3 rd Research/Exper Conference wih Inernaional Paricipaions QUALIT 3, Zenica, B&H, 13 and 14 ovember, 3. EFFECT OF AUTOCORRELATIO O SPC CHART PERFORMACE Vesna Bucevsa, Ph.D. Universiy S. Cyril and Mehodius, Faculy of Economics Sopje, Republic of Macedonia Keywords: Auocorrelaion, SPC chars, saionariy ABSTRACT Saisical Process Conrol (SPC) aims a qualiy improvemens hrough reducion of variances. The bes nown ool of SPC is he conrol char. Over he years he conrol char has proved o be a successful pracical echnique for monioring process measuremens. However, is usefulness in pracice is limied o hose siuaions where i can be assumed ha successive measuremens are independenly disribued, whereas mos daa ses encounered in pracice exhibi some form of serial correlaion. The quesion ha is considered in his paper is wha conrol char mehods should be used o monior serially correlaed daa and how he signals on such chars should be inerpreed. 1. STOCHASTIC PROCESS Classical conrol chars assume no correlaion beween successive observaions of he qualiy characerisic. In his secion we define in a more precise manner wha is mean by correlaion beween repeaedly observed measuremens of a single qualiy characerisic. We need o now how o esimae he serial correlaion, in case i exiss, and how o sudy is effec on classical SPC chars. To achieve hese goals, he concep of a sochasic process is firs necessary. A sochasic process { (), I} is a family of indexed random variables, where I is called he index se. Someimes we will refer o he mechanism generaing he sochasic process simply as he process, which can be undersood in double sense of he underlying sochasic process ha he qualiy characerisic being modeled follows, or as he producion process iself, which in urn generaes he sochasic process (i.e. he qualiy characerisic). In applicaions in his paper, will relae o he discree poins of ime a which an observaion is obained by sampling (i.e. he index se is I {...,, 1,,1,,... }) and he sochasic process is said o be a discree-ime sochasic process. If I { : < < + }, he process is a coninuos-ime sochasic process. For discree-ime sochasic processes, i is cusomary o denoe hem as { }, ha is, a subscrip is used for discree-ime indices. Discree-ime processes can be I idenified by describing he behavior of is h elemen. Thus, we can wrie, for example, μ + ε Implying ha for he given process (Shewhar's model in his case) he equaion holds for all discree poins in ime. In his case, he ime beween observaion h equals Δ. 71
A sochasic process can be hough of as a funcion of wo argumens: namely, { (, w), I, w ς}, where ς is he sample space of he random variables (). I is imporan o poin ou ha for fixed I, (,. ) is a random variable; for fixed w ς, (., w) is a funcion of ime called a realizaion, or rajecory of he process. A ime series is a se of observaions, or realizaion, of a discree-ime sochasic process. In basic probabiliy heory, his is analogous o he noion of a single observed value of a random variable (y) compared o he random variable iself (). The se of all possible realizaions of a sochasic process is called he ensemble of he process. As in many oher areas of saisics, o perform valid saisical inferences from a sochasic process, we need some noion of how repeaable he underlying random experimen is under idenical condiions. For example, one such saisical inference could be o predic o where a qualiy characerisic will move in he near fuure. The only informaion available o us is a single realizaion of a sochasic process, he values recorded of he qualiy characerisic in he pas. If his process is such ha is random variables differ radically a differen poins of ime, no inference could be drawn from a single realizaion since he probabilisic properies of he process during he period of ime when he observaions were aen canno be generalized o oher periods of ime. For he class of models sudied in his paper, he noion ha we need o mae valid inferences is called saionary. A sochasic process is sricly saionary if for any ineger 1, he join disribuion of {,,..., } is idenical o he disribuion of { } 1 + τ, + τ,..., 1 + τ, where i I, i + τ I.Thus he sochasic properies of he process are unaffeced by changes in he ime origin. If in his definiion we loo a he case 1, we noe ha sric saionariy implies ha he disribuion of is he same as he disribuion for for + τ. Therefore, sric saionary implies ha he disribuion of he random variables occurring a differen poins in ime is idenical. 3. MEA AD VARIACE OF A STATIOAR PROCESS { Since for a sricly saionary discree-ime process } he probabiliy densiy funcion f ( ) is he same for all, ha is, we have ha ( ) f ( ) f [ ] yf ( y) [ ] y f ( y) dy μ E dy σ μ Var. Thus μ gives he long-run or asympoic average level of he process a every ime and gives he long run dispersion of he process a every ime. These long-run quaniies are obained over he ensemble of he sochasic process, which is he populaion from which we sample. For example, E [ ] can be hough as he level of he process a ime averaged over an infiniely large number of possible realizaions. The corresponding poin esimaes are given by 1 ˆμ y 1 1 ( ) y ˆ σ 1 where denoes he lengh of he ime series observed. σ 7
4. AUTOCOVARIACE AD AUTOCORRELATIO The prefix auo means a reflexive ac upon oneself, hus auocovariance is he covariance ha γ gives a measure of linear a process has "wih iself". The auocovariance funcion ( ), associaion (covariance) beween wo variables of he same process, and +, ha are separaed period of lags, as a funcion of. For any discree-ime process, we define, [ ] E[ ( E[ ])( E[ ])] γ, for, ± 1, ±,... Cov + + + For a sricly saionary process, he mean auocovariance reduces o [ ] E ( μ)( μ) [ ] γ, γ Cov j, j+ + for, ± 1, ±,... μ is consan for all imes, and he so he auocovariance depends only on he lag. Also, noe ha γ E μ Var. [( )] ( ) σ I is usually beer o scale he auocovariance ino a uniless quaniy by dividing by he process variance. For a saionary process, γ ρ for, ± 1, ±,... γ where 1 ρ 1. Given ha γ γ and ρ ρ (i.e., he auocorrelaion and auocovariance are even funcions), i is common o plo hese wo funcions only for posiive lags. Boh he auocovariance and he auocorrelaion a lag give a measure of he degree of linear associaion beween wo random variables of he same process ha are separaed periods. When considering relaions beween random variables, i is useful o recall he following implicaions: 1. If and + are independen, hey are uncorrelaed (i.e., ρ for all ). If and + are correlaed (i.e, ρ for some ), and + are dependen. The direcion of he implicaions is imporan; uncorrelaed random variables may or may no be independen. Correlaion is a measure of linear associaion, so here migh be some nonlinear associaion beween uncorrelaed variables. I should be poined ou ha he sandard error of he average ( ) is grealy affeced by auocorrelaion. Barle (1946) showed ha σ γ 1 + 1 γ 1 Thus if he sochasic process is compleely uncorrelaed ( γ for all ), he sandard error of he average is he usual σ /. As he (posiive) auocorrelaion increases, so does he sandard error and he poin esimae becomes less reliable. 73
Poin esimaes of he auocovariance funcion are given by he sample auocovariance funcion, compued as c 1 ˆ γ ( y )( y ),1,,... (1.6) + 1 Similarly, he sample auocorrelaion funcion is given by r c ˆ ρ,1,,... c 5. EED FOR STATIOARIT oe ha he poin esimaes, ˆ σ, c, and r are compued based on a single realizaion of he process (i.e., hey are compued based on ime-series daa). Inferences based on a single realizaion are possible because he process is saionary. A differen form of saionariy is called wealy or covariance saionariy. A process is wealy saionary if only is firs wo momens (mean and covariance) are finie and independen of ime (in such case he covariance depends only on he lag (. If each is normally disribued and he process is wealy saionary, he process is sricly saionary. As i urns ou, he ype of sochasic models we sudy are fully characerized by heir firs wo momens, so when we refer o saionariy we are referring o wealy saionariy. Wea saionariy can be beer undersood from an engineering poin of view. If he mean of he process is consan, some process engineers would say ha he process is sable. As will be seen, sabiliy is a propery of a deerminisic dynamic sysem, whereas saionary is a properly relaed o a sochasic process. 