Time Series Approaches to Statistical Process Control

Transcription

1 Inernaional Journal of Innovaive Research in Compuer Science & Technolog (IJIRCST) ISSN: , Volume-1, Issue-, November- 013 Time Series Approaches o Saisical Process Conrol Aji Goswami, Prof. (Dr.) H.Nr. Dua Absrac- In radiional Saisical Process Conrol (SPC) procedure, a sandard assumpion is ha observaion from he process a differen ime poins are independen random variable. However, his independen assumpion is no alwas rue.. In fac, in he las decade, he ime-series approach o Saisical Process Conrol has been a opic of ineres of man quali scieniss. In his paper, an aemp has been made o highligh some of he works in his area and a few models will be discussed o analze he effecs of auocorrelaion on some sandard conrol chars echniques. Index Terms Auocorrelaion, Dependen observaion, EWMA conrol char, SPC conrol o deal wih he auo correlaion in recen imes, he are: (a) Tradiional conrol chars are used, bu mehods used o esimae he process parameers and finall he conrol limis are adjused in order o accoun for he auo correlaion. This mehod is recommended when he level of auo correlaion is no exremel high. (b) A ime series model is fied o he process observaions and he residuals from his model are used in radiional conrol chars I. INTRODUCTION Mos of he indusrial processes ofen have complex behaviors, when successive unis are relaed o previous one. When here are carr over effec from he earlier observaions, he sandard conrol chars ma exhibi an increased frequenc of false alarms. There is an increased likelihood ha he daa will exhibi auocorrelaion in ssems where he process ime is longer han he ime beween sample collecions [1]. Auocorrelaion resuls from man facors- such as work shif, operaor roaions, mechanic or echnician changes. Someimes some processes, inherenl produces auocorrelaed daa. Tradiional Shewhar conrol chars are sensiive o auocorrelaed daa and even a low levels of correlaion, a significan changes ma occur in char properies including shor Average Run Lengh (ARL). Hence in recen imes, sudies on auocorrelaion daa is an imporan area for SPC users and more aenion is being paid b man quali scieniss o sud he behavior of conrol char performance in presence of auocorrelaion. II. EFFECT OF AUTO CORRELATION IN PROCESS DATA When here is significan auocorrelaion in he process daa, i is no advisable o use radiional conrol char echnique wihou modificaion. Two general approaches have been considered b he scholars of quali Manuscrip received November, 013. Aji Goswami, Research Scholar, Dibrugarh Universi, Dibrugarh, Assam, India ( agoswami09@gmail.com). Dr. H. Nr. Dua, Professor, Deparmen of Saisics, Dibrugarh Universi, Dibrugarh, Assam, India ( hnd_sa@rediffmail.com). III. REVIEW OF PAST WORKS ON AUTOCORRELATED PROCESS DATA Dua & Phukan [] reviewed he effec of auocorrelaion on radiional variable conrol chars and oher modified variable conrol chars like Cumulaive Sum Char (CUSUM), Exponeniall Weighed Moving Average (EWMA) conrol char and Mulivariae (T ) conrol char covering he period The, however, did no considered he pas works done b he quali scieniss in he area of auocorrelaed aribue conrol chars. In our presen sud, we shall r o include (as far as possible) mos of he curren research works in hese area, boh for variable (secion A) as well as for aribue conrol chars (secion B). However, considering he fas growing naure of he opic, sudies on auocorrelaion effecs on variable sampling inervals (VSI) conrol chars and non-parameric conrol chars could no be discussed in his secion and i will be repored in a fuure sud. A. Pas Works on Variable Conrol Chars Since 008, man papers have been published b he scholars in suding he effec of auocorrelaion in variable conrol chars and more are offing. Sheu & Lu [3] presens a useful discussion of a mehod ha enables he deecing abili of he EWMA conrol char o be enhanced and shows ha when he observaions are drawn from an AR(1) process wih random error, he EWMA conrol char is far more useful han he Shewhar conrol char in deecing small shifs. The found ha The Generalized Weighed Moving Average (GWMA) conrol char of observaions is shown o be superior o he EWMA conrol char in deecing small shifs in he process mean and variance. The GWMA conrol char of observaions 34

