data sets are used in this work to verify theproposed method: turbofan engine and lithium-ion batteryaging data sets.
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Fig. 9 Results of predicting the RUL at all cycles for 4 engines. a RUL of engine 34. b RUL of engine 41. c RUL of engine 42. d RUL of engine81
Turbofan engine data
The turbofan engine data sets are generated using commer-cial modular aero-propulsion system simulation (C-MAPSS)(Saxena andGoebel 2008). They consist of four training files,
four testing files and four RUL values files. The training filescontain run to failure sensor records of a fleet of enginesgenerated under different combinations of operational con-ditions and fault modes. Each engine is operating normallyand it develops a fault at some point during the operation until
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Table 1 Training data sets with three folds
Fold #1 Fold #2 Fold #3 EOL
B0006 B0005 B0005 168
B0007 B0007 B0006 168
B0026 B0025 B0025 28
B0027 B0026 B0026 28
B0028 B0027 B0028 28
B0030 B0029 B0029 40
B0031 B0031 B0030 40
B0032 B0032 B0031 40
B0034 B0033 B0033 197
B0036 B0036 B0034 197
B0039 B0038 B0038 47
B0040 B0040 B0039 47
B0043 B0042 B0042 112
B0044 B0044 B0043 112
B0045 B0045 B0045 72
B0047 B0046 B0046 72
B0048 B0048 B0047 72
B0050 B0049 B0049 25
B0051 B0050 B0050 25
B0052 B0051 B0052 25
B0055 B0054 B0054 102
B0056 B0056 B0055 102
finally it reaches the system failure and the engine stops. Thetest files are generated in the same way; however, the sensorreadings are omitted prior to system failure. The RUL filescontain vector of trueRULvalues for the test data. Each train-ing and test file contains 26 columns that represent differentvariables. The first two columns represent the engine numberand the time in cycles, respectively. The next three columnsrepresent the operational settings. The last 21 columns, orvariables, represent different time series sensor data such astotal temperature at fan inlet, pressure at fan inlet, physicalfan speed, etc. Each row represents a data snapshot taken dur-ing a single cycle. In thiswork, the datafile “train_FD001.txt”is used for offline training and “test_FD001.txt” is used foronline testing. Each file contains data for 100 engines and theobjective is to predict the number of remaining operationalcycles before failure in the test set. The true RUL values forthe test data are presented in the data file “RUL_FD001.txt”.Variable selection: One of the results of the selection algo-rithm is the pair of sensors number {8,13}, i.e. physical fanspeed and corrected fan speed, respectively (Fig. 8a). Theselected group is interesting as the two variables are corre-lated and both are related to the fan speed. Then, the methodstarts constructing themonotonic trends iteratively from eachpair at each time.
Table 2 Testing data sets with three folds
Fold #1 Fold #2 Fold #3 EOL
B0005 B0006 B0007 168
B0025 B0028 B0027 28
B0029 B0030 B0032 40
B0033 B0034 B0036 197
B0038 B0039 B0040 47
B0042 B0043 B0044 112
B0046 B0047 B0048 72
B0049 B0052 B0051 25
B0054 B0055 B0056 102
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hr)
Fig. 10 Selected pair of variables from the NASA battery B0005
Health indicator: As mentioned before, four features areextracted from each trend at each time and labeled with endof life time to be saved in the offline database. The featuresrepresent the relation between the extracted trends and theengine’s end of life. Figure 8b shows one of the four healthindicators for the NASA training engine number 61. Theindicator is monotonic and shows how the relation betweenthe end of life and the extracted trend changes through thetime. Each health indicator is then saved in offline databaseand labeled with the end of life time and will be used forpredicting the RUL of new sensor data.Prediction results: Figure 9 shows the predicted RUL for 4engines at all cycles. It can be noticed that the accuracy ofthe predictions increases with the time. The prediction errorat the last cycles is less than the errors at the beginning. Toassess the performance of the proposed method, the meanabsolute percentage error (MAPE) is calculated for all 100online predictions:
MAPE(%) = 100 %
n×
n∑
i=1
∣∣∣∣RULi − RUL∗
i
RU Li
∣∣∣∣ (5)
where RUL and RUL∗ are the actual and predictedRULval-ues respectively and n is the number of total predictions. The
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0 20 40 60 80 100 120 140 160 1800
20
40
60
80
100
120
140
160
180
Cycle
RU
L
Pridicted valuesReal values
0 5 10 15 20 25 300
20
40
60
80
100
120
140
160
180
Cycle
RU
L
Pridicted valuesReal values
(a) (b)
Fig. 11 Results of predicting the RUL at all cycles for 2 batteries. a RUL of battery B0005. b RUL of battery B0025
error is calculated only for the last cycles of all 100 test sig-nals. TheMAPEover the 100 test data equals to 12.19%.Andfor comparison, the MAPE over the first 15 test engines is8.7691%, which outperforms the method presented in Kam-ran et al. (2013) in which the MAPE value is 15.5% for the15 test engines.
