Degradation Data Analysis Using Wiener Process and MCMC Approach

Transcription

1 Engneerng Letters 5:3 EL_5_3_0 Degradaton Data Analyss Usng Wener Process and MCMC Approach Chunpng L Hubng Hao Abstract Tradtonal relablty assessment methods are based on lfetme data. However the lfetme data of hgh relablty product are dffcult to obtan even by the accelerated lfe test. In ths paper a very effectve method s presented to assess the relablty va the degradaton data of product where the degradaton path of product s characterzed by mxed effect wener process model. Consderng that the mxed effect degradaton model s very complcated the ayesan Markov Chan Monte Carlo (MCMC method s used to obtan the unknown parameters and the correspondng relablty assessment s carred out. At last a numercal example about laser data s gven to demonstrate that degradaton data can provde more nformaton about the product than lfetme data and pseudo lfetme data. Index Terms Degradaton data Wener process ayesan nference MCMC T I. INTRODUCTION RADITIONAL relablty assessment methods are focused on the use of lfetme data. For the hghly relable product t s dffcult to obtan suffcent lfetme data. Compared wth the lfetme data the degradaton data can provde more lfe nformatve. Degradaton such as wear eroson and fatgue s very common for most mechancal products. In addton degradaton can be descrbed by a contnuous performance process n terms of tme. Consderng that the stochastc process model can flexbly descrbe the falure generatng mechansms and the operatng envronment characterstcs many authors have used the dfferent stochastc processes (.e. Markov chan Gamma processes and Wener processes et al. to model degradaton data such as Sngpurwalla (995 Cox (999 and Aalen (00 et al. Among those stochastc process models Wener process has been wdely studed such as Tseng et al. (003 Lee and Tang (007 Park and Padgett (006 et al. A well-adopted form for the Wener process X ( t t 0 can be ressed as X ( t t ( t ( Ths work was supported by the Humanty and Socal Scence Youth Foundaton of Mnstry of Educaton of Chna (No. 5YJCZH055 the key project of Hube provncal educaton department (No.D0770. C. P. L s wth the Department of Mathematcs Hube Engneerng Unversty Hube Chna. e-mal: lchunpng35@63.com. H.. Hao s the correspondng author wth the Department of Mathematcs Hube Engneerng Unversty Hube Chna. e-mal: haohubng979@63.com. H.. Hao s wth the Hube Key Laboratory of Appled Mathematcs Faculty of Mathematcs and Statstcs Hube Unversty Wuhan Chna. e-mal: haohubng979@63.com. where (t s the standard rownan moton μ s the drft degradaton rate σ s the dffuson coeffcent. The wener process has ndependent and normally dstrbuted ncrements.e. X ( t X ( t t X ( t s ndependent of X( t and X ( t ~ N( t t. Note that the above Wener degradaton process models do not take nto account the dfferences between ndvduals. In fact the dfferences between ndvduals can not be gnored because each tem possbly erences dfferent sources of varatons durng ts operaton. For degradaton model to be realstc the random effects should be ncorporated nto the process to represent the heterogenety. Recently Peng and Tseng (009 Wang (00 and S et al. (0 consdered the random effect Wener process and the relablty assessment of the performance degradaton product are obtaned. However they only used the MLE method to obtan the estmaton of unknown parameters and they only utlzed the current degradaton data wthout consderng the pror nformaton about the unknown parameters. In ths paper Wener process model wth mxed effect s proposed to characterze the performance degradaton path of product and ayesan nference method s used to obtan the estmaton of the parameters. Consderng the complexty of mxed effect degradaton model the estmatons of unknown parameter are obtaned by the ayesan Markov Chan Monte Carlo (MCMC method and goodness of ft measures s gven. The results show that MCMC method s better than the MLE method and the uncertanty s smaller than the MLE method. II. DEGRADATION MODEL ASED ON WIENER PROCESS Assume that the degradaton path of a product s governed by Equaton ( and ξ s predefned threshold. Gven the threshold value the product s lfetme T can be defned as T nf{ t 0 X (0 0 X ( t } ( It s well known that the lfetme T follows an nverse Gaussan dstrbuton wth probablty densty functon (PDF as ( t ft ( t 3 (3 t t Then the relablty at tme t can be ressed as t t Rt ( t (4 t To capture the dfference between ndvdual we assume that μ and (t are ndependent and assume that μ follows N(. Then we can get the mxed effect model as followng (Advance onlne publcaton: 3 August 07

