SPC Procedures for Monitoring Autocorrelated Processes

Transcription

1 Qualiy Technology & Quaniaive Managemen Vol. 4, No. 4, pp , 007 QTQM ICAQM 007 SPC Procedures for Monioring Auocorrelaed Processes S. Psarakis and G. E. A. Papaleonida Ahens Universiy of Economics and Business, Deparmen of Saisics, Greece (Received June 005, acceped December 006) Absrac: The inference abou he saisical properies of qualiy conrol mehodologies is based on he assumpions of normaliy and independence. In real indusrial environmens hough process daa is ofen correlaed or exhibis some serial dependence affecing he efficiency of Saisical Process Conrol (SPC) mehodologies. New echnology gives managers he opion of using more sophisicaed SPC models which more accuraely reflec he process being moniored, by relaxing some of he assumpions. The aim of his paper is o presen, o apply and o evaluae conrol chars ha are designed o accoun for auocorrelaion. Keywords: Auocorrelaed processes, conrol chars, SPC, ime series. 1. Inroducion A basic assumpion in radiional applicaion of Saisical Process Conrol (SPC) echniques is ha he observaions from he processes under invesigaion are normally and independenly disribued. When hese assumpions are saisfied, convenional conrol chars may be applied. However, he independence assumpion is ofen violaed in pracice. In discree as well as in coninuous producion process daa ofen shows some auocorrelaion, or serial dependence. Auocorrelaion is presen in he daa generaed by mos coninuous and bach process operaions since he value of he paricular parameer under monioring is dependen on he previous value of ha parameer. Coninuous produc manufacuring operaions such as he manufacure of food, chemicals, paper and oher wood producs ofen exhibis serial correlaion. This phenomenon can also be presen in monhly series of survey qualiy daa. I is more apparen for daa colleced wih frequen sampling bu can also be due o he dynamics of he process. For insance, observaions from auomaed es and inspecion procedures where every qualiy characerisic is measured on every uni in ime order of producion, or measuremens of process variables from anks, reacors and recycle sreams in chemical processes are ofen highly correlaed. Auocorrelaion can also be eviden in daa arising from compuer inrusion deecion (Ye e al. [163] and Ye e al. [164]). Even, small levels of auocorrelaion beween successive observaions can have big effecs on he saisical properies of convenional conrol chars. Many auhors have considered he effec of auocorrelaion on he performance of SPC chars. Johnson & Bagshaw [6] and Bagshaw & Johnson [17] derived approximae run lengh disribuion for he Cummulaive Sum Conrol Char (CUSUM) char when he process follows an Auoregressive Process AR(1) or a Moving Average Process MA(1) model. They sae ha incorrec conclusions can be drawn by using convenional CUSUM schemes in he

2 50 Psarakis and Papaleonida presence of daa correlaion. Harris & Ross [48] discussed he impac of auocorrelaion on he performance of CUSUM and Exponenially Weighed Moving-Average (EWMA) chars, and showed ha he average and median run lenghs of hese chars were sensiive o he presence of auocorrelaion. Alwan [] discussed he masking effec of special causes by he auocorrelaion of he daa, and demonsraed ha in he presence of even moderae levels of auocorrelaion, an ou of conrol poin of he char does no necessarily indicae a process change. Padge e al. [99] invesigaed Shewhar chars when he correlaion srucure of he process can be described by an AR(1) model plus a random error and found ha his ype of auocorrelaion effecs he false alarm rae. Alwan [3] discussed he capabiliy of he Shewhar chars when he observaions follow a general Auoregressive Moving Average ARMA(p,q) char. Alwan e al. [4] discussed he effec of auocorrelaion for he frequenly advocaed supplemenary runs rules. Schmid and Schone [16] proved heoreically ha he run lengh of he auocorrelaed process is larger han in he case of independen variables provided ha all he auocovariances are greaer han or equal o zero. Prybuok e al. [107] found ou ha no diagnosing and considering he correlaion in he daa leads o a decrease in he average ime o signal as he amoun of correlaion increases. Boyles [4] also gave an esimaion mehod for firs-order auoregressive common-cause model ha disinguish models of auocorrelaed common-cause variaion from he acual baseline daa. The main effec of auocorrelaion in he process daa wihin SPC schemes is ha i produces conrol limis ha are much igher han desired. This causes a subsanial increase in he average false alarm rae and a decrease in he abiliy of deecing changes on he process. Consequenly, he average number of observaions aken before an ou of conrol signal is riggered, namely he Average Run Lengh (ARL) of he conrol char is no calculaed properly. Early deecion of he occurrence of assignable causes ensures ha necessary correcive acion can be aken before a large quaniy of non-conforming produc is manufacured. Thus, when i is verified ha auocorrelaion is presen in he daa, acion should be aken o avoid is effec in he performance of he SPC echniques. A simple idea ha can break up auocorrelaion is sampling from he process daa sream less frequenly. The inefficien use of available daa hough, can lead o a decrease of he performance of conrol charing since wih limied daa i may ake much longer o deec a real process shif han wih all he daa. Beyond he simple approach of using less frequen samples, wo general approaches for consrucing conrol chars in he case of correlaed processes are developed. The firs approach uses sandard conrol chars, and adjuss boh he conrol limis o accoun for he auocorrelaion and he mehod of esimaing he process variance, so ha he rue process variance is being esimaed (see e.g. Vassilopoulos and Samboulis [146], VanBrackle and Reynolds [143], Schmid [13]. The second approach fis a ime series model o he process daa so ha forecass of each observaion can be made using he previous observaions and hen applies o he residuals radiional conrol chars or some slighly modified versions of hose (see e.g. Alwan and Robers [1], Harris and Ross [48], Mongomery and Masrangelo [90], Masrangelo and Mongomery [87], Lu and Reynolds [78]). The raional of using residuals chars is ha assuming ha he correc ime series model is fied o he daa, he residuals will be independenly and idenically disribued random variables. All he assumpions of radiional qualiy conrol will hen be me, and