2 Time Series Approaches o Saisical Process Conrol requires less ime o deec small process mean and/or variance shifs as he level of auocorrelaion declines. Keoagile [4] considers he problem of monioring a process in which he observaions can be represened as a firs-order auoregressive model following a heav ailed disribuion. He propose a char based on compuing he conrol limis using he process mean and he sandard error of he leas absolue deviaion for he case when he process quali characerisics follows a heav ailed -disribuion. Chang & Wu [5] developed a general and unified approach based on he use of discreizaion and he finie Markov chain imbedding echnique o invesigae he run lengh properies for various conrol chars when he process observaions are auocorrelaed. Also numerical resuls are presened for illusraive purposes. Suriaka e.al [6] derived an explici formula for he characerisic of EWMA conrol char for rend saionar exponenial AR (1) processes. The compare he resuls for Average Run Lengh (ARL) obained from he explici formula wih values obained from he inegral equaion and found ha he new resuls are simple, eas o programming, which make i aracive o be used in pracice b performers. Karaoglan & Bahan [7] compued ARL performances of conrol chars for peroxide daa from wo baches, for which rend saionar firs order auoregressive (rend AR(1) for shor) model is a represenaive model. B. Auocorrelaed Aribue Conrol Chars To our knowledge lile aenion has been given o he developmen of conrol chars in he case of correlaed aribue daa. A few work in his area are Deligonul and Mergen [8], Bha and Lal [9]. The assumed a wo-sae Markov chain model for auo correlaed aribue daa. Harve and Fernandes [] and Wisnowski and Keas [11] shows ha correlaed coun daa can be modeled wih a EWMA approach.. Simson and Masrangelo [1] sudied he monioring of seriall dependen processes wih aribues daa obained from mulisaions of producion. Lai e al. [13] 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. [14] also sudied he problem of process monioring when he process is of high quali and measuremen values possess a cerain serial dependence. Nembhard e.al [15] sudied a demeris conrol chars (U-char) for auocorrelaed daa. Their sud is relaed o injecion-modeling producion lines produced b various models of leak proof plasic conainers. Tang and Cheong [16] proposed a conrol scheme ha is effecive in deecing changes in fracion nonconforming for high ield processes wih correlaion wihin each inspecion group. Shepherd e al. [17] proposed wo conrol char schemes. These conrol chars are based on a sequence of random variables ha are used o classif an iem as conforming or nonconforming under a saionar Markov chain model and 0% sequenial sampling. IV. TIME SERIES MODEL To appl conrol char for residual, we can modeled quali characerisics as follows p p (1) Here, is a p h order auoregressive or AR (p) Process where, and (-1 < <1) are unknown consan and i is normall and independenl disribued wih mean 0 and sandard deviaion. If we modeled 1 1 () hen i is called firs order auoregressive AR (1) model; he observaions from such a model have mean /(1 ), sandard deviaion /(1 ) 1/ and he observaions ha are k periods apar ( ) have correlaion coefficien k k. Suppose ha is an esimae of, obained from analsis of sample daa from he process, and is he fied value of. Then he residuals e are approximael normall and independenl disribued wih mean zero and consan variance. Convenional conrol chars could now be applied o he sequence of residuals. Similarl, he second order auoregressive model AR () will be (3) 1 1. Anoher possibili is o model he dependenc hrough he random componen. A simple wa o do his is 1 (4) This is called a firs-order moving average model. In his model, he correlaion beween and 1 is p and is zero a all oher lags. Thus, he 1 /(1 ) correlaive srucure in onl exends backwards one ime period. Someimes combinaions of auoregressive and moving average erms are useful. A firs order mixed model is 35

3 1 1 Inernaional Journal of Innovaive Research in Compuer Science & Technolog (IJIRCST) ISSN: , Volume-1, Issue-, November- 013 (5) We also encouner he firs-order inegraed moving average model (6) 1 1 in some applicaions. Whereas he previous models are used o describe saionar behavior (ha is wanders around a fixed mean), he model in equaion (6) describes non-saionar behavior (he variable drifs as if here is no fixed value of he process mean). V. CALCULATION OF AUTOCORRELATION The auocorrelaion coefficien for daa ha are k ime period apar r k is defined as r k nk 1 ( )( k ), k 0,1,,3,.. n ( ) 1 where n is he oal number of observaions in he daa se. The sandard error a lag k is sek (7) 1/ n ;k=1 (8) = k1 1/ n(1 ri ) i1 ; k>1 (9) CL z LCL z 3 UCL z 3 [1 (1 ) [1 (1 ) ] ] (11) where he esimae of he process variabili,, picall is esimaed using he same mehod as for he individual conrol char. VII. ANALYSIS To analze he conrol char model in presence of auocorrelaion, we have sudied he finished produc of Formalin from a chemical facor in Assam. I ma be menioned here ha formalin produc of he chemical facor is se as 37 ± 0.5 % weigh of formaldehde gas. If he finished Produc is below 36.5%, he cusomers don accep i. If he finished produc is above 37.5%, i is no affordable o he managemen so far is cos benefi margin is concerned. To analze he daa, we have colleced 68 se of raw daa of formalin and deal wih using saisical process conrol ools. Firs, we calculae he auocorrelaion funcion (ACF) of he formalin (chemical) produc daa which will indicae he presence of he auocorrelaion in he daa. Graph of ACF and PACF are shown in figure 1 and respecivel. VI. EWMA CONTROL CHART Robers inroduced exponeniall weighed moving average char in 1959 [18]. This char is popular for he conrol of indusrial processes where he individual observaions arrive one b one. The EWMA, is compued sequeniall as a linear inerpolaion beween he presen observaion, he previous EWMA 1 ( 1 ) 1 z and z () Auocorrelaion Fig.1. Auocorrelaion Funcion (ACF) of Puri of Chemical Produc (Formalin) Where is a consan 0 1. Huner [19] has shown ha for independen and normall disribued daa, he conrol limis for he EWMA are given b 36

