Multivariate Reliability Modelling with Empirical Bayes Inference

Transcription

1 Multvarate Relablty Modellng wth Emprcal Bayes nference John Qugley and Lesley Walls Department of Management Scence, Unversty of Strathclyde, Glasgow G1 1QE, Scotland Abstract Recent developments n technology permt detaled descrptons of system performance to be collected and stored. Consequently, more data are avalable about the occurrence, or nonoccurrence, of events across a range of classes through tme. Typcally ths mples that relablty analyss has more nformaton about the exposure hstory of a system wthn dfferent classes of events. For hghly relable systems, there may be relatvely few falure events. Thus there s a need to develop statstcal nference to support relablty estmaton when there s a low rato of falures relatve to event classes. n ths paper we am to show how emprcal Bayes methods can be used to estmate a multvarate relablty functon for a system by modellng the vector of tmes to realse each falure root cause. 1. ntroducton The motvaton for ths research s based on experence of modellng system relablty n collaboraton wth the UK aerospace ndustry. The focus of our work has focussed upon relablty assessment wthn new product development. A modellng framework has been developed to provde decson support about relablty decsons durng system desgn and development (Walls, Qugley and Marshall, 2006). Our premse s that the ntal desgn specfcaton wll be nformed by nformaton from the performance of hertage systems. Ths s the case for the evolutonary desgn processes common n the aerospace ndustry. For example, changes to an exstng desgn may be n response to a weakness experenced n an earler generaton of the system, or may be motvated by the need to develop mproved functonalty through, for example, technologcal nnovaton. For an evolutonary desgn process, the demand for change may consttute a mx of reactve and proactve motvatons and hence relablty modellng wll be nformed by nformaton from the n-servce hstores of operatonal systems as well as knowledge about the lkely mpact of nnovaton on the new desgn. Durng the development phase, addtonal decsons wll be made to change the desgn n terms of, for example, component and materals selecton, board layout, manufacturng process, and mantenance polcy. Thus the model needs to capture the engneerng desgn knowledge as well as relevant event hstory data about potental faults, often referred to as concerns, wthn the new desgn that may result n relablty problems n servce f not removed or the effects mtgated. A key characterstc of our model s the de-couplng of the engneerng concern, whch may or may not be realsed as a fault, and the condtonal dstrbuton of the tme untl realsaton of the concern assumng t to be a fault wthn the desgn. The former s nferred from expert elctaton processes to obtan pror probablty dstrbutons, whle the latter s nferred from data for hertage systems that are smlar n desgn or expected envronmental exposure. Wthn the modellng process we map each engneerng concern to a root cause, whch descrbes the characterstcs of the falure should t occur. As such we requre a probablty 1

2 dstrbuton for each class of root cause. Typcally there are many classes and few events. Fully Bayesan approaches to such a problem would requre engneers to assess, not only the lkelhood of an dentfed concern beng realsed, but also the tme untl t wll be realsed n operatonal age. Our experence has shown the elctaton of such tmes to be problematc, wth engneerng experts beng vague or uncomfortable makng such assessments. Ths s partly due to the way n-servce performance data s fed back to the desgn engneer. Here we explore an emprcal based soluton to ths problem. An emprcal Bayes based methodology provdes a sound bass for constructng condtonal dstrbuton functons for each root cause. Broadly, ths method ntally assesses a dstrbuton for the tme to falure for all causes by poolng all data and subsequently assessng a covarance structure between root causes to permt adustments from the pool for each specfc root cause. Ths leads to each root cause havng a unque dstrbuton functon. The covarance structure s obtaned by constructng what could be consdered emprcal pror dstrbutons, whch are updated n the usual Bayesan manner to adapt to the specfc root cause. Specfcally, we use a multnomal dstrbuton to capture the samplng varablty, where for each root cause s parttoned nto tme nto ntervals wth each nterval assgned a parameter to measure the lkelhood that should such a fault exst wthn a desgn then t would be realsed n that tme nterval. The set of probabltes for any root cause are constraned to le wthn a smplex. The Drchlet dstrbuton s regarded as a convenent generc pror for the vector of probabltes wthn each root cause. Moreover, we assume that the vectors of probabltes across root causes are ndependent and dentcally dstrbuted from a Drchlet dstrbuton. Thus the number of events realsed for a specfc root cause are condtonally ndependent n relaton to another root cause. Takng the expectaton of the multnomal dstrbuton wth respect to the Drchlet measure provdes a probablty measure for each root cause whch s ndependent and dentcally dstrbuted. Usng ths dstrbutonal form a lkelhood functon s constructed from whch parameter estmates and confdence ntervals can be constructed for the Drchlet pror dstrbuton. For each root cause a posteror dstrbuton s obtaned by updatng the emprcally estmated pror n the usual Bayesan manner. An llustratve example based on an ndustral case s descrbed. We explore the proposed method for constructng the relablty functons for each root cause and combnng ths wth the expert udgement descrbng the engneerng concerns. We dscuss approprate relablty statstcs and demonstrate the usual decson support. Fnally we reflect on the proposed methodology by examnng ssues concernng the classfcaton of falures and the mpact on assessng the system relablty and we dscuss the problems concernng the double countng of data wth respect to pont and nterval estmates. We consder the proposal from a practcal perspectve, wth reference to cogntve lmtatons of experts n provdng fully specfed pror dstrbutons and the need for emprcally based solutons, at least n part. 2. The Model The applcaton of Emprcal Bayes (EB) wthn the context of rsk or relablty s not new: Martz and Waller (1991) dscuss the technque generally; Vesely et al (1994) dscuss an applcaton to emergency desel generators for bnary data; Vauro (2002, 2005) use EB for estmatng the rate of common cause falures; Ferdous et al (1995) use EB to support nference for the Webull dstrbuton wthn a software relablty growth context; Grabsk and Sarhan (1996) combne splne densty estmates for pror dstrbutons wth Emprcal Bayes for nference wth the exponental dstrbuton; Vauro and Jankala (2006) use EB 2