3 SPC Procedures for Monioring Auocorrelaed Processes 503 hus any of he radiional SPC chars can be applied. Once a change of he mean and/or variance in he residuals process is deeced, i is concluded ha he mean and/or variance of he process iself has been changed. Thus, ploing he residuals on a conrol char provides a mechanism for deecing a process change. However, many people seem o agree ha he residuals chars do no have he same properies as he radiional chars i.e., he chars for he original observaions and ha he abiliy of a char o deec a mean shif depends on he model ha is assumed o describe appropriaely he daa. This was firs demonsraed by Longnecker and Ryan [74] and Ryan [11] where an AR(1) model is used. Addiional models were considered in Longnecker and Ryan [75] where he performance of a X-char of he residuals from an AR(1), AR() and ARMA(1,1) model is invesigaed. They poined ou ha he X-char of he residuals may have poor capabiliy o deec he process mean shif and showed ha a residuals char has high probabiliy of deecing a mean shif as soon as i occurs, bu if i fails o deec he shif immediaely, here is low probabiliy ha he shif will be deeced laer especially for an AR(1) process wih posiive auocorrelaions. To he same conclusion came Wardel e al. [150] afer deriving he run lengh disribuion of residuals char and more recenly Zhang [166], Lu and Reynolds [78] and ohers. Harris & Ross [48] also sudied he response of residuals of an Auoregressive Inegraed Moving Average ARIMA(0,1,1) and an AR(1) processes o sep shif in process mean and concluded ha he residual analysis is insensiive o he mean when he process is posiively auocorrelaed and recommended radiional charing mehods for residuals monioring. Up o our knowledge lile aenion seems ha has been given o he developmen of conrol chars in he case of correlaed aribue daa. Deligonul and Mergen [38] as well as Bha and Lal [0] assumed a wo-sae Markov chain model for auocorrelaed aribue daa. Harvey and Fernandes [49] claim ha correlaed coun daa can be modeled wih an EWMA approach. In he same conclusion came also Wisnowski and Keas [154]. Simson and Masrangelo [131] sudied he monioring of serially dependen processes wih aribues daa obained from mulisaions of producion. Lai e al. [69] examined conrol procedures based on he conforming uni run lengh applied o near-zero-defec processes in he presence of serial correlaion. Lai e al. [70] sudied he problem of process monioring when he process is of high qualiy and measuremen values possess a cerain serial dependence. Tang and Cheong [133] proposed a conrol scheme ha is effecive in deecing changes in fracion nonconforming for high yield processes wih correlaion wihin each inspecion group. Finally Shepherd e al. [17] proposed wo conrol char schemes. These conrol chars are based on a sequence of random variables ha are used o classify an iem as conforming or nonconforming under a saionary Markov chain model and 100% sequenial sampling. Seminal works o he presen review originally appeared in Papaleonida [10] and Papaleonida and Psarakis [103]. There, an exensive review on procedures for monioring auocorrelaed processes is presened. Laer, Knoh and Schmid [65] gave a review focused on heir work including a comparison sudy of some modified and residual EWMA and CUSUM schemes. The aim of his paper is o presen, o apply and o evaluae conrol chars ha are designed o accoun for auocorrelaion. Firs in secion some opics from Time Series heory are presened. Conrol chars ha are designed for monioring he mean of correlaed daa are presened in secion 3, while in secion 4 conrol chars for monioring he variance are provided. In secion 5 some resuls concerning he performance of he

4 504 Psarakis and Papaleonida proposed conrol chars are presened. Secion 6 presens mulivariae mehods for auocorrelaed daa and finally, in secion 7 some general conclusions along wih some opics for furher research are presened.. Some Time Series Models The correlaion in a process can be capured using ime series models. An imporan class of ime series models are he saionary processes, which assume ha he process remains in equilibrium around a consan mean. This ype of models can provide a framework for seeking saisical conrol when monioring auocorrelaed processes. Throughou his paper he following basic ime series models are going o be used. The firs order auoregressive moving average process ARMA(1,1) is described as follows: X = μ(1 ϕ) + ϕx + α θα, 1 1 where X is he observaion a ime, α is he random error erm a ime, ϕ is he auoregressive parameer, θ is he moving average parameer, and μ is he mean of he process. The variance of he ARMA(1,1) process is 1 ϕθ + θ σ x = σ a. 1 ϕ and he process is saionary if 1<φ<1 and inverible if 1<θ<1. When θ = 0 he above model reduces o an AR(1) process, which is saionary if φ saisfies he condiion 1<φ<1 and he variance of he process is α σα 1 ϕ σ σ x = = 1 ρϕ 1 A more complicaed scheme is he AR(1) wih an addiional random error model. The process can be described by X = μ + ε, where he ε s are independen normal random errors wih mean 0 and variance σ ε. The mean μ is no consan bu ime wandering and can be inerpreed as he random process mean a ime. I follows an AR(1) process wih mean ξ, namely μ = (1 ϕξ ) + ϕμ 1 + γ, = 1,,..., where γ are independen and normally disribued random errors wih mean 0 and variance σ γ while φ is he auoregressive parameer saisfying φ <1. If i is assumed ha he saring value μ 0 follows a normal disribuion wih mean ξ and variance σ μ = σγ /(1 ϕ ), hen his implies ha he disribuion of X is consan wih mean μ and variance σ x = σμ + σε for all =1,,...,. If σ ε = 0 he model reduces o he simple AR(1) model. The AR(1) process wih addiional random error is equivalen o an ARMA(1,1) process where ϕ is he same auoregressive parameer as in he AR(1) model used before o describe he wandering of he mean. There are equaions for expressing he parameers φ, θ,σ a in he ARMA(1,1) model in erms of he parameers φ, σ γ in he AR(1) plus a random error model and vice versa as shown in Box e al. []. In paricular, if φ>0 and σ > 0, hen he ARMA(1,1) model parameers can be derived from he AR(1) plus he ε.

