An mproved segmenaon-based HMM learnng mehod for Condon-based Manenance T Lu 1,2, J Lemere 1,2, F Carella 1,2 and S Meganck 1,3 1 ETRO Dep., Vre Unverse Brussel, Plenlaan 2, 1050 Brussels, Belgum 2 FMI Dep., Insue for Broadband Technology - IBBT, Gason Crommenlaan 8 (box 102), B-9050 Ghen, Belgum 3 COMO Lab., Vre Unverse Brussel, Plenlaan 2, 1050 Brussels, Belgum E-mal: ngng.lu@vub.ac.be, an.lemere@vub.ac.be, francesco.carella@vub.ac.be, sn.meganck@vub.ac.be Absrac. In he doman of condon-based manenance (CBM), perssence of machne saes s a vald assumpon. Based on hs assumpon, we presen an mproved Hdden Markov Model (HMM) learnng algorhm for he assessmen of equpmen saes. By a good esmaon of nal parameers, more accurae learnng can be acheved han by regular HMM learnng mehods whch sar wh randomly chosen nal parameers. I s also beer n avodng geng rapped n local maxma. The daa s segmened wh a change-pon analyss mehod whch uses a combnaon of cumulave sum chars (CUSUM) and boosrappng echnques. The mehod deermnes a confdence level ha a sae change happens. Afer he daa s segmened, n order o label and combne he segmens correspondng o he same saes, a cluserng echnque s used based on a low-pass fler or roo mean square (RMS) values of he feaures. The segmens wh her labelled hdden sae are aken as evdence o esmae he parameers of an HMM. Then, he esmaed parameers are served as nal parameers for he radonal Baum-Welch (BW) learnng algorhms, whch are used o mprove he parameers and ran he model. Expermens on smulaed and real daa demonsrae ha boh performance and convergence speed s mproved. 1. Inroducon In modern ndusry, he demands of low cos, hgh relably and human safey are hghly ncreasng, herefore effecve manenance sraeges o ncrease profably and compeveness play an mporan role n ndusry. Condon Based-Manenance (CBM) s a decson-makng sraegy based on real-me dagnoss of mpendng falures and prognoss of fuure equpmen healh [1]. Dagnoscs s an assessmen of he curren saus of a sysem. I deecs errors and fauls based on he observed abnormaly of he sysem. Prognoscs deal wh he faul and degradaon predcon before hey occur. Dagnoscs and prognoscs can be performed separaely, however a combnaon of boh asks can decrease coss and mprove effcency and accuracy of he resuls, herefore dagnoscs can be used as a bass and pre-processng sep for prognoscs n hybrd approaches. In hs paper, we focus on daa drven dagnoscs whch am a ransferrng he daa colleced by he sensors no relevan models. A sascal approach - Hdden Markov Model (HMM) has been successfully appled o several applcaons n boh academc and engneerng felds [2]. Mos ofen, an HMM s bul and raned wh randomly nalzed parameers, whch ncreases he rsks of naccuracy by geng rapped no local maxma. The proposed mehod esmaes nal parameers of HMM whch are assumed close o he opmal ones. By he naure of ndusral machnes and equpmen, he healh saes can be safely assumed o reman sable (.e. perssen) durng he whole lfe me, whereas n oher applcaon domans of HMM, saes can flucuae que heavly. In hs paper, we focus on perssen saes. Based on hs assumpon, he sgnal can be spl no several large segmens by a segmenaon algorhm, where