2 Engneerng Letters 5:3 EL_5_3_0 X ( t t ( t (5 ~ N( where the parameter μ s a random effect representng between tem varaton and σ s a fxed effect that s common to all tems. ased on the Equaton (5 when the drft parameter μ s a random varable the PDF of the lfetme T can be reconstructed by the total law of probablty as follow ft( t ft( t d ( t 3 (6 ( t t ( t t where ( s the probablty densty functon of the standard normal dstrbuton. Then the relablty at tme t can be ressed as t Rt ( 4 t t t ( t (7 t t where ( s the dstrbuton functon of the standard normal dstrbuton. III. AYESIA INFERENCE AND MCMC APPROACH The ayesan nference s a method of estmatng the unknown parameters of a gven dstrbuton by combnng the prevous knowledge of these parameters wth the new nformaton contaned n the observed data. The prevous nformaton of these parameters s reflected by the pror dstrbuton and the new nformaton s ncorporated through the lkelhood functon then the posteror dstrbuton s obtaned about the unknown parameters. The lkelhood functon pror and posteror dstrbutons are descrbed n the followng sectons. A. Lkelhood functons of the unknown parameters Let X ( t denote degradaton measurements of product at tme t for = N j = M. In general the degradaton data can be ressed as the followng form X( t X( t L X( tm X ( t X ( t X ( tm X L (8 M M O M X N ( tn X N ( tn L X N ( tnm Set X ( t X ( t X ( t( j t 0 X( t0 0 (9 0 Accordng to the ndependent ncrement property of the Wener process X ( t has the followng dstrbuton X t N t t (0 ( ~ ( where t t t( j s the drft degradaton rate of the product. Therefore the condtonal PDF of X ( t s ( X ( t t g( X ( t t t ( When the uncertanty of s taken nto account the PDF of X ( t s gven by ( X ( t t g( X ( t t ( t ( t t ( and the cumulatve dstrbuton functon (CDF of X ( t s X ( t t G( X ( t (3 t ( t where ( s the dstrbuton functon of the standard normal dstrbuton. Then the log-lkelhood functon of unknown parameters and can be gven as N M ( X ( t t L( X j t ( t ( t t. Pror dstrbuton For smplcty we assume that the unknown parameter (4 has normal pror dstrbuton. Consderng that the unknown parameters and are postve quanttes a natural choce for the pror of each parameter has gamma pror then ( ( ( ( ( ( (5 ( ( ( ( where 3 and 3 are chosen to reflect pror knowledge about unknown parameters and. Note that when 0 3 t s correspondng to the case of non-nformatve prors. C. Posteror dstrbuton Let ( denote the unknown parameters and X denote the degradaton data. The jont posteror dstrbuton ( X s obtaned by combnng the jont pror dstrbuton of wth the lkelhood L(X η accordng to ayes theorem (Advance onlne publcaton: 3 August 07

3 Engneerng Letters 5:3 EL_5_3_0 ( LX ( ( X ( L( X ( L( X d (6 where N M ( X ( t t L( X j t ( t ( t t ( ( ( ( 3 Consderng that the jont posteror dstrbuton s very complcated the MCMC smulaton technques mplemented n ths study to numercally evaluate the posteror dstrbutons of the parameters. D. MCMC approach MCMC approach s a smulaton technque when the analytcal posteror dstrbuton s dffcult to be computed. A Markov chan s generated by samplng the current pont based on the prevous one. MCMC method works successfully n ayesan computng. y usng MCMC method t s possble to generate samples from the posteror dstrbuton and to use these samples to estmate the desred features of the posteror dstrbuton. The MCMC technques ncludng the Metropols Hastngs (M H algorthm [8 9] and the Gbbs sampler [0 ] have become very popular methods for generatng a sample from a complcated model n recent years. The Gbbs sampler s a specal case of MCMC algorthm. It generates a sequence of samples from the full condtonal probablty dstrbutons of two or more random varables. Gbbs samplng requres decomposng the jont posteror dstrbutons nto full condtonal dstrbutons for each parameter n the model and then samplng from them. From Equaton (6 we know that ( X L( X ( ( ( ( ased on the Equaton (7 the posteror nference for parameters can be obtaned but t s not easy to get the detaled results. Therefore the MCMC method wth the Gbbs sampler to carry out ayesan nference s used for the model parameters. Let (-j denote some vector wthout the jth component. Then the full condtonal can be wrtten as ( j ( L( X ( ( L( X( ( (8 ( j ( ( L( X( ( ( j ( 3 3 We used the ayesan software package OpenUGS [3] to carry out the Gbbs samplng after whch we estmated the model parameters. IV. NUMERICAL EXAMPLE In ths secton a numercal example about laser data [] s used to demonstrate the valdty of the proposed method and results. The performance characterstc of a laser devce represents ts operatng current. When the operatng current reaches at a predefned threshold level ths devce s consdered to be faled. Table I shows the plot of operatng current over tme for 5 tested unts. The measured frequency of ts current s 50 hours and the erment s termnated at 4000 hours. For example the degradaton response for the 0th unt whose degradaton s the fastest and observed every 50 hours from 0 to 4000 hours s The falure threshold ξ s 0. That s although a laser s workng at 0 t s stll perceved as beng faled. From the Table I the degradaton curves of the lasers are approxmate lnear and there are obvous dfferent degradaton path of all test unts therefore we use the mxed effect Wener process wth lnear drft model to ft the degradaton data. Pseudo lfetmes can be obtaned by fttng lnes to each degradaton curve and calculatng the tmes TALE I THE LASER DATA Operatng current when the ftted lnes reach the falure threshold (Meeker and Escobar 998; Tseng Hamada and Chao 995.If the degradaton path s descrbed by the Wener process the pseudo lfetmes are follow the nverse Gaussan dstrbuton. A. Parameters estmaton and data analyss Frstly we use the mxed effects model to ft the degradaton data. ased on the MCMC method we can get the ayesan estmaton of the unknown parameters. Table II presents posteror estmator summares for μ β σ β σ based on the samples ncludng the mean and standard devaton as well as the quantles. To test the goodness-of-ft we frstly obtan each unt s pseudo falure tme whch s the tme of degradaton path to threshold ξ =0. The emprcal CDF and the CDF obtaned TALE II POSTERIOR ESTIMATOR SUMMARIES ASED ON DEGRADATION DATA σ R( Quantles (Advance onlne publcaton: 3 August 07

4 Engneerng Letters 5:3 EL_5_3_0 from the estmated nverse Gaussan dstrbuton are smultaneously dsplayed n Fg.. From the Fgure we can fnd that the estmated falure tme dstrbuton based on the Wener process agrees well wth the emprcal dstrbuton. Then we can use the posterors estmaton results to nference about the relablty functon R(t at each tme t. For example from the posteror estmator summares the relablty at 4500 hours s ts medan value s and ts 95% credble nterval s ( respectvely. From Fg. we can see that the PDFs of the estmated lfetme for MCMC method and the MLE method have a lttle TALE III THE ESTIMATION RESULTS VIA DIFFERENT METHODS μ β σ β σ MCMC MLE dfference and the estmated PDF under the MLE method of the lfetme covers a wder range that s to say ts uncertanty s larger than the MCMC method. Fg. The emprcal CDF and the CDF of the laser data.. Comparson wth the MLE method We use the laser data to compare our methods wth the work of Peng and Tseng [] n whch the MLE s used to obtan the unknown parameters. For comparson we summarze the correspondng estmaton results of the parameters n the Table III. From the Table III we can fnd that our estmaton results are slghtly dfferences from the results n Ref []. Furthermore we obtan the PDFs of the lfetme T at the dfferent estmaton method as shown n Fg. and the correspondng relablty curves are shown n Fg.3. Fg.3 The relablty curves of the lfetme va dfferent methods C. Comparson wth the lfetme data In ths secton by usng the laser data we compare the degradaton data wth the lfetme data to confrm that degradaton data can provde more nformaton about the product. When the threshold ξ =0 we can fnd that there are three unts (names unt unt 6 and unt 0 faled and the lfetme data consst of three nterval censored observatons ( ( and ( respectvely. Note that the degradaton s descrbed by the mxed effect Wener process and the lfetme dstrbuton s gven by the TALE VI POSTERIOR ESTIMATOR SUMMARIES ASED ON LIFETIME DATA Quantles σ R( Equaton (6. A ayesan analyss method usng the same prors for μ β σ β and σ as above the posteror summares presented n Table IV as follow: Fg. The PDF curves of the lfetme va dfferent methods Comparng the Table IV and the Table II we can fnd that the 95% credble ntervals of the unknown parameters under the lfetme data are wder than under the degradaton data. For example the 95% credble ntervals for R(4500 s now ( Ths llustrates the ncreased uncertanty as compared wth those obtaned from degradaton data. (Advance onlne publcaton: 3 August 07

5 Engneerng Letters 5:3 EL_5_3_0 D. Comparson wth the pseudo lfetme data In ths secton we compare the degradaton data and the pseudo lfetme data to verfy that degradaton data generally can provde more nformaton. When the threshold ξ =0 we can fnd that twelve unts (names unts do not reach the falure threshold. From Ref [ 6] the pseudo lfetme can be obtaned by the falure threshold ξ and the rate of degradaton ntensty μ β. Note that TALE V POSTERIOR ESTIMATOR SUMMARIES ASED ON PSEUDO LIFETIME DATA σ R( the pseudo lfetme dstrbuton s gven by the Equaton (6. Smlarly a ayesan analyss method usng the same prors for μ β σ β and σ as above the posteror summares presented n Table V as follow: Comparng the Table V and the Table II we can fnd that the 95% credble ntervals of the unknown parameters under the pseudo lfetme data are wder than under the degradaton data. For example the 95% credble ntervals for R(4500 s now ( Ths also llustrates the ncreased uncertanty under the pseudo lfetme data. V. CONCLUSION In ths paper we have shown that the degradaton data can be modeled by a Wener process model wth mxed effects and we llustrate the advantages to assess relablty va degradaton data. y usng the ayesan MCMC approach the unknown parameters of the complcated degradaton model can be obtaned and the correspondng relablty assessment s carred out. At last a numercal example about laser data s gven to demonstrate that degradaton data can provde more nformaton about the product than lfetme data and pseudo lfetme data. REFERENCES Quantles [] W. Q. Meeker and L. A. Escobar Statstcal method for relablty data John Wley & Sons New York NY USA 998. [] W. Nelson Accelerated Testng: Statstcal Models Test Plans and Data Analyss John Wley & Sons New York NY USA 990. [3] R. Pan A herarchcal modelng approach to accelerated degradaton testng data analyss: A case study Qual. Relab. Eng. Int. vol. 7 no. pp Mar. 0. [4] M. J. Zuo R. Y. Jang and R. C. M. 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