5 SPC Procedures for Monioring Auocorrelaed Processes 505 random error parameers by γ + (1 + ) ε 1 γ + (1 + ) ε φσ ε φσ ε σ φ σ σ φ σ θ = 4 and σ φσ ε α =. θ Alernaely, if φ>0 hen he AR(1) plus he error model parameers can be obained from he ARMA(1,1) model parameers using he following equaions α σ ( φ θ)(1 φθ) σ γ =, φ σ θσα ε =. φ No all he ime series models encounered in pracice are saionary. There are series ha exhibi explosive or evoluionary behaviour, which is far away from saionariy, and ohers ha exhibi some homogeneiy and by supposing some suiable difference of he process can be viewed as saionary. An imporan class of linear non-saionary models are hose which aren saionary bu for which he d h difference is a saionary auoregressive moving average model. These processes are called Auoregressive Inegraed Moving Average Processes or briefly ARIMA( p, q, d ). For more informaion concerning he usage and he descripion of ime series models he reader should refer o he lieraure (see e.g. Box e al. [], Box and Luceno [3], Brockwell and Davis[5]). 3. Conrol Chars for Monioring he Mean In his secion, some of he conrol chars for monioring he mean ha aemp o deal wih he problem of auocorrelaion are presened. Firs, radiional conrol chars of he original observaions wih modified conrol limis o accoun for he auocorrelaion are implemened and hen he charing of he residuals from he fi of a ime series model o he daa along wih some modificaions of his procedure is discussed Conrol Chars Based on he Observaions A simple bu someimes effecive way o deal wih auocorrelaed daa is o implemen radiional conrol chars of he observaions using he correc variance of he process when calculaing he conrol limis. Firs he appropriae ime series model mus be fied o he daa. In he following we discuss he modified Shewhar, CUSUM and EWMA chars as well as some oher approaches The Modified Shewhar X-Char To monior he sabiliy of a process { X, T} a ime, he individuals Shewhar char compares observaion X wih he arge value of he mean μ ο. If he absolue value of he difference a ime is considered large hen he process is said o be ou-of-conrol i.e., E(X ) μ 0. Oherwise he process is said o be in-conrol and he nex observaion is aken and examined. Saisically speaking, a large magniude is always defined in erms of he sandard deviaion and hus he process is said o be ou-of-conrol if X μ > c σ 0 X, where σ X is he sandard deviaion of he process. The main idea behind he modified X-char is ha he deviaion of he observaion from he arge value of he mean is compared wih he sandard deviaion of he ime

6 506 Psarakis and Papaleonida series model ha describes he process. This sandard deviaion for a paricular ARMA(p, q) model can be calculaed following he suggesions of Box and Luceno [3] using some previous observaions from he ime he process was supposed o be in-conrol. The oher issue concerning he implemenaion of he Shewhar char of he observaions of a correlaed process is he selecion of he appropriae value of c. This can be done in erms of he desired average run lengh ARL of he char. According o Kramer and Schmid [67] for auocorrelaed processes seing he in-conrol ARL equal o a specified value, he quaniy c does no only depend on his value bu on he parameers of he process as well. For he case of modeling he process by an AR(1) ime series model Schmid [13] gave ables of c for an in-conrol ARL of 500. Kramer and Schmid [67] suggesed he use of his able when he level of correlaion is large. When he correlaion is low or moderae hey recommended he choice of c as in he independen and idenically disribued (i.i.d.) case. Wardell e al. [149] have considered an individuals char for a sequence of auocorrelaed observaions {X } when he correlaion srucure is modeled by an ARMA(1,1) ime series model. Since he process is auocorrelaed, is sandard deviaion σ Χ depends on he parameers of he ime series model, and as saed a Box and Luceno [3] i is equal o σ X = 1 φθ + θ 1 φ σ, α where σ α is he sandard deviaion of he random error erms α and φ, θ are he model parameers. The conrol limis for he X-char are μ ± c σ, 0 X where c is a consan. When he process is uncorrelaed φ, θ equal o zero, hus σ X =σ α and he above limis are he radiional Shewhar individuals char. Wardell e al. [149] recommended choosing c equal o 3 as in he i.i.d. case. For achieving he desired in-conrol ARL value of 370, 4 Wardell e al. [150] used simulaion o se he limis and found ou ha c should range from ±.45 o ±3.03 sandard deviaions of he observaions o adjus for auocorrelaion. Reynolds e al. [11] modified he X-char o accoun for he auocorrelaion for he AR(1) plus a random error model which is he naural way o view he auocorrelaion as an inheren characerisic of he process. This model is a possible model for processes in which he variabiliy of observaions has boh shor and long erm componens. One can hink of σ μ as represening long erm variabiliy and σ ε as represening a combinaion of shor-erm variabiliy and measuremen errors. The objecive of he saisical conrol in his case is o deec a change in he overall mean ξ E ( μ ). Reynolds e al. [11] considered he case where only one observaion is available a each sample, and he mean of he process is a is arge value ξ 0. They give values for c o achieve an in-conrol ARL of For low o moderae level of auocorrelaion hese values are very close o 3, which is he value ha gives in conrol ARL of for he case of independen observaions. Recenly Crowder and Eshleman [37] developed a conrol char wih greaer sensiiviy han he sandard Shewhar char. They invesigaed an adapive filering approach o process monioring wih a possibly shor ime series of auocorrelaed daa. Wrigh e al.