4 Time Series Approaches o Saisical Process Conrol Parial Auocorrelaion Normal Probabili Plo for Residual Mean: SDev: 5.09E Percen Fig.. Parial Auocorrelaion Funcion (PACF) of Puri of Chemical Produc (Formalin) From he visual inspecion of he figure 1, we can easil conclude ha here is auocorrelaion in he original se of daa. Also, from he ACF plo fig 1., i is clear ha he lag(s) is significanl differen from zero and he series is no whie noise i.e he daa has auo correlaion. A. Removing Auocorrelaion from he Observed Daa To achieve an independen, normall disribued daa se, Mongomer [1] recommends modeling he correlaive srucure and conrol charing he residuals direcl. For he formalin daa, he prediced puri of formalin (chemical) produc a period ime is (from he fig 1) (1) Onl four poins from he previous daa were used because of he high auocorrelaion coefficien for lags 1-4 (fig-6.1). To deermine he parameers of his model muliple linear regression can be performed. Using Miniab Sofware, he regression are calculaed which is given parameers 1,, 3 and 4 below. = = =971 3 = =-116 To check he model, we show in figure 3, a normal plo of he residuals, and in figure 4, a plo of he residuals in ime order. Boh plos indicae ha he model fis he daa well. The ACF and he PACF of he residual provide a furher check. Ideall, if he model fis well, all auocorrelaion would have been removed from he daa and he residual behave like whie noise. Figure 5 and 6 show he ACF and he PACF for he residual afer fiing he AR (4) model o he formalin daa. Boh he ACF and he PACF are esseniall zero for all lags Daa Fig. 3. Normal Plo of Residuals afer Fiing an AR (4) Model o he Formalin Daa Residuals Auocorrelaion Index 0 00 Fig. 4. Time Series Plo of he Residuals Fig. 5. The ACF of he Residuals afer Fiing an AR (4) Model o he Formalin Daa Parial Auocorrelaion Fig. 6. The PACF of he Residuals afer Fiing an AR (4) Model To he Formalin Daa 37

5 Inernaional Journal of Innovaive Research in Compuer Science & Technolog (IJIRCST) ISSN: , Volume-1, Issue-, November- 013 B. EWMA Conrol Char for Residual In using he inflaed limis for he individuals conrol char, we emphasized he imporance of reducing he false alarm rae, and making he char eas o inerpre. However, his approach desensiizes he char and will likel increase he average run lengh (ARL) o signal an alarm in case of a real change. For he curren process, we could use an individual s conrol char, a cumulaive sum (CUSUM) char or an EWMA char. The residuals are no on a meaningful scale. Hence he pracical inerpreaion argumen for using he individuals conrol char no longer applies. We herefore sugges using an EWMA char. EWMA Sample Number 0 UCL=39.35 CL=38.57 LCL=37.78 Fig. 7. An EWMA of Observaions from he Original Formalin Daa EWMA Sample Number 0 UCL=0. CL=0 LCL= Fig. 8. An EWMA of Observaions from he Residual Formalin Daa Using equaion (11), he original formalin daa is ploed for EWMA char using Miniab 11 version wih he pical defaul value = 0.. Fig 7 shows an EWMA for original formalin daa se. We see ha he process wih his char appears o be ou of conrol. Bu afer removing he effec of auocorrelaion when we use he EWMA conrol char for he residual formalin daa, he process is found in saisical conrol. (fig.8) VIII. CONCLUSION In modern approach of applicaion of saisical process conrol, he effec of auocorrelaion is increasingl becoming a fac of life and mus no be ignored. In our sud, we have ried o explain wih a chemical daa how o deec auocorrelaion; illusraed i s consequences for sandard conrol char EWMA char. Oher conrol char can be used. As demonsraed, modern sofware packages such as MINITAB, make i relaivel eas o perform he compuaions needed when dealing wih auocorrelaed processes and using AR ime series models. REFERENCES [1] Mongomer, D.C., Inroducion o Saisical Quali Conrol, 6 h Ediion. John Wile and Sons-New York.,009 [] Dua, H.Nr. & Phukan, A, Performance of Some Conrol Char in Presence of Auocorrelaion: A Review and Surve of Lieraure, Assam Saisical Review, 008,, 1-, [3] Sheu, S, S. Lu., The effec of auocorrelaed observaions on a GWMA conrol char performance, Inernaional Journal of Quali & Reliabili Managemen., 009, Vol. 6 Iss: pp [4] Keoagile, T., Conrol Char for Auocorrelaed Processes wih Heav Tailed Disribuions. Economic Quali Conrol., 0, Volume 3, Issue, Pages , ISSN (Online) [5] ] Chang, Y.M., & T. Wu, On Average Run Lenghs of Conrol Chars for Auocorrelaed Processes, Mehodol Compu Appl Probab, 011,13: [6] Suriaka, W, Y. Areepong, S. Sukparungsee and G. