A Review on Trend Tests for Failure Data Analysis

Transcription

1 A.H.C. Tsag: A Revew o Tred Tests for Falure Data Aalyss 4 ISSN The West Ida Joural of Egeerg Vol.35, No., July, pp.4-9 A Revew o Tred Tests for Falure Data Aalyss Albert H.C. Tsag Departmet of Idustral ad Systems Egeerg, The Hog Kog Polytechc Uversty, Hug Hom, Hog Kog, Cha; E-mal: albert.tsag@polyu.edu.hk (Receved Jue ; Revsed Jauary ; Accepted 6 March ) Abstract: Tred detecto s a mportat task falure data aalyss both relablty ad mateace. Usually, detectg certa possble tred falure evets of systems or tems s the frst step of data hadlg, ad the result of tred detecto dcates some drectos for further statstcal aalyss of these data. Avalable statstcal tests commoly used for detectg treds falure evets collected over tme as well as the uderlyg hypotheses beg tested are revewed ths paper. Drectos for future research o statstcal tred tests are suggested at the ed of the paper. These suggestos wll address ssues cocerg aalyss of falure data obtaed from sgle ad multple systems. Keywords: Falure data aalyss, statstcal tred tests, homogeeous Posso process, mootoc tred, multple systems. Itroducto Relablty evaluato ad mateace decso modelg typcally volve characterzato of a physcal asset s lfetme dstrbuto. A rage of techques, such as Webull aalyss, are avalable to estmate a tem s lfetme dstrbuto. These procedures are developed o the premse that the process geeratg the falure evets s stable. That meas, statstcally speakg, all the falure tmes observed are depedetly ad detcally dstrbuted (d). I realty, ths codto may ot apply. For example, tervals betwee falures observed a reparable system may show a tedecy to decrease wth tme due to the cumulatve effect of mperfect repars successve repar cycles. A repar s mperfect f t smply returs the tem to a operatoal state stead of the as-good-as-ew codto. I aother scearo, these tervals may have show a creasg tred that could be the effect of cremetal desg mprovemet troduced successve repar cycles. It s a commo practce to model a reparable tem s tme to falure as a statoary dstrbuto of ts usage sce the last repar acto,.e., the dstrbuto s varat from oe repar cycle to aother. However, ths approach to modelg wll be approprate f the falure tme data are geerated from a ustable process. Thus, t s ecessary to detect treds the rate of occurrece of falures before ay attempt s made to characterze a tem s lfetme dstrbuto as a statoary process. Ths paper revews the commoly used graphcal techques ad statstcal tests for detectg treds sequece of falure evets, ad the uderlyg hypotheses beg tested by these statstcal procedures. Practcal ssues of applyg statstcal tred tests o sgle as well as multple systems are also dscussed. Drectos for future research o statstcal tred tests that wll address these practcal ssues are suggested at the ed of the paper.. Tred Tests. Graphcal techques Smple graphcal techques are used to help determe whether the relablty of a system s mprovg or deteroratg. These techques are partcularly useful for detfyg the salet features of the data ad for checkg the assumptos made fttg formal models to data. (Ascher ad Fegold, 984) (a) Plottg cumulatve falures versus cumulatve tme o lear paper A tred of creasg ter-arrval tme of falure evets a system dcates a mprovg system; a tred of decreasg ter-arrval tme of falure evets a system dcates a deteroratg system. (b) Estmatg average ROCOF successve tme perods Rate of occurrece of falure (ROCOF) s the rate of arrvals of falure evets occurrg at a partcular tme of operato defed as follows: N( t ) N( t ˆ( ν t) ) t t I the above formula, t < umber of falures observed from T to the ed of the th terval. < t t ad N(t ) s the total

2 A.H.C. Tsag: A Revew o Tred Tests for Falure Data Aalyss 5 (c) Duae plots Cumulatve mea-tme-betwee-falures (CMTBF) s defed as follows: ^ t CMTBF N() t where N(t) s the total umber of falures observed from T to the ed of the observato perod (T t). Duae observed that a plot of the CMTBF versus cumulatve operatg (or test) tme o log-log paper s lear, ad the resultg graph s kow as a Duae plot (O Coor, ). Ths plottg techque s typcally appled to aalyze falure data obtaed from product developmet tests coducted to expose desg weakesses, ad desg mprovemets are troduced to address these weakesses before the product s resubmtted for further testg. Thus, relablty ofte grows wth the testg