7 SPC Procedures for Monioring Auocorrelaed Processes 507 [156] also invesigaed he use of join esimaion oulier deecion mehod as a saisical process conrol mehod for shor-run auocorrelaed daa. Finally Liu e al. [73] sudied he effec of correlaed daa on he economic design of warning limi X chars. Based on heir sudy, i is observed ha among he parameers in he economic design he run lengh is affeced by he correlaed daa The Modified EWMA Chars The previous approach can be used o consruc he conrol limis for a radiional EWMA and CUSUM char of he observaions. The analyical reamen of modified conrol chars includes he calculaion of he variance aking ino consideraion he correlaive srucure of he process and he deerminaion of he conrol limis. The EWMA conrol char based on he original observaions is defined by Z = λx + (1 λ) Z 1, = 1,,..., and he conrol limis are ξ ± c λ σ X, λ where σ X is he sandard deviaion of he observaions, calculaed aking ino consideraion he correlaive srucure of he process. Wardell e al. [149] implemened he EWMA char for an ARMA(1,1) process. They noed ha he abiliy of he EWMA char o deec shifs in process mean is quie robus o daa correlaion. Schmid [14] inroduced a modificaion of he radiional EWMA conrol char. I is based on he idea of he Rober s [113] scheme. In his case, he difference beween he auocorrelaed process and he mean μ 0, and he variance of his process are compared. Schmid [14] gave several ables for he criical values of he modified EWMA schemes for he AR(1) model. VanBrackle and Reynolds [143] inroduced a modificaion of he EWMA conrol char for he AR(1) plus a random error model when only one observaion is available a each sample, and he mean of he process is a is arge value ξ 0. They also provided a able wih ARL values, for he design of he EWMA char of he observaions, aking ino consideraion he auocorrelaion of he process. Lu and Reynolds [78] also gave values of c for achieving an in-conrol ARL of for he EWMA char of he observaions. The opimal values of λ are considered for deecing specific shifs in he mean. When he level of auocorrelaion is low, choosing λ=0. would provide a saisfying overall performance. The value of λ should be increased as he level of auocorrelaion increases, especially when deecing large shifs. Zhang [167] proposed an exension of he radiional EWMA conrol char for monioring sep shifs in he mean applied o any saionary process daa, he so called EWMA for saionary processes (EWMAST) char. He assumed ha { X, T} is a discree saionary process wih consan mean and auocovariance funcion and showed ha he EWMA of X defined as Z = λx (1 λ) Z 1 is asympoically saionary, and ha for large he approximaion of he variance σ z of Z is M k M k { k } ( ) z = Z = X + k= 1 σ var[ ] [ λ/( λ)] σ 1 ρ( )(1 λ) [1 (1 λ) ], where M is an ineger and ρ(k) is he auocorrelaion of X a lag k.

8 508 Psarakis and Papaleonida The EWMAST char is consruced by charing Z. The limis are μ ± cσ z, where σ z is he sandard deviaion of Z. Assuming no change of he auocorrelaion of {X }, he EWMAST will signal changes of he process mean. In pracice, μ and σ z are esimaed based on some hisorical daa of X when he process is in conrol, and by replacing μ by he sample mean and σ X and ρ(k) by heir sample esimaes. Zhang [167] suggess ha he size of he hisorical daa se should conain a leas 100 observaions and ha for λ 0. Μ should be 5. A reasonable choice of λ is 0. or 0.1. Winkel and Zhang [153] concluded ha he EWMAST char should be used insead of he EWMA char in he case of posiive auocorrelaion of he daa. Jiang e al. [56] developed a new charing echnique based on he auoregressive moving average (ARMA) saisic, he ARMA char which is a general case of he EWMAST char. The successive values o be ploed on an ARMA char wih parameers ( ϕ, θ ) implemened for saionary processes are 1 k 1 θ0 α φ k k= 1 Z = X + x, whereα = ϕθ0 θ.the ARMA char signals if Z > Lσ z.assuming ha he auocorrelaion funcion of {X } is ρ( τ) = γ( τ)/ γ(0)whereγ ( τ ) = cov[ X, X + τ ], he seady sae variance can be calculaed o be k 1 α φα k 1 σ z = θ0 + + θ 0α + φ ρ( k) σ, x 1 φ 1 φ k= 1 where k = 1 φ ρ( k) converges because ρ (k) < 1, (k>0). The parameers of he ARMA char (ϕ and θ ) could be chosen o achieve a cerain ARL performance. Jiang e al. [56] proposed a guideline for choosing he parameers whereas Jiang and Tsui [58] provided how he conrol limi affecs he ARL performance of he ARMA char. Furhermore, Jiang [57] developed a Markov chain model for evaluaing he run lengh performance of he ARMA char applied o an ARMA(p,q) process. Ramjee e al. [109] inroduced a modificaion of he EWMA conrol char, he Hyperbolically Weighed Moving Average (HWMA) ha is designed o deec changes in a long-memory process. Amin and Li [9] discussed how auocorrelaion affecs he confidence level of olerance limis of a modified EWMA char, he maxmin EWMA, inroduced by Amin e al. [8], based on he smalles and larges observaion in each sample. A sudy on he effecs of auocorrelaion on olerance limis was firs presened in Amin and Lee [7]. Knoh and Amin [64] discussed how o consruc olerance limis o cover a specified proporion of he populaion when auocorrelaion is presen in he process. They also gave recommendaions for choosing he opimal esimaor of he process variabiliy for he consrucion of olerance limis The Modified CUSUM Chars Yashchin [16] evaluaed he performance of radiional CUSUM chars applied o auocorrelaed daa. He considered charing he raw daa direcly when he auocorrelaion is low. When he auocorrelaion is high he considered he use of ransformed observaions.

9 SPC Procedures for Monioring Auocorrelaed Processes 509 Schmid [15] inroduced modified CUSUM schemes whenever he arge process is an arbirary Gaussian process. These schemes are slighly differen from CUSUM for he independen case. VanBrackle and Reynolds [143] and Lu and Reynolds [80] discussed he modificaion of he CUSUM conrol char based on he observaions from an AR(1) plus a random error process for deecing a shif in he mean from is arge value ξ 0. The one-sided upper and lower CUSUM chars based on he original observaions use correspondingly a is h observaion (or sample for n>1) he conrol saisic C = max {0,C + ( X ξ Κ) }, C = min {0,C-1 + ( X ξ0 Κ) }, where he saring values C, C are consans usually aken o be zero. The consan K>0 is he reference value and is a parameer of he char expressed in Lu and Reynolds [80] as + rσ X. A signal is given for he upper CUSUM if C falls above an upper conrol limi cσ X and for he lower CUSUM if C falls bellow a lower conrol limi c σ X, where c is a + consan. The wo-sided CUSUM char uses boh he C and he C saisics simulaneously and signals if eiher saisic signals. If i is desirable o deec a shif from ξ 0 o ξ 1 hen choosing r=δ/ will minimize he ARL, where X ξ1 ξ0 δ =. σ VanBrackle and Reynolds [143] provided a able for he design of he CUSUM char of he observaions, aking ino consideraion he auocorrelaion of he process. Timmer e al. [138] presened a es for monioring he AR(1) process. The es is a modified CUSUM conrol char and is based on a sequenial implemenaion of he change-poin hypohesis esing problem o deermine if he level parameer of an AR(1) process has changed. Luceno and Box [81] examined he ARL for a one sided CUSUM char as a funcion of he sampling inerval beween consecuive observaions, he decision limi for he CUSUM saisic and he amoun of auocorrelaion beween successive poins. Sparks [130] also, discussed some modificaions of he CUSUM char for daa ha follows approximaely he AR(1) process. Aienza e al. [16] proposed a CUSUM scheme ha direcly uilizes he auocorrelaed observaions in deecing a sep change in he mean of a process ha follows a saionary ARMA(p,q) model. Assuming ha he ime when he shif occurred is =j, 1 j n hey esed for all j he hypoheses ha a shif has occurred a ime j agains he alernaive ha no shif has occurred for no j. They formed a Backward Cumulaive Sum (BCUSUM) n j n BCUSUM = x, j = 1,,..., n, and esed he presence of a sep shif in he mean by signaling if i= j i n * j n j+ 1 BCUSUM zv n j+ 1, j = 1,,..., n,