each segmen corresponds o a regme wh relavely saonary behavour. In general, he saes are dfferenaed by he dfferences of her observaons,.e., P( O S ) P( O S ), and represen dfferen saes. In order o denfy he dfferen regmes based on her dfferen behavours,.e., deermne changes of regmes, change deecon echnque s requred. The Suden -es [3] s a smple paramerc echnque o deec a sgnal change, bu only addresses he changes of he mean value. Regresson analyss [4] can also be appled for change deecon bu s dffcul o deal wh small devaons. Bayesan analyss [5] s an opon as well however requres he mahemacal model of he daa known n advance. All he assumpons made by he paramerc echnques canno be always me n real applcaons. Therefore, he non-paramerc echnques such as Mann-Kendall [6] and cumulave sum chars (CUSUM) [7] are preferable and more suable n sequenal analyss [8]. Mann-Kendall echnque evaluaes he sgn of all parwse dfferences of observed values and s wdely used n clmae change analyss, whle CUSUM deecs sysemac changes over me and has he advanages of s smplcy, he graphcal nerpreaon of resuls and s able o deec abnormal paerns, ec [8]. CUSUM has been successfully used n faul deecon and change deecon n mechancal sysems [7]. Here we use change-pon analyss mehod proposed by Taylor [9] whch uses a combnaon of CUSUM and boosrappng echnques. I deecs change-pons when a change occurs n a sgnal, where he change-pons can be seen as sae ranson pons. Based on hs mehod, he observaons can be segmened a he deeced change-pons. Moreover, n order o combne he homogenous segmens, cluserng algorhms are used on he low-pass flerng or roo mean square (RMS) values of he feaure daa. As a resul, each segmen s clusered and labelled, represenng a dfferen hdden sae. These learned hdden saes can be used o compue he nal parameers of an HMM model. The remander of he paper s as follows. Secon 2 gves an overvew of he basc heores of HMM and he problems wh he learnng mehod. Secon 3 shows he basc deas of he proposed mehod and summarzes he approach mehodology of hs paper. Secon 4 shows he deals of he algorhm of change-pon analyss and he esmaon of nal parameers. Smulaon and expermenal resuls are llusraed and analysed n Secon 5 and Secon 6. In he end, conclusons are gven n Secon 7. 2. Hdden Markov Model 2.1. Theory A Hdden Markov Model (HMM) s a doubly sochasc process. The underlyng process s characerzed by a Markov chan and unobservable (hdden) bu can be observed hrough anoher sochasc process whch ems he sequence of observaons. An example of an HMM s shown n shown n fgure 1. Fgure 1. A Hdden Markov Model. An HMM s descrbed by he followng parameers [2]:, he number of saes n he model. The ndvdual saes are denoed as S { S1, S2,, S };
M, he number of dsnc observaon symbols per saes. The ndvdual symbols are denoed as V { V1, V2,, V M }; A { a } he sae ranson probably dsrbuon, where a P( q S q S ),1,. (1) 1 B { b( Vk)}, he observaon symbol probably dsrbuon n sae S, where b ( V ) P( o V q S ),1,1 k M. (2) k k { }, he nal sae dsrbuon, where P( q S ),1. (3) where o and q are he observaon and sae a me respecvely. Le { o1, o2,, o T } denoe a sequence of all observaon symbols up o me T where an observed pon o s aken a me and o V(1 T). The acual sae sequence up o me T s denoed by{ q1, q2,, q T }, where a sae q S(1 T). For convenence, a compac noaon ( AB,, ) s used o represen an HMM model. Varous ypes of HMMs exs, and n hs paper, only ergodc HMMs wh dscree symbol observaons are consdered. In real-world applcaons, hree basc problems relaed o HMMs have been denfed and solved [2]. Evaluaon: Gven an HMM and a sequence of observaonso { o1, o2,, o T }, wha s he probably PO ( ) ha he observaons are generaed by he model? Decodng: Gven an HMM and a sequence of observaonso { o1, o2,, o T }, wha s he mos lkely sequence of saesq { q1, q2,, q T }, n he model ha produced he observaons? Learnng: Gven a model and a sequence of observaons, how o adus he parameers ( AB,, ) n order o maxmze he probably of he observaons gven hs model PO ( )? For he learnng problem, here s no analycal soluon; however, a locally maxmzed parameer can be acheved wh an erave procedure such as he Baum-Welch (BW) mehod (.e. Expecaon- Maxmzaon algorhm). The probably of beng n sae S a me and he sae S a me 1 gven he model and observaon sequence s defned by [2]: (, ) P( q S, q S O, ) Where 1 1 ( ) ab ( o 1) 1( ) PO ( ) 1 1 ( ) a b ( o ) ( ) Thus, he probably of beng n sae S a me s: 1 1 ( ) a b ( o ) ( ) 1 1 1 ab o 1 ( ) ( ) ( ) (5) ( ) a b ( o ) ( ) (6) 1 1 1 (4)