6. EFFECT OF AUTOCORRELATIO O SPC CHART PERFORMACE As menioned before, SPC conrol chars assume ha if in conrol, he process has a consan mean and is compleely uncorrelaed. An imporan pracical quesion is o invesigae wha happens wih he performance of SPC chars as he process exhibis more and more serial correlaion. As one process engineer said "almos every producion process exhibis auocorrelaion." The effec and implicaions of auocorrelaion have been opics of frequen discussion and debae in SPC lieraure. For a deailed review please refer o Knoh and Schmid (1). These issues are usually acled numerically: ae for insance he invesigaions by Johnson and Bagshaw (1974), Alwan (199), Maragah and Woodall (199), Wardell, Mosowiz and Plane (1994), Runger, Willemain and Prabhu (1995), Van Bracle III and Reynolds Jr. (1997) and Lu and Reynolds Jr. (1999). These and several oher papers have ables and graphs usually referring o he ARL, o provide evidence ha he performance of he appealing conrol schemes is severely compromised by he presence of serial correlaion. Auocorrelaion ofen reflecs increased variabiliy. Thus, he firs wo opions o consider should be o remove he source of he auocorrelaion or o use some ype of process adjusmen scheme such as hose discussed by Box and Luceno (1997) and Huner (1998). Conrol charing can be used in conducion wih process adjusmen schemes, and Box and Luceno (1997) emphasized ha he wo ypes of ools should be used ogeher. Only if hese 74
wo opions prove infeasible should one consider using sand-alone conrol chars for process monioring such as hose discussed by Lu and Reynolds (1999), Lin and Adams (1996) and Adams and Lin (1999). Posiive auocorrelaion a low lags is common because given he advances in sensor echnologies, measuremens are aen closer ogeher in ime. In discree-par manufacuring, his someimes implies ha every par is measured. Observaions ha were generaed close in ime will end o be similar, hence posiive correlaion a low lags will resul. The following wo examples illusrae he effec of auocorrelaion on he performance of SPC chars. Example: Suppose ha a process is described by Z ε + λz + λ λ Z + λ 1 λ ( ) ( ) Z... 1 1 3 + μ + Z where λ 1 and is he qualiy characerisic. There are wo exreme cases: if λ, he process is jus Shewhar's process, where he observaions are compleely uncorrelaed. If λ 1, he process is Z Z μ + μ + ε + 1 which evidenly is highly correlaed wih μ Z 1 + 1 Vander Weil considers his process as λ increases from o 1. In he realizaions of Figure 1.13b and d, a susained shif in he mean of magniude 5σ was induced a ime 1; ha is, he mean of he process changes according o μ μ μ + δσ if if > ` where in his example δ 5, 1, and μ 1. This ype of shif suddenly changes he mean of he process a ime. As can be seen from he figures, for λ, deecion should be almos immediae for any SPC char and false alarms should be infrequen. For λ. 8, he shif becomes indisinguishable from he auocorrelaion srucure. Such posiive auocorrelaion will no necessarily worsen he deecion capabiliies of he chars as measured by he ARL ou performance crierion (Goldsmih and Whifield, 1961; Lu and Reynolds, 1999, 1), bu he number of false alarms will cerainly increase. A char ha gives frequen false alarms will soon be abandoned. Example : The impac of auocorrelaion on SPC char performance was sudied by Mragah and Woodall (199), who consider he process μ + φ 1 + ε (1.8) where 1 < < 1 process. If φ in equaion (1.8) we obain a Shewhar's uncorrelaed process. The auocorrelaion funcion of his process is ρ φ, so he process is saionary as long as φ is a parameer ha allow us o define he auocorrelaion in he { } 75