10 510 Psarakis and Papaleonida where v n j+ 1 γ ( h) is he auocovariance a lag h and h = 1 γ ( h), h< n j+ 1 n j + 1 The wo-sided BCUSUM is given esing eiher j = 1,,..., n, * z is a pre-specified conrol limi. or BCUSUM n * j n j+ 1 BCUSUM zv n j+ 1, n j n j+ z ν 1 n j + 1, j = 1,..., n. Using he characerisics of he symmeric funcions and especially ha he CUSUM n j and BCUSUM n n j + 1 are symmeric a CUSUM n / one can implemen he CUSUM mask o BCUSUM and vice versa. Thus, he BCUSUM scheme described above can be implemened using or n n j n j n CUSUM z ν n j + CUSUM, j = 1,..., n. The wo-sided CUSUM is given esing eiher n n j n j n CUSUM z ν n j + CUSUM, n n j n j n CUSUM z ν n j + CUSUM, j = 1,..., n. The CUSUM 0 0 is assumed o be 0. For an AR(1) process Aienza e al.[16] used Mone Carlo simulaion o provide a Nomogram for choosing z* given he φ and he required ARL Some Oher Approaches When a process is conaminaed by periods of exernal disurbances he ARIMA model may be incorrecly specified and hus he conrol limis incorrecly placed. Wes e al. [151] used a ransfer funcion model which akes ino accoun hree sources of variabiliy: he dynamic inpu erm which represens an impulse funcion, he inervenion erm which idenifies periods of imes when assignable causes are presen in he process and he classical ARIMA model. Dyer e al. [41] proposed a new forecas based monioring scheme, he Reverse Moving Average conrol char, which rely on he fac ha he average of all forecas errors afer a shif has occurred a he process mean provides a good indicaor of he presence of a disurbance. Of course he ime period when a forecas accuracy changes is unknown bu one can calculae a series of moving averages consising of parial averages of las forecas errors. Timmer and Pigniaelo [138] presened hree change poin esimaors for use when

11 SPC Procedures for Monioring Auocorrelaed Processes 511 conrol chars are applied o auocorrelaed daa as modeled by AR(1) processes. Each of hese esimaors can be applied afer a signal from a conrol char indicaes he presence of special cause source of variabiliy. Ben Gal e al. [18] inroduced a new conex-based SPC mehodology for monioring varying lengh sae-dependen processes. The idea is o compare he reference ree ha represens he in-conrol reference behaviour of he process wih a moniored ree, generaed periodically from a sample of sequenced observaions. Pawlak e al. [105] developed a nonparameric sequenial deecion rule based on a clipping median ha is designed o deec srucural changes in paricular jumps in ime series. Han and Tsung [47] proposed a reference free cumulaive score (RSCuscore) char ha can race and deec dynamic mean changes (i.e. nonconsan, ime-vaying shifs in he mean) wihou knowing he reference paern. 3.. Conrol Chars Based on he Residuals Alwan and Robers [1] firs suggesed o direcly model he correlaive srucure of he process wih an appropriae ime series model and o apply aferwards conrol chars o he independen idenically disribued sream of he residuals. Many oher auhors have also proposed conrol chars based on he residuals The Alwan and Robers Mehod and Is Modificaions For saisical process monioring, a saionary process, which has consan mean and consan variance, is a naural exension of he case of an i.i.d. sequence, hus he appropriae model should be a saionary ime series model. Alwan and Robers [1] suggesed he implemenaion of wo basic chars raher han one: I. Common Cause Char (CCC), which is a char of forecased values ha are deermined by fiing he correlaed process wih an ARIMA model. II. Special Cause Char (SCC), which is a radiional conrol char of he residuals or one sep ahead predicion errors. The Common Cause char assumes ha no special causes have occurred, and i is no really a conrol char since i has no conrol limis. I serves as guidance in viewing he curren level of he process and is evoluion hrough ime. This char essenially accouns for he sysemaic variaion of he process. The Special Cause char is essenially a Shewhar individuals, EWMA or CUSUM char for he residuals (i.e., he difference beween he acual process values and heir forecass). A Shewhar individual char of he residuals is a ime-ordered plo of he residuals wih c σ conrol limis, and will be referred o as residuals char, X-char of he residuals. Since he residuals ha are he daa used for he Special Cause char are i.i.d. random variables all supplemenal guidelines, such as run rules, are applicable. In pracice, he ime series modeling needed o implemen he above procedure migh be awkward. To ackle his difficuly Mongomery and Masrangelo [90] suggesed he modeling of every process wih an ARIMA(0,1,1) ime series model. They suggesed he use of a X-char for he residuals, which assumes an ARIMA(0,1,1) ime series model for all he process and uses he predicion errors. They uilized he fac ha by replacing λ wih 1 θ he EWMA saisic is equivalen o he ARIMA(0,1,1). As shown by Box and Luceno [3] he EWMA wih λ = 1 θ is he opimal one-sep-ahead predicion for he