Lithium-ion battery data
These data are collected on 34 lithium-ion batteries runthrough different operational profiles (e.g. charge, dischargeand impedance) at different temperatures (Saha and Goebel2007). In this work only charge and discharge data are used.Each data set, corresponding to one experiment, consists of11 variables such as charging voltage, charging current, tem-perature, discharging current, discharging voltage and capac-ity. The aging of the batteries was accelerated and the exper-iments continued until the batteries reached their end of lifetime. Each cycle is presented by the mean value to reduce theprocessing time. In order to validate the proposed method,a threefold cross-validation is applied, i.e. the available datasets are partitioned into three groups of equal size. Eachgroupis then divided into training and testing data set as depictedin Tables 1 and 2, respectively. Only 31 battery data sets areused in this experiment as three batteries, namely B0018,B0041 and B0053, do not have any similar data sets with thesame end of life.Variable selection: One of the results of the selection algo-rithm is the pair of variables {6, 11}, i.e. the voltage mea-sured at discharge and the capacity of the battery (Fig. 10).The selected pair is interesting because the two variables arecorrelated. Indeed, the capacity is related to the battery healthas the decrease in the capacity indicates health degradation.Health indicator:Four features are extracted from each trendat each time and labeledwith end of life time to be saved in theoffline database. Figure 5 shows two of the four health indi-
Table 3 Mean absolute percentage error for the NASA battery data sets
Fold #1 Fold #2 Fold #3 Average
28.0493% 26.3089% 28.3536% 27.5706%
cators for the battery B0005. The indicators are monotonicand show how the relation between the end of life and theextracted trend changes through the time.Prediction results: To assess the performance of the pro-posed method, MAPE is calculated for all cycles of eachbattery (Fig. 11). The average MAPE per fold is calculatedas follows:
MAPE f = 1
n×
n∑
i=1
MAPEi, f (6)
where MAPE f is the average MAPE for a complete fold,MAPEi, f is the MAPE for test battery i in fold f . The finalresults are calculated and summarized in Table 3.
Figure 11a shows plot of the predicted RUL for all cyclesfor battery B0005. It can be seen that the prediction accuracyincreases with time, i.e, the longer the test trend is the higherthe predication accuracy. Figure 11b shows a plot of the RULpredicted for the battery B0025. Only 10 cycles were con-sidered as late prediction. However, the error was decreasingat the later cycles.
Conclusion
In this paper a data driven method for RUL prediction basedon a Bayesian approach is proposed. The method buildson unsupervised selection of interesting variables from theinput offline signals. It constructs representative features thatcan be used as health indicators. The method represents the
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current status of the online signals as well as the uncertaintyabout the predictions in a probabilistic form.
The performance of the proposed method is evaluatedusing two data sets, namely, turbofan engines and lithium-ionbattery data downloaded form theNASAprognostic center ofexcellence website. The prediction results show low MAPEerror for both applications.
For future work, the proposed method should considerthe data sets with no training samples in the database, suchas the case with battery data sets. Also, it should be testedusing data sets with variable operating conditions and afterintroducing maintenance interventions. Different classifica-tion/regression models should be tested in the proposedframework.
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