φ <1. oe ha for φ 1, he process urns ou o be a random wal. In he case of posiive auocorrelaion ( < φ <1), he movemen of he process is smooher han ha of an uncorrelaed process. In he case of negaive auocorrelaion ( 1 < φ < ), he process shows a sawooh paern, which compared o Shewhar's process is much more crumpled. Maragah and Woodall (199) compued by simulaion he average number of ou-of-conrol poins ha a Shewhar char for individuals will generae i 5 observaions of process (1.8) wih.9 φ. 9 are used o se he char limis. A char for individuals or a char is one in which no subgroups are formed. The conrol limis are compued using he moving-range esimaor 1 ˆ σ MR / d, where MR i i i 1. Table 1 shows some of heir resuls. Table 1: umber of Poins Ouside Limis Generaed by a Shewhar Char for Individuals Used o Monior 5 Observaions of an AR(1) Process, Obained by Simulaion φ Average Sd. Dev. -.9.1.35 -.7.4.63 -.3.7.166 -.1.58.41..94.316.1.135.377.3.31.61.7 1.87 1.887.9 4.81 3.331 Source: Maragah and Woodall (199) oe ha no shif in he mean was inroduced while simulaing he AR(1) process. From he able i can be seen ha he number of ou of conrol signals increases wih increasing posiive auocorrelaion. For φ. 9, for example, an average of abou four signals will occur in 5 samples. This is a much higher signal rae han he adverised average of one false alarm every 37 observaions for Shewhar chars. egaive auocorrelaion reduces he number if signals, bu posiive auocorrelaion is much more common in pracice as menioned before. egaive auocorrelaion a low lags inflaes he variance and hence he conrol limis widh becomes oo large, so deecing acual shifs becomes more difficul. The eviden problem in Example is ha he limis were compued based on a variance esimae, which assumes ha he process is uncorrelaed. If φ > (posiive auocorrelaion), adjacen observaions will end o be similar and he moving-range esimaor will underesimae he variance of he process. This will resul in limis ha are oo narrow, producing many alarms compare o he uncorrelaed vase. This resul is quie general: The wihin-sample variance esimaor will underesimae he process variance in a posiively auocorrelaed process. If φ < (negaive auocorrelaion), he moving-range esimaor will overesimae he rue variance, resuling in limis ha are oo wide, so very few signals will be obained as opposed o he uncorrelaed case. oe ha his will have an impac on he deecion capabiliies of real shifs (i.e., wide limis will mae he char ae much longer o deec shifs in he process mean). 1 This esimaor was used because i is he mos common esimaor in conrol chars for individuals, bu as menioned below, i is no recommended for auocorrelaed daa. 76
7. COCLUSIOS In recen years, saisical process conrol (SPC) for auocorrelaed processes has received a grea deal of aenion, due in par o he increasing prevalence of auocorrelaion in process inspecion daa. Wih improvemens in measuremen and daa collecion echnology, processes can be sampled a higher raes, which ofen leads o daa auocorrelaion. I is well nown ha he run lengh properies of common SPC mehods lie CUSUM and X chars are srongly affeced by daa auocorrelaion, and he in-conrol average run lengh (ARL) can be much shorer han inended if he auocorrelaion is posiive. The mos widely researched mehods of SPC for auocorrelaed processes are residual-based conrol chars, which involve fiing some form of auoregressive moving average (ARMA) model o he daa and monioring he model residuals (i.e., he one-sep-ahead predicion errors). If he model is exac, hen he model residuals are independen. Consequenly, sandard SPC conrol chars can be applied o he residuals wih well undersood in-conrol run lengh properies. 8. REFERECES 1. Box, G. E. P., and Luceno, A., Saisical Conrol by Monioring and Feedbac Adjusmen, ew or, 1997. Mongomery, D. C., Inroducion o Saisical Qualiy Conrol (3rd ed.), ew or, 3. Rao, C. R., Saisics and Truh: Puing Chance o Wor, Fairland, MD. Inernaional Co- Operaive Publishing House, 1989 4. Ryan, T. P., Saisical Mehods for Qualiy Improvemen (nd ed.), ew or, 5. Shewhar, W. A., Economic Conrol of Qualiy of Manufacured Produc, ew or, 1931 6. Weherill, G.B. and Brown, D.W., Saisical Process Conrol: Theory and Pracice, Chapman and Hall, London, 1991. FIGURE 1. SIGLE REALIZATIO (DARKER LIE) COMPARED TO OTHER POSSIBLE REALIZATIOS THAT MAKE UP THE ESEMBLE OF A SIGLE STOCHASTIC PROCESS 77
FIGURE. TIME-SERIES PLOT AD SAMPLE AUTOCORRELATIO FUCTIO OF THE DATA I EXAMPLE 1, CHEMICAL PROCESS DATA FIGURE 3. TIME-SERIES PLOT AD SAMPLE AUTOCORRELATIO FUCTIO OF THE DATA I EXAMPLE, MACHIED PARTS DATA FIGURE 4. PROCESS WITH O SUDDE SHIFTS (a,c) AD WITH A SUDDE SHIFT I THE MEA OCCURRIG AT TIME 1 (b, d), EXAMPLE 3. GRAPHS (a) AD (b) DISPLA A UCORRECTED (λ); GRAPHS (c) AD (d) DISPLA A HIGHL AUTOCORRELATED (λ.8) SERIES 78