The expeced number of ransons from S s where T 1 1 ( ) (, ) (7) T 1 1 S s () ; he expeced number of ransons from S o 1 (, ). The re-esmaed parameers of an HMM can be calculaed by formulas as below: a () (8) 1 T 1 (, ) 1 T 1 1 T () k 1 ( k ) T b V ( ) ( o, V ) 1 1 1 () (9) (10) ( ) max[ ( ) a ] b ( o ),2 T,1 (11) 2.2. Falure of learnng mehod The dea of he BW learnng mehod s ha he nal parameers are randomly generaed a frs and hen he parameers are eravely re-esmaed as long as he new model has a beer log-lkelhood han he prevous one,.e., P( O ) P( O ), unl converges. Rabner [2] poned ou ha has he rsk ha he log-lkelhood converges o local maxma, whch s shown n fgure 2. The learnng s repeaed 10 mes, and each me random nal parameers are chosen. Fgure 2. Convergence of log-lkelhood values.
In hs expermen, he ranng daa conssed of 10 sequences of 1000 observaons generaed by a randomly generaed HMM. The parameers of he generaed HMM model are shown n able 1. Table 1. Parameers of he generaed HMM model. Inal probables Transon probables Observaon probables 0.33 0.33 0.33 0.990 0.010 0 A 0.005 0.990 0.005 0.001 0.001 0.998 0.950 0.050 B 0.478 0.522 0.244 0.756 The daa se s always he same one for each run, however due o he randomness of he nal parameers generaed by BW learnng algorhm; he resuls are dfferen each me. Some runs sar wh beer nal parameers and end earler, and some sar wh bad parameers whch end laer. In fgure 2, here s one run whch falls no local maxma wh a log-lkelhood around -0.68, whle oher 9 runs reach a beer resul around -0.55. 3. Overvew of Mehodology 3.1. Assumpons The assumpons made by he proposed mehod are gven below: The HMM model s assumed o be an ergodc HMM, meanng ha he hdden saes are fullyconneced, n oher words, each sae can say a he same sae or ransfer o any oher sae a he nex me samp. The ergodc HMM allows revsng prevous saes. Ths s a proper assumpon when dealng wh machnes or equpmen whch can be repared durng her lfeme. The observaons daa are dscree bu non-caegorcal. Saes are assumed o be perssen (.e., P( S S 1 ) s relavely hgh), nsead of flppng frequenly. Ths assumpon s reasonable for real ndusry machnes or equpmen whch s generally conssen and keeps a one sae for a whle. Ths propery wll be exploed o denfy he regmes ha correspond o hdden saes. The number of saes s assumed o be known, for our es we use 3 saes, whch sands for normal, bad and serous damage condons of machnes. The proposed mehod works for an arbrary number of saes. 3.2. Basc dea: recognzng regmes As menoned n he prevous secon, each hdden sae of an HMM model s assumed o be perssen (.e. self-ranson probably P( S S 1 ) s hgher han a hreshold, for example, 0.9) n hs paper, herefore, he sgnal can be spl no mulple large regmes. In order o recognze varous regmes, one way s o deec where he changes occur (.e., P( O S ) P( O S ) ), where and represen dfferen saes and s a hreshold, meanng he observaon dsrbuons change when movng from one sae o anoher). Segmenaon echnques are used o deec such changes n behavour, where each segmen s supposed o belong o one sae. Moreover, n order o combne he regmes ha belong o he same saes, a cluserng mehod s appled n order o group he regmes ha belong o he same sae. Fnally, he learned pah of regmes can be seen as a hdden sae pah, whch can be furher used o learn he parameers of an HMM model. Fgure 2 already shows ha beer nal parameers wll converge faser o he opmal resul, herefore, our mehod whch learns