program. A Duae plot wth a postve slope demostrates relablty growth. A larger slope dcates faster rate of chage of relablty after each desg mprovemet. (d) TTT Plot (Kvaløy ad Ldqvst, 998) The Total Tme o Test (TTT) plot s the most well kow graphcal techque. Barlow ad Davs (977) dscussed a TTT plot for data from reparable systems based o the NHPP (o-homogeeous Posso Process) model. The plot s based o the depedet NHPP s wth commo testy fucto λ() t s observed, ad all observatos are cotaed the tme terval [, T]. The NHPP dffers from the HPP oly that the rate of occurrece vares wth tme rather tha beg a costat. t represets the tme of the th arrval the superposed process; p(u) represets the umber of processes uder observato at tme u. t Τ () t p( u) dudeotes the total tme o test from tme to tme t. The scaled TTT plot for NHPP s s a plot of the scaled total tme o test statstc,.e., T T p( udu ) T T p( udu ) versus scaled falure umber (o the abscssa). No tred correspods to a TTT plot located ear the ma dagoal of the plot. The shape of the TTT plot dcates the type of tred that exsts the falure data (see Fgure ). The graphcal techques revewed ths secto are maly appled practce to detect possble treds a tme seres of falure evets. Whle they eable vsual judgmet of treds, these graphcal techques do ot provde statstcal evdece to cast doubt o the otred exsts hypothess,.e., the ull hypothess.. Statstcal tred tests I falure data aalyss, aother category of tred tests covers the statstcal tests for ull hypotheses agast ther alteratves. Table shows the taxoomy of these tred tests as proposed by Ascher ad Fegold (984). Fgure. TTT plots from NHPP s wth decreasg, creasg ad bathtub shaped testy fucto (from top), respectvely. Table Taxoomy of tred tests Null Hypothess HPP (homogeeous Posso process) (H) Reewal process (R) Geeral: Statoary sequece (S) Alterate Hypothess Mootoc tred (M) No-mootoc tred (N) Let X be the ter-arrval tme betwee the ( )-th ad the -th falures. A sequece of falure evets s a homogeeous Posso process (HPP) f the X s are depedet ad detcally expoetally dstrbuted,.e., the ROCOF s costat. A o-homogeeous Posso process (NHPP) dffers from HPP that the ROCOF vares wth tme.

3 A.H.C. Tsag: A Revew o Tred Tests for Falure Data Aalyss 6 Let F be the cumulatve dstrbuto fucto X (x) of X. A sequece of falure evets exhbts mootoc tred f t satsfes the codto F, for X ( x) > ( < ) F ( x) X j every, every j >, ad every x >, X ad are X j chroologcally ordered but depedet radom varables. A sequece of falure evets s sad to exhbt o-mootoc tred f there s a tedecy for successve X s to decrease or crease eve though the equalty F ( s ot fully met. X x ) > ( < ) FX j ( x j ) A reewal process s a geeralzato of HPP. That s, the X s are depedet ad detcally dstrbuted (IID). Sce X s ot restrcted to be expoetally dstrbuted, the hazard rate, h X, geeral s a ( x ) fucto of x. Whe h for all x >, the X () < ( > ) h X ( x ) system s sad to be good (bad)-as-ew after each repar. A sequece of X s s sad to be statoary f the jot dstrbuto of ay j of the ter-arrval tmes s varat uder a jump from oe falure evet to aother, for ay j. Most of the tred tests are desged for categores H-M ad R-M the Ascher ad Fegold Taxoomy. These tests are revewed as follows. (a) Category H-M Tests proposed for dscrmatg betwee a HPP, or H short, ad a process wth mootoc tred clude Laplace (Bartholomew, 955) ad MIL-HDBK-89. These two tests are troduced below. Laplace s test Fgure shows the tme seres of falure evets (dcated as dots o the tme le) observed a tmetermated test. Fgure. Tme-termated Laplace Test The test statstc for the Laplace tred test appled to tme-termated data s: t U N( t ).5 T N( t) t deotes the rug tme of a reparable tem at ts falure, where,,. Let N( t ) of falures observed up to tme t th be the total umber, ad the observato termates at tme T whe the tem s ot the fal state. Whe the observato termates at a falure evet, the test statstc for the tred test appled to such faluretermated data s: t U N( t ).5 t N( t ) U s ormally dstrbuted wth mea ad stadard devato f the ter-arrval tmes of falure evets are geerated from a HPP. Whe U s sgfcatly small (egatve), the ull hypothess of HHP s rejected, dcatg evdece of relablty growth; whe U s sgfcatly large (postve), the ull hypothess of HPP s rejected as well, dcatg evdece of relablty deterorato (Jarde ad Tsag, 6). Suppose the sgfcace level, α, of