3 wthn a Posson modellng framework; Sohn (1999) makes use of the methodology wth response surface modellng of categorcal qualty characterstcs of possble desgns; Qugley et al (2007) use t to model the rate of occurrence of ralway accdents; and Bedford et al (2006) use t to estmate probablty wthn a fault tree model. We develop a model for the tme to falure of an tem, where we assume that an tem fals due to the realsaton of an engneerng concern. Should a concern be realsed, t s consdered a fault. Each concern s classfed a pror nto a mutually exclusve and exhaustve set of root causes. The operatonal tmes experenced untl realsaton of a fault are assumed to be statstcally ndependent. The operatonal tmes untl realsaton of a fault wthn a root cause class are assumed dentcally dstrbuted. We denote the number of root cause classes by J. The operatonal tme to realsaton of faults s parttoned nto mutually exclusve and exhaustve parttons. t s assumed that there are N faults n the desgn assocated wth root cause class. We seek a pror dstrbuton on the xj matrx, denoted by P, whose (,) element s the probablty that a fault assocated wth root cause class wll be realsed n tme perod, whch s denoted by p. Therefore, we can express the probablty that an tem wll not fal by tme t 0 condtoned on the matrx P and the vector N ( N1,..., N J ) as: J P TS N, P 1 p 1 1 N (1) nference for the model s supported through hstorcal data analyss on smlar tems and through expert engneerng udgment. The expert udgement s to construct pror dstrbutons for the vector N through elctng engneerng concerns and assessng the lkelhood each wll be realsed as a fault n operaton. 3. nference We assume falure event data are avalable from smlar desgns. Denote the number of faults that were realsed n tme perod for root cause as m and denote M as the correspondng matrx of data. We obtan the followng lkelhood functon for root cause, whch s a functon of the vector P ( p1,..., p ) : m L P 1 p 1 m m (2) t s assumed that the multnomal dstrbuton can be used to represent the number of tmes to frst realse a fault wthn a desgn that wll be classfed as root cause. The Emprcal Bayes (EB) methodology wll be appled by assumng that pror to wtnessng any data, the pror dstrbuton for the vector P s exchangeable for all. Specfcally, we assume the pror dstrbuton to be the followng Drchlet dstrbuton: 3

4 a p p p p p a 1,..., 1 a 1, 0, 1, 1 1 a 0 1 (3) We apply the Drchlet pror dstrbuton (3) and take the expectaton of P wth (2) for the th root cause and obtan a new Lkelhood functon whch s a functon of the parameters n the pror dstrbuton: a a m L a a a 1 1 1,...,, 0 a a m (4) 1 1 As the dstrbuton of the number of faults exposed by root cause s exchangeable, the lkelhood functon for the data becomes the followng: J a a m L a,..., a, a a a m (5) 1 1 We then seek the Maxmum Lkelhood Estmate (ML) of denote by a. a for all usng (5) whch we 3.1 Posteror and Predctve Dstrbuton Snce the posteror dstrbuton s unque for each root cause class we use the subscrpt. However snce each class belongs to the Drchlet famly of dstrbutons, we substtute the estmates a nto the posteror to obtan the followng: a m,...,, 0, 1, 0 1 a m 1 p1 p M p p p a 1 1 a m 1 (6) The model descrbng the aleatory uncertanty wthn any root cause class s the multnomal dstrbuton. Takng the expectaton of a generc multnomal dstrbuton wth respect to the posteror dstrbuton (6) to obtan a predctve dstrbuton for the th root cause gves: 4

5 a m n a m n n1... n 1 a a m n m 1 Pr N n,..., N n N n (7) Ths can be represented as: a m n k n. 1 k1 Pr N1 n1,..., N n N n., a 0 n. n 1... n a m n k k 1 1 (8) 3.2 Combnng Expert Judgement Assume the pror dstrbuton descrbng the number of faults wthn each root cause, denoted N n has been fully specfed by an expert. Takng the expectaton of (7) wth by Pr.. respect to ths pror gves a predctve dstrbuton uncondtonal of the number of faults wthn the desgn. Care must be taken as there are restrcton on the number of realsatons wthn any tme nterval as they are constraned to sum to n.. We evaluate the probablty that the tem fals for the frst tme due to a fault wthn root cause class after tme t 0 assumng three dfferent parametrc forms: the Bnomal dstrbuton as an example of low dsperson; the Posson for medum dsperson; and the Negatve Bnomal for large dsperson. To evaluate the probablty the tem fals after tme t 0 we multply the probabltes for each root cause class. Note that as t 0 tends to nfnty for each of these classes, the probablty of tem survval tends to the probablty that no faults are n the desgn. To present a succnct closed form expresson of the probablty that the tem fals for the frst tme due to a fault wthn root cause class after tme t 0 we provde frst order approxmatons. Bnomal Pror Dstrbuton Assume the expert has specfed a Bnomal pror dstrbuton for the number of faults that wll be realsed as root cause. We express ths pror as: k n n PN n q 1 q, q 0, n 0,1,..., k n (9) Usng (1) we consder the expectaton wth respect to N to obtan the probablty condtonal on the parameters P: n 5

6 PT t P E 1 p 0 N 1 1q 1 p k N (10) From prevous results, t s known that the posteror dstrbuton for P s has a Drchlet dstrbuton. We are nterested n the dstrbuton of the convoluton of p whch has a Beta dstrbuton. Takng the expectaton of (10) results n the followng: k PT t M E 1 q p 0 p 1 1 r k k r q E t p 0 r0r p k r1 k r 1 q r0 r y0 t y a m y a m 1 Ths expresson can be approxmated by substtutng the mean of the convoluton of the probabltes drectly nto the expresson for the expectaton to gve: PT t q 1 0 M 1 1 a m a m k (11) The formula n (11) has an ntutve appeal. The rato a m a m s the MLE of 1 1 the probablty a fault wll be realsed before or durng tme t 0 and q s the probablty a concern wll be realsed as a fault. Thus the product s the probablty a concern wll be realsed as a fault wthn the frst t 0 tme perods. As there are k concerns assumed wthn a desgn, the expresson provdes an estmate of the probablty that all k concerns are realsed after tme t 0. Posson Pror Dstrbuton We consder evaluatng the probablty the tem fals for the frst tme due to a root cause fault after tme t 0 assumng the expert has provded a Posson pror dstrbuton wth mean gven P: 6

7 PT t P E 1 p 0 N 1 e p 1 N (12) As wth the Bnomal example, we approxmate the expectaton of (12) by substtutng the mean of the convoluton of p s: p 1 PT M E t e 0 p 1 e a m 1 a m 1 (13) The probablty of a fault beng realsed wthn the frst t 0 tme perods s expressed by the exponent of the exponental functon,.e. a m a m and t s multpled by the 1 1 expected number of faults wthn the desgn provdng the expected number of faults realsed wthn the frst t 0 tme perods. As such, the resultng formula s qute ntutve. Negatve Bnomal Dstrbuton The Negatve Bnomal dstrbuton can be obtaned through mxng a Posson dstrbuton wth a Gamma dstrbuton (Greenwood and Yule (1920) as cted n Johnson et al (1993)), as the followng demonstrates: n e e n! 0 P N n d n n 1, 0, 0, n 0,1,2,... n! 1 1 (14) To obtan an approxmaton of the probablty the tem wll not fal wthn the frst t 0 tme perods due to root cause, we treat n (13) as though t were a Gamma random varable and take the expectaton. 7