beer nal parameers wll mprove he convergence speed of he mehod and lkely also avods local maxma. 3.3. Scheme Wh he above dea, he learned parameers can be used as nal parameers of BW learnng algorhm o avod he fauls caused by sarng wh random parameers. The nenon of hs preprocessng sep s o opmze he BW learnng and lower he probably of geng suck n local maxma. The whole assessmen scheme s shown n fgure 3. Fgure 3. Scheme of performance assessmen. Frsly, useful feaures of he sensors' sgnals are exraced. These feaures are no only used as ranng and esng daa, bu also appled for he parameers esmaon. The selecon of he nal parameers follows he seps as below: Segmenaon: uses Taylor [9] change pon analyss o deermne wheher a change has aken place, hen segmen he observaon daa where here s a change. Feaure exracon: mean values of he low-pass flered values or he RMS values nsde each segmen are used as he feaures o base he cluserng on. Cluserng: segmens wh K-means, hus each cluser represens a hdden sae. Learn parameers: learn he parameers of he HMM defned by he number of saes resuled from he cluserng algorhm. Fnally, a comparson beween HMM models wh and whou random nalzaons are esed and compared wh log-lkelhood as an ndcaor of he performance. 4. Algorhm deals 4.1. Change-pon deecon Taylor [9] proposed a change-pon analyss mehod whch s capable of deecng mulple changes. I eravely uses a combnaon of cumulave sum chars (CUSUM) and boosrappng echnques o deec he changes. CUSUM s defned as: le X1, X 2,, X represen he daa pons and S0, S1,, S be he cumulave sums. Le X be he average values of he whole daa se, and he sar of he cumulave sum a zero be S 0. The cumulave sums are calculaed by addng he dfference beween he curren value and he average of he prevous sum, whch s S S 1 ( X X ), where 1,2,,. A sudden urn (.e. peak) n CUSUM char ndcaes a change n he daa ses, and boosrap analyss s used o deermne he change (or nflexon) wh a confdence level. An esmaor of he magnude of he change s defned by he dfference beween he maxmum ' ' ' and mnmum values of he CUSUM resuls,.e., Sdff Smax Smn. Le X1, X 2,, X be a ' ' ' boosrap sample whch s generaed by reorderng he orgnal values. Le S0, S1,, S represen he
CUSUM of he boosrap. The dfference of he boosrap CUSUM s denoed by S ' S ' S '. A large number of boosraps are performed and le X denoe he number of boosraps for whch ' X Sdff Sdff. The confdence level s calculaed by 100 %, and ypcally 90% and 95% confdence s requred. To deermne when he change occurred quanvely, wo esmaors proposed by Taylor [9], whch are: CUSUM esmaor: Sm max S (12) where m 0, S s he pon furhes from zero n he CUSUM char. Mean square error (MSE) esmaor: m 2 2 1 2 1 m1 (13) MSE( m) ( X X ) ( X X ) Afer a change s deeced, he daa se s spl no wo segmens where on boh sdes, he same procedure s repeaed. As a resul, mulple changes are deeced and he daa se s dvded no several segmens. An example s shown n fgure 4. dff max mn Fgure 4. CUSUM based change-pon deecon In fgure 4, he sold lne on he op represens he hdden saes. The sold lne n he mddle represens he bnary observaons whch have dfferen probables whn varous regmes. In hs case, he values are weghed values n order o show more clearly n he graph. The doed lne a he boom s he CUSUM char of he observaons. A sudden urn of he CUSUM means ha a change of regmes occurs. The red doed vercal lnes are he changes deeced by change-pon analyss mehod. 4.2. Cluserng Afer he observaons are segmened, k-means cluserng s appled o combne and label each segmen. In he es, funcon kmeans ( X, k ) of Malab s used, where X s he npu marx and k s he number of pre-defned clusers. These segmens are assumed o be he hdden saes of an HMM, based on whch he nal parameers can be calculaed by smple counng mehod. K-means