the test s set at 5%, the lower ad upper bouds of the test statstc for a two-sded test are.96 ad.96, respectvely. If the U value s wth ths rage, a Posso model ca be used to characterze the ter-arrval tmes, X s, of the observed falure evets. Laplace s test s optmal agast the alterate hypothess of o-homogeeous Posso process (NHPP) whch: ρ() t exp( α + α t) ρ () t s the ROCOF, also kow as the perl rate, of the NHPP. α, α are o-egatve parameters. MIL-HDBK-89 test (98) Ths test s for trackg relablty growth developed by the US Army Materel Systems Aalyss Actvty (AMSAA) (Ukle ad Vekatarama, ). It assumes that the ROCOF of falure evets has a perl rate of β ρ (Crow, 974). λ s a o-egatve ( t) λβt parameter, ad β s kow as the growth parameter. The hypotheses to be tested are: H : β,.e. the falure evets are geerated from a HPP, or ρ ( t ) λ H : β (NHPP) I ths test, the test statstc for falure data obtaed from a tme trucated test s: χ, where the growth parameter s ^ β. ˆβ T l t

4 A.H.C. Tsag: A Revew o Tred Tests for Falure Data Aalyss 7 t < t < < t < T. deotes the total umber of falure observatos. Uder the ull hypothess of HPP (o growth), χ has a Ch-square dstrbuto wth degrees of freedom. The test statstc for the falure trucated test s χ( ), where the growth parameter s ˆ β ˆ β t l t Ths test statstc s Ch-square dstrbuted wth ( ) degrees of freedom whe the ull hypothess of HPP s true. The ull hypothess s rejected for small (large) values dcatg a mprovg (deteroratg) system. The statstc ^ β estmates the growth parameter β. I the case of o growth, β s equal to ; whe β <, the process dcates relablty growth, whe β >, the process dcates relablty deterorato. (b) Category R-M The ull hypothess ths category s more geeral tha the oe of HPP, ad ts tests are desged to dstgush betwee a reewal process ad a mootoc tred. The Ma Reverse Arragemets Test (Ma, 945) Ths s a o-parametrc test that does ot requre ay assumpto about the falure process, other tha beg a reewal process. By coutg the reverse arragemets amog the chroologcally ordered depedet terarrval tmes X, X..., X. a reverse arragemet s sad to exst wheever X < X j for < j. Defe R m as the total umber of reverse arragemets for I,, ad j,,. Rm s calculated by comparg every ter-arrval tme wth every later ter-arrval tme. Sce a total of ( )/ comparsos are to be made, ( ) E( R reewal) 4 Uder the ull hypothess of reewal, there wll be o tedecy for earler ter-arrval tmes to be less tha, or greater tha, later oes. The varace of R m was ( + 5)( ) Var( R reewal) 7 Ma showed that uder the reewal hypothess, s approxmately ormally dstrbuted for R ( ) R * ad tabulated Pr R R for,,, ( ) /. The oe sded probablty of obtag or less R reverse arragemets wth ter-arrval tmes ca be obtaed from Ma s Table (Ma, 945). The Par-wse Comparso No-parametrc Test (PCNT) Ths s a modfed verso of the Ma Reverse Arragemet Test proposed by Wag ad Cot (5). The test statstc s: R ( ) / 4 U p ( + 5)( ) 7 R s as defed the Ma Reverse Arragemet Test. The ull hypothess that the observed evets are geerated from a reewal process s ot rejected f z α / U p z where α s the α / sgfcace level of the test. The Lews-Robso Test (Lews ad Robso, 974) Ths test s a modfcato of the Laplace s test, wth the test statstc U U LR CV ^( X ) ^ U s the Laplace test statstc, ad CV ( X ) s a estmate of the coeffcet of varato of the X s (the ter-arrval tme of the -th falure evet). 3. Tred Test for Data from Multple Systems Cox ad Lews (966) proposed a tred test for falure data collected from multple systems. Suppose multple depedet seres of falure observatos are avalable, whch ' " t, t, LL, deote the arrval tmes of the -th falure the varous seres, ' ",,... deote the umbers of evets observed the dfferet seres, the observatos o the varous seres are termated at ' " t, t, LL, respectvely, oe of whch s the tme of a falure evet. The hypothess of ths test s that the rates of occurrece, ρ(t), of evets observed all the tme seres have treds that follow the same model: ρ ( t) exp( α + βt). The test statstc for the ull hypothess of o tred,.e., β s: ' " ' ' " " ( t + t + L) ( t + t + L) U ' ' " " ( t + t ) + L Whe the ull hypothess of o tred s true, U s approxmately ormally dstrbuted wth zero mea ad ut varace. Whe U s sgfcatly larger tha, there s evdece that the rate of occurrece of falures some or all the seres may crease wth tme,.e., β > some of the models for ρ(t).