8 P T M E e a m 1 a m a m a m (15) 4. llustratve Example Ths example ams to show how the proposed methods can be used to estmate the relablty of a new desgn, whch s a varant of an exstng tem. The relablty statstc of nterest s the duraton of the falure free operatng tme. The exstng desgn had 171 faults exposed durng operaton. These have been classfed nto 8 dfferent root causes. The data have been obtaned from a fleet of 200 tems and the tme to the frst occurrence of a fault has been extracted for analyss. There have been a consderable number of modfcatons on the old desgn to produce the new desgn. An extensve elctaton exercse has been conducted on the new desgn, whereby several engneerng concerns have been dentfed and assessed for lkelhood of beng realsed as a fault n operaton and an assocated root cause class dentfed should a falure be realsed. Three of the eght root cause classes have been used for concerns. Tme has been parttoned nto fve ntervals. Frst we consder the pror and posteror estmates and then consder the predctve dstrbuton for the falure free operatng tme of the desgn. 4.1 Pror and Posteror Dstrbuton The MLE s of the parameters have been solved usng equaton (5) and are gven n Table 1. Table 1 MLE of parameters for pror dstrbuton Parameter MLE a a a a a

9 Table 2 provdes a summary of both the EB pror estmates of the probablty that a fault wll be realsed n the fleet wthn each of the tme ntervals and the emprcal estmate for each of the root causes classes. Table 2 Comparson of emprcal estmates and Emprcal Bayes estmates Tme Perod Emprcal Bayes Pror Estmates As the engneerng concerns that were elcted were mapped to only three root cause classes (1, 4 and 8), we shall develop the posterors for these classes only. Fgure 1 llustrates the means of the posteror dstrbutons for the probablty of realsng a fault wthn each of the tme ntervals. The pror probabltes are ncluded for comparson. Fgure 1 shows that ths approach to nference does not mpose a monotonc functon on the rate of occurrence of falures but allows natural characterstcs to be revealed through the data, such as the mode n tme perod 3. Fgure 1 Posteror and pror probabltes for realsng a fault wthn the fleet wthn each tme nterval for each root cause class 4.2 Predctve Dstrbuton Falure Free Operatng Tme We seek nference on the duraton of falure free operatng tme for the tem. We have pror dstrbuton for each of the three root cause classes. Durng the elctaton process a Posson 9

10 dstrbuton was agreed for each of the pror dstrbutons. For comparson and senstvty analyss we wll also consder the Negatve Bnomal and the Bnomal dstrbutons. Note that for ths paper we use the approxmatons presented earler for convenence. Posson Pror Table 3 summares the probablty that no tem n a fleet of 200 wll fal due to each of the root causes by the end of each of the tme perods. Table 3 shows that root cause 8 s less of an ssue that the concerns assocated wth root cause 1 and 2, whch have very smlar profles n comparson. Table 3 Probablty no tem n the fleet wll fal due gven each root cause by end of tme perod. Tme Perod As the new tem s not beng used across a fleet of 200 tems we should convert the analyss to represent the probablty of a sngle tem survvng for a specfed perod tme. Assumng each tem functons ndependently of each other we acheve ths through calculatng the 200 th root of the survval probabltes for each root cause, as the tme that the fleet frst detects a fault s a mnmum order statstc from a sample of 200. The survval probabltes for a sngle tem are summarsed n Table 4. t s clear that approxmately 4% of the fleet wll not survve the frst tme perod but two thrd of the fleet wll survve the frst 4 tme perods. Table 4 Probablty of an tem survvng each tme perod by root cause and overall Tme tem Negatve Bnomal Pror & Bnomal Pror n order to assess the senstvty of the results to the Posson pror dstrbuton, analyss s conducted wth a Negatve Bnomal pror assumng the same mean and a varance 10 tmes greater than the varance of the Posson, as well as usng a Bnomal pror dstrbuton wth the same mean but wth a varance equal to a 10 th the varance of the Posson pror. The dfference between these prors s shown n Table 5 where we record the dfference n the expected number of tems n a fleet of 200 that would be operatng beyond the specfed tme. The results show that usng the Negatve Bnomal pror wll ncrease the expected number of tems survvng although not greatly. Wth the excepton of tme perod 4, the dfferences were less than 1 tem. For the Bnomal pror dstrbuton the reverse occurs whereby the number of tems expected to survve s slghtly fewer than under the Posson model. 10