cluserng echnque s used here o label he saes o denfy smlar saes. An example of applyng a k-means cluserng wh 3 saes s shown n fgure 5. Fgure 5. K-means cluserng wh 3 saes. The daa se s generaed wh one sequence of 1426 daa pons, wh 3 saes and bnary observaons. Here n fgure 5(b) he daa was ploed wh weghed values n order o be seen clearly. Based on he mean values of he low-pass flerng of each segmen, K-means cluserng s appled. The resulan 3 clusers are shown n fgure 5(a). Ths cluserng resul s used o label each segmen, resulng a learned (assumed) pah of he hdden saes shown a he boom of fgure 5(b). oe ha wh general K-means cluserng he nal parameers are randomly generaed whch mgh lead o ncorrec cluserng and gves compleely wrong clusers. For example, he op wo nodes n he cluser 2 n he mddle are somemes wrongly clusered no cluser 3 on he op. In he proposed mehod, one exra sep s added n he K-means cluserng o check he resuls. If he maxmum dsance nsde each cluser s larger han wce he mnmum dsance beween clusers, hen we redo he K-means cluserng from he begnnng. Ths check sep s repeaed a maxmum number of mes, here we used maxmum 5 repeons. Expermens show ha each me he maxmum dsance nsde each cluser s smaller han he mnmum dsance beween clusers; herefore, usng wce he mnmum dsance for checkng s no crcal n our es. 4.3. Esmaon of nal parameers HMM parameers (.e. probably marces) can be calculaed by smple counng. Le S represens he curren sae, S represens he nex sae. Le S represen he se of all saes and O he se of observaons. The symbol #( ) represens he number of. The nal parameers can be calculaed as below: Inal sae dsrbuon: #( S1 1) #( S1 2) #( S1 3) (14) #( S1) #( S1) #( S1) Transon probably dsrbuon: #( S 1 S 1) #( S 2 S 1) #( S 3 S 1) #( S 1) #( S 1) #( S 1) #( S 1 S 2) #( S 2 S 2) #( S 3 S 2) A (15) #( S 2) #( S 2) #( S 2) #( S 1 S 3) #( S 2 S 3) #( S 3 S 3) #( S 3) #( S 3) #( S 3)
Observaon probably dsrbuon: #( O 1 S 1) #( O 2 S 1) #( O 3 S 1) #( S 1) #( S 1) #( S 1) #( O 1 S 2) #( O 2 S 2) #( O 3 S 2) B #( S 2) #( S 2) #( S 2) #( O 1 S 3) #( O 2 S 3) #( O 3 S 3) #( S 3) #( S 3) #( S 3) (16) 5. Smulaon sudy Smulaed sgnals are generaed by randomly-generaed HMM models and used o evaluae and compare he assessmen performances of HMM models wh random and non-random nal parameers. For smplcy, he number of saes are se o be hree and he observaon daa are generaed wh bnary daa (.e., only 0s and 1s). To make sure he saes are sable, we predefne he range of self-ranson probables are above 0.95. Tweny sequences of 500 observaons are generaed as ranng daa by samplng he HMM. The parameers of he generaed HMM model s shown n able 2: Table 2. Parameers of he generaed HMM model. Inal probables Transon probables Observaon probables 0.33 0.33 0.33 0.97 0.02 0.01 A 0.00 0.99 0.01 0.03 0.01 0.96 0.76 0.24 B 0.46 0.54 0.43 0.57 To learn he daa se and compare he resuls wh boh he radonal HMM learnng and segmenaon-based HMM mehod, he es was repeaed 5 mes, he resuls are shown n fgure 6: Fgure 6. Comparson of he learnng mehods. In fgure 6, he log-lkelhood of he 5 runs usng segmenaon-based nalzaon are ploed wh asersks, whle he radonal HMM learnng are ploed wh crcles. The number of repeons and log-lkelhood values are shown n able 3.