5 A.H.C. Tsag: A Revew o Tred Tests for Falure Data Aalyss 8 The value of β the model for ρ(t) that apples to falure observatos o a sgle seres ca be estmated by solvg the followg equato (Cox ad Lews, 966): t + t β exp( βt) Ths estmate of β s approxmately ormally dstrbuted wth a stadard error of: β t e β t ( ) exp( β t) Applyg the above results to test the ull hypothess of β β, the case of falure observatos o a sgle seres, the test statstc s: t + t β exp( β t) U t e β t β ( ) exp( β t) The ull hypothess that the observed evets are geerated from a process wth β β ts perl rate s ot rejected f z where α s the α / U zα / sgfcace level of the test. 4. Dscusso 4. Graphcal Versus Statstcal Tred Tests Graphcal methods ted to mask local varatos, eve whe the sample sze s relatvely large. It s dffcult to recogze the treds due to radom varatos the plot. Furthermore, these techques caot be used to estmate the cofdece terval of the tred estmate; other words, the tred detfed from these techques s ot quatfed statstcally. Statstcal tred tests, as compared to graphcal techques, place more emphass o testg sgfcace. The lterature revew preseted ths paper dcates that most of pror studes the theory of tred testg were publshed decades ago. Recet publcatos o tred testg are o applcatos of tred tests revewed ths paper, such as Musa (996). The power for detectg very specfc types of tred or for detectg a rage of alteratve hypotheses should be a crtero for selecto of the statstcal test to be used. More research o the power of tred tests wll help relablty practtoers makg formed decsos o selecto of statstcal tred tests. 4. Statstcal tred tests for data from multple systems It s oted that there are very few papers o statstcal tests for detecto of treds falure data observed o multple systems, a ssue commoly ecoutered dustry. Two categores of multple systems commoly ecoutered dustry are: (a) Falure data from smlar tems wth a Seral Number Items are maufactured ad delvered to customers. Suppose the frst falure of each tem s covered by warraty. Thus, the tmes to frst falure of these tems wll be kow whe they are retured to the customer servce ceter for repar. A fully fuctoal computerzed mateace maagemet system (CMMS) that allows for mateace formato o each tem to be recorded wll facltate the maagemet of such data. The sequece of maufacture of the retured tems ca be tracked f ther seral umbers are kow. However, mssg data ofte exst seral umber problems because some customers may ot ask for servce whe the purchased tem fals. Future research s therefore proposed to develop tred tests that ca be appled to hadle seral umber problems wth mssg data. (b) Falure data from detcal multple systems Systems are cosdered detcal whe ther desg specfcatos as well as ther workg evromets are smlar. However, stuatos commoly exst practce may cast doubts o whether these systems are deed detcal, such as: ) Desg revews o the system may result desg mprovemets. Hece, systems maufactured dfferet perods may ot be detcal relablty performace. ) Multple systems of the same desg are used dfferet workg evromets, such as drve motors used mg equpmet may also be used escalators of offce buldgs. It s dffcult to judge that these systems are detcal wthout testg. 3) Imperfect repar(s) made o systems the group. Proscha (963) suggests a meas of poolg test results by combg the test results for two or more systems. Except for the case wth HPP as ull hypothess, t s ot approprate to pool falure tmes from multple systems because the probablstc model for the pooled data s ukow, except asymptotc cases. Eve uder the HPP hypothess, t s advsable to pool test statstcs, rather tha falure tmes, because of the lack of robustess agast other ull hypotheses (e.g., Lews ad Cox (966) ad Clfford (98)). Lews ad Cox (966) argue that f a pooled test s requred, t s best to take as ull hypothess that the seres dvdually are smple Posso processes, possbly wth dfferet perl rates for dfferet seres. They used the techque troduced Secto 3 of ths paper to aalyze the falure data of ar-codtoers a umber of arcraft. After applyg the Laplace s tred test o falure data of ar-codtoer dvdual arcraft, falure data from ar-codtoers multple arcraft were pooled to test the ull hypothess that falure data for each ar-codtoer follows a