11 Table 5 Arthmetc dfference n expected number of tems n a fleet of 200 survvng specfed tme perods Tme Perod Dfference Negatve Bnomal Pror Dfference Bnomal Pror E Summary and Conclusons A methodology has been presented for estmatng the relablty of a varant desgn based upon the ntegraton of hstorcal data for the operatonal experence of the orgnal desgn and expert engneerng udgement to express the dfferences between the desgn varants through the dentfcaton of potental faults and assocated lkelhoods. The problem whch we consder s one where few observed falures are recorded for operatonal tems and hence we are challenged to fnd robust nference. An Emprcal Bayes methodology for supportng statstcal nference has been developed. The lterature suggests that the estmates resultng from the EB methodology are more accurate than tradtonal statstcal methods. The methodology s consdered approprate for the specfed problem due to the multtude of possble root causes that may exst. Ths leads to the possblty of poolng data for accuracy through the constructon of emprcal prors, whle adustng the pooled estmate for each root cause separately to result n a unque dstrbuton for each root cause. The approach we develop s an mprovement over usng the raw data on each root cause because of the ntrnsc smoothng performed on the data. Consder the raw probablty estmates n Table 2 where seven of the eght root causes had no observatons beyond the fourth tme perod. Ths creates a sharp fnte support emprcal dstrbuton for the tme to realse faults wthn these root causes, whle two of the root causes would only permt faults to be realsed wthn one tme perod. The usefulness of smoothng data to mprove nference s supported n the lterature. t can be argued that the EB approach developed s an mprovement over parametrc modellng of the rate of occurrence of faults wthn operaton because most models wthn the lterature propose a smooth monotoncally changng ntensty functon, whle we propose a non-parametrc model through the multnomal dstrbuton. The llustratve example provdes evdence of a b-modal ntensty functon where tme perod 1 and 3 appear to be peaks wthn the realsaton process. The accuracy of the nference supported by ths EB methodology ncreases as the stochastc behavour of the root causes becomes more homogeneous. Hence the next stage n developng ths methodology to develop data analyss technques for homogensng the root causes. References 1. Bedford T, Qugley J and Walls L. (2006) Fault Tree nference Through Combnng Bayes and Emprcal Bayes Methods. Proceedngs of the ESREL Conference, Estorl, Portugal. 11

12 2. Ferdous J, Borhan Uddn M and Pandey M (1995) Relablty Estmaton wth Webull nter Falure Tmes, Relablty Engneerng and System Safety, 50, pp Grabsk F and Sarhan A (1996) Emprcal Bayes Estmaton n the Case of Exponental Relablty, Relablty Engneerng and System Safety, 53, pp Greenwood M and Yule GU (1920) An nqury nto the nature of frequency dstrbutons representatve of multple happenng wth partcular reference to the occurrence of multple attacks of dsease or of repeated accdents, Journal of the Royal Statstcal Socety A, 83, pp Johnson N., Kotz S and Kemp A (1993) Unvarate Dscrete Dstrbutons 2 nd Edton, John Wley and Sons 6. Martz H. and Waller R (1991) Bayesan Relablty Analyss, Kreger Publshng Company. 7. Vesely WE, Uryasev SP and Samanta PK (1994) Falure of Emergency Desel Generators: A Populaton Analyss Usng Emprcal Bayes Methods, Relablty Engneerng and System Safety, 46, pp Qugley J, Bedford T and Walls L. (2007) Estmatng Rate of Occurrence of Rare Events wth Emprcal Bayes: A Ralway Applcaton. Relablty Engneerng and System Safety, 92, pp Sohn SY (1999) Robust Parameter Desgn for ntegrated Crcut Fabrcaton Procedure wth Respect to Categorcal Characterstc, Relablty Engneerng and System Safety, 66, pp Vauro JK (2002) Extensons of the Uncertanty Quantfcaton of Common Falure Rates, Engneerng and System Safety,78, pp Vauro JK (2005) Uncertantes and Quantfcaton of Common Falure Rates and Probabltes for System Analyses, Relablty Engneerng and System Safety, 90, pp Vauro JK and Jankala KE (2006) Evaluaton and Comparson of Estmaton Methods for Falure Rates and Probabltes, Relablty Engneerng and System Safety, 91, pp Walls L, Qugley J, Marshall J (2006) Modelng to Support Relablty Enhancement Durng Product Development wth Applcatons n the UK Aerospace ndustry, EEE Transactons n Engneerng Management, 53, pp