The resuls show ha n general segmenaon-based HMM learnng has a beer log-lkelhood and needs fewer eraon seps han randomly nalzed one. Moreover, here s one whn he randomly nalzed HMM learnng whch s rapped no a local maxma of -0.6746 whn 5 seps. Table 3. Comparsons of log-lkelhood and number of repeons. Repeons 1 2 3 4 5 Log-lkelhood HMM -0.5522-0.5523-0.6746-0.5535-0.5510 Log-lkelhood Seg-HMM -0.5389-0.5389-0.5389-0.5389-0.5389 Log-lkelhood Dfference -0.0133-0.0135-0.1357-0.0147-0.0121 um Ieraons HMM 20 20 12 20 20 um Ieraons Seg-HMM 8 8 7 8 7 um Ieraons Dfference 12 12 5 12 13 To furher compare he resuls, a smlar es s conduced wh varyng amoun of daa szes and random parameers. The amouns of sequences are se from 10 o 50 wh an ncreasng sep of 10, and whn each sequence, he szes of he daa pons are se from 100 o 500 wh an ncreasng sep of 100. Therefore, n oal 25 daa ses are generaed randomly wh 3 saes and bnary observaons. Fgure 7 shows he comparson of he resuls wh boh mehods: Fgure 7. Comparsons wh varyng amoun of daa szes. From fgure 7, we can see ha n general segmenaon-based HMM uses fewer number of eraons and has beer log-lkelhood values. The average values of he number of eraons and log-lkelhood are calculaed and compared n he able below: Table 4. Comparsons of log-lkelhood and number of repeons. Mehods HMM Seg-HMM Dfferences Log-lkelhood -0.5846-0.5503-0.0343 um of Ieraons 17.12 9.24 7.88 6. Expermenal sudy The proposed mehod was raned and esed on he bearng daa of ASA [10]. Four bearngs were nsalled on one shaf. The angular velocy was kep consan a 2000 rpm and a 6000 lb radal load
was appled ono he shaf and bearngs (fgure 8). On each bearng wo acceleromeers (one horzonal X and one vercal Y) were nsalled for a oal of 8 acceleromeers o regser he acceleraons generaed by he vbraons, where he samplng rae was fxed a 20 khz [11]. Fgure 8. Bearng es rg and sensor placemen llusraon [12]. Bearng 3 n es 1 s consdered faled a he end of s assocaed hsory. In he es, four condon monorng daa hsores relaed o bearng 2 and 3 n es 1 are used as ranng se. The monorng hsory relaed o bearng 3 n es 1 s used as es se. The es resul s shown n fgure 9. To ransform he connuous sgnal no dscree one, he vbraons hsores are processed wh roo mean square values (fgure 9(a)) and quanzed (.e. rounded) o he neares neger (fgure 9(b)). Quanzed daa are hen used as observaons, based on whch boh radonal HMM learnng and segmenaonbased HMM learnng mehods are appled o learn he hdden saes, shown n fgure 9(c) and fgure 9(d) respecvely. Fgure 9(c) shows a frequenly flucuaed pah whle fgure 9(d) shows a raher perssen pah. oe ha for all models he numbers of he saes are assumed o be hree. Fgure 9. Bearng performance assessmen. Tradonal HMM learnng uses 8 eraons and acheves a log-lkelhood of -0.0443, whle segmenaon-based HMM learnng uses 4 eraons and ges a log-lkelhood of -0.0135. Ths shows ha he proposed mehod has a beer log-lkelhood and a relavely faser speed for learnng.
7. Concluson The proposed mehod conans he followng hree seps: 1. esmaon of nal parameers (wh segmenaon and cluserng echnques), 2. learnng and 3. evaluaon. Frsly, o exac useful feaures from he daa obaned from he sensors, change-pon analyss mehod whch uses a combnaon of cumulave sum chars (CUSUM) and boosrappng echnques s appled for daa segmenaon. I deermnes wh a confdence level ha a sae change happened nsead of an accdenal change n behavor. The resulan segmens can be seen as correspondng o dfferen saes of he equpmen. To label and combne segmens belongng o he same sae, a cluserng algorhm s used on he lowpass flerng or roo mean square (RMS) values of he feaure daa. These feaures relaed o her hdden saes are aken as evdence o esmae he parameers of an HMM. Moreover, he esmaed parameers are served as nal npu model parameers for he radonal Baum-Welch (BW) learnng algorhm, whch s used o re-esmae parameers and ran he model. In he las sep, he learned model s used o assess he equpmen condons for furher manenance. The benefs of hs approach s ha pre-processes he daa wh an nellgen bu smple way o learn nal parameers, whch s more accurae han regular BW learnng mehods wh randomly generaed nal parameers. Log-lkelhood values are used o descrbe he dagnosc performances of he models and compare he models. Moreover, he exraced feaures, on whch he learnng can be based on, provde evdence for he learned model whle radonal learnng s only based on a global score. We have shown ha for sysems wh perssen saes, segmenaon of he observaons s possble by change-pon deecon snce sysems behavor changes from one sae o anoher. Based on segmenaon, nal parameers can be esmaed. The resuls have been emprcally proven on boh a es case as well as an ndusral benchmark. I s shown ha he proposed mehod o selec nal parameers speeds up he learnng of he HMM models wh fewer eraons and beer log-lkelhood values n general. Beer log-lkelhood means beer accuracy. In he fuure work, we would lke o es on dfferen HMMs, such as Hdden sem-markov Model [13, 14, 15]. The resuls wll be verfed o see wheher reaches global maxma n he learnng algorhm. Moreover, oher performance ndcaors oher han log-lkelhood wll be esed as well. Acknowledgmens The research repored n hs paper was suppored n par by he PhD-VUB scholarshp awarded by VUB-IRMO. Ths research was parly funded by he Prognoscs for Opmal Manenance (POM) proec (gran nr. 100031; www.pom-sbo.org) whch s fnancally suppored by he Insue for he Promoon of Innovaon hrough Scence and Technology n Flanders (IWT-Vlaanderen). References [1] Peng Y, Dong M and Zuo M J 2010 In. J. Adv. Manuf. Technol. 50 297-313 [2] Rabner, L R 1989 Proc. IEEE. 77 257-286 [3] Iman R L and Conover W J 1983 A Modern Approach o Sascs (ew York: John Wley and Sons) p 498 [4] Mongomery D C and Peck E A 1982 Inroducon o Lnear Regresson Analyss (ew York: John Wley and Sons) [5] Perreaul L, Berner J, Bobee B and Paren E 2000 J. Hydrology. 235 221-241 [6] Kendall M G 1975 Rank correlaon mehods (London: Charles Grffn & Company Lmed [7] Manly B and Mackenze D 2000 Envronmercs 11 151-166 [8] Alpp C and Rover M 2006 IEEE In. Symp Crc S (Kos: Greca) p5752-5755 [9] Talyor W A 2000 In www.varaon.com/cpa/ech/changepon.hml, onlne n 2011 [10] SF-I/UCRC Cener for nellgen manenance sysems prognosc daa reposory: Bearng daa se. In hp://.arc.nasa.gov/ech/dash/pcoe/prognosc-daa-reposory/, onlne n 2010 [12] Tobon-Mea D A, Medaher K and Zerhoun 2011 Prognosc and Sysem Healh Managemen Conference (Shenzhen: Chna) [13] Yu S Z and Kobayash H 2006 IEEE Trans Sg Proc. 54 1947-1951
[14] Dong M and He D 2007 Mach. Sys. Sg. Proc. 21 2248-2266 [15] Yu S Z 2010 Arf. Inell. 174 215-243