12 51 Psarakis and Papaleonida ARIMA(0,1,1). The parameer of he EWMA can be deermined by minimizing he sum of squares of he EWMA one-sep-ahead predicion errors. If X ˆ + 1 () is he forecas for he observaion in period + 1 made a he end of period, hen he opimal forecas is he value of he EWMA calculaed in period, and he one sep ahead residuals are: Xˆ () Z, + 1 = e = X Xˆ ( 1). If he underlying process is really an ARIMA(0,1,1) he one sep ahead predicion errors given above are i.i.d. wih mean zero and sandard deviaion σ, which can be esimaed based on some hisorical daa. Therefore, conrol chars can be applied o he residuals. The EWMA char can also be used as he basis of a saisical process monioring procedure ha is an approximaion of he exac ime-series approach. The procedure consiss of a conrol char where he one-sep-ahead predicion errors are ploed, and a run char of he original observaions on which he EWMA forecas is superimposed. The residuals char deecs unusual random shocks bu is no as sensiive o slow rending shifs in he mean. Therefore, Mongomery and Masrangelo [90] provided a modificaion of he above procedure ha combines informaion abou he sae of saisical conrol and process dynamics on a single conrol char. The modified procedure acually moniors simulaneously he forecas and he forecas errors. Since he EWMA saisic is a suiable one sep ahead predicor, Z could be used as he cenerline on a conrol char for period +1 wih conrol limis a CL = Z ± 3 σ, and he observaion X +1 will be compared o hese limis o es for saisical conrol. This procedure is called Moving Cenerline EWMA conrol char or for simpliciy MCEWMA. The MCEWMA is an exac procedure for he ARIMA(0,1,1) process, bu is a good approximaion as well for many oher ime series, provided ha he value of λ is seleced appropriaely. I is a good approximaion, especially for a process in which he mean exhibis slow drifing behavior and he observaions a low lags appear o be posiively auocorrelaed. To improve he sensiiviy of he SCC char Runger e al. [118] applied he one-sided CUSUM char on he residuals of an auoregressive process {X, =1,,...,} which follow an AR(p) ime series model wih consan mean μ of order p. The one-sided CUSUM accumulaes wih he saisic C upward derivaions of he residuals e from μ 0 developed as follows: C = [ C + e ( μ + K)] = max{0,[ C + e ( μ + K )]}, where he saring values are C 0 = 0 and K> 0 is a parameer of he char. The value of K is usually chosen close o he midway poin beween μ 0 and he ou of conrol value of he mean μ 1. When he shif in he process mean is a single sep change of size δ (where δ is measured in erms of he sandard deviaion of he process), for he special case of an AR(1) process, he mean of he firs residual afer he shif is δ, and he mean of subsequen residuals is δ(1 φ). Therefore, since every residual bu he firs has expeced value δ(1 φ) Runger e al. [118] proposed he modified guideline K= σ(1 φ)/ which akes explici accoun of he auocorrelaion. The char signals if C exceeds he decision inerval H ha

13 SPC Procedures for Monioring Auocorrelaed Processes 513 usually has a value five imes he process sandard deviaion σ. Dooley and Kapoor [40] suggesed a scheme based on monioring he residuals auocorrelaion n n k = k+ 1 k = 1 r = e e e, k = 1,,..., a various lags for he laes n observaions. The variance of he r k could be simply assumed o be equal o (n k)/n(n+). The plo named as SACC (Sample Auocorrelaion Char) could be similar o he sample auocorrelaion funcion as described in he lieraure for ime series models. When a change occurs in he process which could be eiher in is mean or is variance bu could also be in he model parameers, here should be a discrepancy beween he assumed ime series model and he behavior of he process ha will lead o auocorrelaed residuals. Thus, an r k significanly differen han zero is an indicaion of a possible special cause in he process. To make he SACC more sensiive in deecing changes in he auocorrelaion of a process Aienza e al. [15] implemened an EWMA o he sample auocorrelaions: S = λ + λ ( rk) (1 ) S 1( rk) rk, k = 1,,..., m = 1,,..., where λ is he exponenial smoohing consan and m is he number lags. The conrol limis are given by 0 ± D σ r( k) λ/( λ), where D is he muliple of σ used o se he conrol limi. Runger and Willemain [119] proposed a disribuion free conrol char based on Unweighed Bach Means (UBM) for monioring auocorrelaed daa. The UBM char breaks successive groups of sequenial observaions ino baches wih equal weighs assigned o every poin in he bach. They showed, using he ARL crierion in AR(1) model, ha ploing averages of baches of he raw daa can be an effecive alernaive o ploing residuals. To implemen residuals conrol chars in a live environmen he ime series model ha describes he process daa needs o be esimaed. In real applicaions he rue model of he process is never known and he errors from model idenificaion as well as from parameer esimaion are hardly avoidable especially in he presence of assignable causes of variaion. Kramer and Schmid [67] sudied via simulaion he effecs of esimaing he parameers of he process and found ha chars perform much beer when he parameers are precisely esimaed. If he model used o describe he process is accurae he residuals are as menioned above uncorrelaed, hus all SPC echniques can be applied and he usual in-conrol lengh properies of he sandard conrol chars can be expeced. However, when a mean shif occurs a he process, he saisical properies of he chars aren clear and he resuling shif in he mean of he residuals doesn necessarily have he same form as he one occurred a he original observaions. Apley and Shi [1] saed ha an auocorrelaed process given by y () = x () + μ f() ha undergoes a mean shif a ime τ of 0, < τ f () = 1, τ, where x() follows an ARIMA(p,d,q) model inroduces a mean shif in he residuals ha experiences some ransien dynamics before seling down o a seady sae value (called

14 514 Psarakis and Papaleonida faul signaure) which depends on he ARIMA model ha describes he process and he magniude of he shif. The faul signaure is given by Φ ( B )(1 B f () ) f (), Θ( B) where Β is he back shif operaor and Φ(Β), Θ(Β) are polynomials of degree p and q respecively. To ake advanage of he ransien dynamics of he faul signaure Apley and Shi [1] used he Generalized Likelihood Raio Tes (GLRT) and implemened i for SPC purposes. They used he saisic o es he hypoheses ha no faul has occurred versus he hypoheses ha a faul has occurred a ime k+1 (k=1,,...,n) where N is a user defined quaniy ha resrains he ess o he N mos recen residuals. I is shown ha he GLRT saisic is given by 1/ k k σ α i= 1 i= 1 Tk () = f () i e( k+ i) f (), i where f ( ) is he faul signaure for a faul occurring a ime sep 1. T k () can be considered as measure of he correlaion beween he acual residuals and he faul signaure. The higher his correlaion is, he larger he magniude of T k () and he larger he likelihood ha he faul occurred a ime k+1. The GLRT involves comparing G () max T(), k= 1,..., N k d wih a user defined hreshold chosen o provide a desirable ARL. An alernaive mehod for deecing a known faul signaure in a series of observaions is he Cuscore char. The one-sided Cuscore saisic is { } Q = max Q + r ( e k),0, = 1,,..., 1 where r is he deecor. Shu e al. [18] showed ha if a faul is o occur a some known ime τ he deecor akes he same value as he faul signaure i.e., r = f τ. In order for he Cuscore char o ake full advanage of he faul signaure dynamics Shu a al. [18] proposed a CUSUM riggered Cuscore char which reduces he discrepancy of he deecor coefficiens and he faul signaure. In addiion hey used he GLR saisic o esimae τ and achieved even beer performance. Yang and Makis [158] inroduced a mehod for sudying residual response o disurbances in general conrolled auocorrelaed processes. They concluded ha residuals are always independen, wih zero means when he process is in conrol and when a disurbance occurs he residual means are fully deermined by he disurbance sequence and by he auocorrelaion srucure of he process. Runger [116] using a physically realisic disurbance model also concluded ha he signal generaed from he disurbance evenually disappears for he conrol chars of he residuals. Apley and Lee [11] proposed a mehod for designing residual-based EWMA chars or ARMA processes, under he consideraion of uncerainy in he esimaed model

15 SPC Procedures for Monioring Auocorrelaed Processes 515 parameers. The mehod is designed o use wider EWMA conrol limis based on a wors case design approach. They suggesed using as an esimae of he variance of he process he upper boundary of an upper one-sided 1 α confidence inerval. The resuling EWMA conrol limis are widened by an amoun ha depends on a number of facors including he level of model uncerainy. Tesik [135] also, used a similar mehod o widen he EWMA conrol limis for AR(1) processes when he model parameer is esimaed. As an alernaive o common sandardized residuals Terpsra e al. [134] recommended applicaion of radiional conrol chars o eiher sudenized ime series residuals or sudenized linear regression residuals since hey carry much more informaion abou he error disribuion. Casagliola and Tsung [7] recenly invesigaed he effec of skewness on convenional auocorrelaed saisical process conrol echniques. They proposed a mehod called scaled weighed variance o handle he non-normaliy in an auocorrelaed process if is disribuion is heavily skewed. Wang [148] examined he case where he daa under sudy are negaively correlaed. In ha case he proposed a simple ransformaion ha changes he auocorrelaion coefficien of he daa se o he opposie sign. Snoussi e al. [19] proved ha he Q-saisics inroduced by Quensberry [108] in conjuncion wih residual conrol chars is an appropriae ool for shor run auocorrelaed daa Conol Chars Based on he Residuals from he AR(1) Plus a Random Error Model Lu and Reynolds [78] used he AR(1) model wih an addiional random error o describe he auocorrelaion of he process. They developed an individuals and an EWMA conrol char based on he residuals from he forecas values of he model for monioring he mean of he process. They assumed ha each sample conains only one observaion and ha he mean of he process is a is arge value ξ 0. Lu and Reynolds [80] on he oher hand developed a CUSUM char for he residuals. When he process is in conrol he minimum mean square error forecas made a ime 1 for ime is for he AR(1) model plus a random error, or he equivalen ARMA(1,1) model Xˆ = ξ + φ( X ξ ) θe, where e = X ξ0 ϕ( X 1 ξ0) + θe 1 is he residual a ime. The mean of he residuals is consan and hus o achieve an in-conrol ARL of he mehods for independen observaions can be used. When here is a shif in he mean of he process he mean of he residuals varies wih ime. Suppose ha here is a sep change from ξ 0 o ξ 1 in he process mean beween ime =Τ 1 and Τ. The expecaions of he residuals for various imes are and for =Τ+l, l=1,,..., 0 = T 1, T,... Ee ( ) =, ξ ξ = T 1 0 l l l i 1 θ ( φ θ) φ + 1 Ee ( ) = θ + (1 φ) θ ( ξ1 ξ0) = ( ξ1 ξ0). i = 1 1 θ The residual immediaely afer he shif has is larges mean ha decreases aferwards

16 516 Psarakis and Papaleonida and asympoically ges o 1 φ Ee ( ) = ( ξ 1 ξ 0). 1 θ The residuals are uncorrelaed and normally disribued wih variance σ a. A Shewhar individuals char based on he residuals plos he residuals e and uses heir sandard deviaion σ a as appearing in he ARMA(1,1) model for consrucing he conrol limis ± cσ a where c is a consan and σ a is he sandard deviaion of he residuals from he ARMA(1,1) model. When he process is in conrol he residuals are independen and have consan mean and hus using c=3 gives an in-conrol ARL of An EWMA char of he residuals for monioring he process mean uses he conrol saisic Z =λe +(1 λ)z -1, =1,,..., and he conrol limis are of he form ± c [ λ /( λ)] σα where c is a consan and σ α is he sandard deviaion of he residuals from he ARMA(1,1) model. The residuals are independen when he process is in conrol and have consan mean, hus he conrol limis of he EWMA saisic can be deermined using mehods for independen observaions. In paricular, using c=.859 will give ARL 0 =370.4 when λ=0.. Lu and Reynolds [78] invesigaed he opimal choice of he smoohing parameer λ of he EWMA and concluded as expeced ha small values of λ are beer for deecing small shifs while large values of λ are beer for deecing large shifs. When he auocorrelaion is a is lowes level a relaively small value of λ, such as λ =0., would work well across a wide range of shifs. When he level of auocorrelaion becomes higher very small values of λ are opimal for deecing small shifs, bu hese values of λ perform poorly for large shifs. Lu and Reynolds [80] presened he CUSUM char based on he residuals from he AR(1) plus a random error model. The wo-sided CUSUM char of he residuals plos simulaneously he + + C = max {0,C- 1 + ( e r σ a )}, C = min{ 0,C ( e rσ a )}, where σ α is he sandard deviaion of he residuals of he equivalen ARMA(1,1) model and r is he reference value. The char signals if eiher of he saisics exceeds he decision inerval H= c σ a. The saisics are he same as hose of he wo-sided CUSUM char of he observaions, excep ha he residual e is ploed insead of he observaion X. The mean of he residuals is zero so he in-conrol mean ξ 0 is omied from he above equaions. Lu and Reynolds [80] discussed he choice of he parameers for he CUSUM char of he residuals. The residuals are independen hus he opimal value of r should depend on he expeced values of he residuals afer he shif. Afer he shif hough, he mean of he residuals is no consan and alhough he residuals are independen, he values of he parameers of he char canno be deermined using he mehods for independen observaions. For he case of a small shif he opimal value of r is close o half he sandardized asympoic mean of he residuals. This is reasonable because a small shif is no easily deeced and hus many residuals will be ploed before he char signals. In his way, he mean of he residuals will be close o is asympoic mean E(e ). A reasonable compromise for good overall performance of a CUSUM char of he residuals is a moderae value of r

17 SPC Procedures for Monioring Auocorrelaed Processes 517 such as Conrol Chars for Monioring he Variance In radiional char mehodology, a char for monioring he mean is usually used in conjuncion wih a char for monioring he process variance. The observaions are assumed independen and normal, and he objecive of process monioring is o deec special causes ha can produce a change in he process mean and variance. In his case, here are wo process parameers: he mean and he variance. However, in he case of auocorrelaed observaions, he monioring problem is more complicaed since he model used for he process observaions conains more parameers. The following secions deal wih correlaion beween samples and conrol chars for daa modeled by he AR(1) plus a random error process The MacGregor and Harris's Approach MacGregor and Harris [8] considered wo conrol chars for monioring he process variance: one based on an Exponenially Weighed Mean Squared deviaion from he arge, he EWMS, and anoher based on an Exponenially Weighed Moving Variance in which he curren process mean is esimaed using an EWMA char of he observaions, he EWMV char. To monior he variance of auocorrelaed processes, MacGregor and Harris [8] considered he siuaion in which only one observaion X is aken from he process a sampling poin and he correlaive naure of he process is described by he AR(1) plus a random error model and is equivalen ARMA(1,1) form which were presened in secion. The raio of he variance σ ε of he random errors o he oal variance of he process σ X will be needed laer and is expressed by σ ε (1 φθ)( φ θ) = 1 ρ1 φ = 1 σ X φ(1 θ φθ) where ρ 1 is he firs lag auocorrelaion of he X process and φ, θ he auoregressive and he moving average parameers correspondingly of he ARMA(1,1) model The Exponenially Weighed Mean Square Char The Exponenially Weighed Mean Square Char, or briefly he EWMS char, plos he saisic k = μ + 0 = 1 + μ k= 1 S r(1 r) [ X ] (1 r) S (1 r) S r[ X ], where X is an individual observaion of he process aken a sample ime, S is an iniial esimae of he mean squared error (usually aken o be he hisorical in-conrol value), r is a weigh (0<r 1) ha conrols he rae of exponenial discouning of pas daa. The sum of weighs is given by r r(1 r) + (1 r) = 1, k= 1 k A modificaion of he EWMS conrol char is he Exponenially Weighed Roo Mean Square Char, or briefly he EWRMS char, which is acually he square roo of he EWMS and plos he saisic S., 0

18 518 Psarakis and Papaleonida When he process mean is on arge and he in-conrol variance is σ 0 (where σ 0 is obained from hisorical daa) he conrol limis for he EWRMS are 0 1 α/ v α/ v 0 ( σ χ ( ) σ, σ χ ( ) σ ), where χ1 α /() v and χ α /( v ) are he (1 α /)100% and ( α /)100% perceniles respecively, of he chi-square disribuion wih ν degrees of freedom. The degrees of freedom ν for an auocorrelaed process ha can be described by he above model, are dependen from he parameers ϕ, σ / σ X and r The Exponenially Weighed Moving Variance Char ε The Exponenially Weighed Moving Variance Char, briefly he EWMV char, is obained by replacing he overall mean of he process μ wih he ime varying μ and hus plos he saisic S = (1 r) S + r[ X μ ]. 1 The conrol limis for he EWMV conrol char when he process follows he model described by MacGregor and Harris [8] are given by ( σ C σ, σ + C σ ), where he consans C 7 and C 8 are calculaed from he Johnson [61] curves approximaion or a ransformaion of he chi-square disribuion. 4.. The Lu and Reynolds s Mehod Lu and Reynolds [79] invesigaed he problem of monioring he process variance when here is correlaion beween samples, and he sample size is n=1. They also considered he problem of simulaneously monioring he process mean and variance. Similarly o MacGregor and Harris [8], hey used he AR(1) plus a random error and he equivalen ARMA(1,1) models o describe he correlaion srucure of he process, for he case of posiive auocorrelaion The EWMA of he Logs of he Squared Residuals Char Lu and Reynolds [79] used he AR(1) wih an addiional random process o model he correlaion srucure of he process. The variance of he ime wandering mean using he equaions in Βox e al. [] is σ μ = σγ /(1 ϕ ) and he variance of he process is σ γ Χ = μ + ε = +, ε σ σ σ σ 1 φ A change in he process variance σ X could be caused by an increase in σ μ and/orσ ε. When he auoregressive parameer φ in he AR(1) plus a random error model is considered fixed, he increase in σ μ is caused by an increase in σ γ. Equivalenly, when boh he auoregressive and he moving average parameer in he ARMA(1,1) model are considered consan a change in σ α would change boh σ ε and σ γ hrough he parameer relaionships given in secion. Lu and Reynolds [79] supposed ha beween samples T 1 and T, σ γ increases from he in-conrol value σ γ 0 o σ γ 1 and σ ε increases from he in-conrol value σ ε 0 o σ ε1. The residuals afer he shif are correlaed normal random variables wih mean zero and variance a =T and a =T+l α0 ε1 ε0 γ1 γ0 Var( e ) = σ + ( σ σ ) + ( σ σ ), T