6 A.H.C. Tsag: A Revew o Tred Tests for Falure Data Aalyss 9 expoetal dstrbuto,.e., β. Procedures for estmatg the β values for dvdual seres of falure evets ad for pooled data, as well as those for determato of cofdece terval for the β estmates were also demostrated. Sce falure data may ot be expoetally dstrbuted practce, more geeral methods for tred detecto data sets obtaed from multple systems are desred. Exteso of the method proposed by Cox ad Lews (966) could be a soluto for such applcatos. 5. Cocluso Graphcal techques ad statstcal tests for detecto of treds falure data aalyss are revewed. The curretly avalable tred tests commo use are adequate for testg hypotheses categores H-M ad R-M of the Ascher-Fegold Taxoomy. Research ageda for tred tests has bee detfed wth a vew to provdg relablty practtoers wth useful procedures for selecto of statstcal tests that are sestve to detect specfc types of tred. Future research efforts that focus o tred tests of falure data obtaed from multple systems are also recommeded. Ackowledgmet The author would lke to express hs scere thaks to the Research Commttee of The Hog Kog Polytechc Uversty for facal support of the research work preseted ths paper (Project No. G-U379). Refereces: Ascher, H. ad Fegold, H. (984), Reparable Systems Relablty, Modelg, Iferece, Mscoceptos ad Ther Causes, Marcel Dekker, Ic., p.74, 78 Barlow, R.E. ad Davs, B. (977), Aalyss of Tme Betwee Falure for Reparable Compoets, I: Fussell, J.B. ad Burdck, G.R. (eds), Nuclear Systems relablty Egeerg ad Rsk Assessmet. SIAM, Phladelpha, pp Bartholomew, D.J. (955), Dscusso of Cox (955), Joural of the Royal Statstcal Socety. Seres B (Methodologcal), Vol.7 No., pp Bates, G.E. (955), Jot dstrbutos of tme tervals for the occurrece of successve accdets a geeralzed Polya scheme, The Aals of Mathematcal Statstcs, Vol.6, pp Clfford, P. (98), Some geeral commets o odetfablty, I: LeCam, L. ad Neyma, J. (eds), Probablty Models ad Cacer, North-Hollad, New York, pp.8-8 Cox, D.R. (955), Some statstcal methods coected wth seres of evets, Joural of the Royal Statstcal Socety - Seres B (Methodologcal), Vol.7, No., pp Cox, D.R. ad Lews, P.A. (966), The Statstcal Aalyss of Seres of Evets, Methue, Lodo, pp.45-5 Crow, L.H. (974), Relablty aalyss for complex, reparable systems, Relablty ad Bometry, SIAM, pp Jarde, A.K.S. ad Tsag, A.H.C. (6), Mateace, Replacemet, ad Relablty: Theory ad Applcatos, CRC Press, Boca Rato Kvaløy, J.T. ad Ldqvst, B.H. (998), TTT-based tests for tred reparable systems data, Relablty Egeerg ad System Safety Vol.6,pp.3-8 Lews, P.A. ad Robso D.W. (974), Testg for a mootoe tred a modulated reewal process, I: Proscha, F. ad Serflg, R.J. (eds), Relablty ad Bometry, SIAM, Phladelpha, pp Ma, H.B. (945), Noparametrc tests agast tred, Ecoometrka, Vol.3, pp MIL-HDBK-89 (98), Relablty Growth Maagemet, Headquarters, U.S. Army Commucatos Research ad Developmet Commad Musa, J.D. (996), Software relablty-egeered testg, Computer, Vol.9, No., pp O Coor, P.D. (), Practcal Relablty Egeerg, 4th ed. Heyde, Lodo, pp Proscha, F. (963), Theoretcal explaato of observed decreasg falure rate, Techometrcs, Vol.5, pp Ukle, R. ad Vekatarama, R. (), Relatoshp betwee Webull ad AMSAA models relablty aalyss: A case study, Iteratoal Joural of Qualty ad Relablty Maagemet, Vol.9, Nos.8/9, pp Wag, P. ad Cot, D.W. (5), Reparable systems relablty tred tests ad evaluato, Proceedgs of the Aual Relablty ad Mataablty Symposum, Alexadra, Vrga, USA, Ja. 4-7, pp.46-4 Author s Bographcal Notes: Albert H.C. Tsag s Seor Teachg Fellow of the Departmet of Idustral ad Systems Egeerg at The Hog Kog Polytechc Uversty. He has provded cosultacy ad advsory servces to eterprses ad dustry support orgazatos maufacturg, logstcs, publc utltes, healthcare ad govermet sectors o matters related to qualty, relablty, mateace, egeerg asset maagemet, performace maagemet ad assessmet of performace excellece these are also areas of hs research terest. He s the author of WebullSoft, a computer-aded self-learg package o Webull aalyss. Apart from publshg papers varous teratoal refereed jourals, he s also the author / co-author of three books o varous aspects of egeerg asset maagemet, ad two books o dustral applcatos of RFID.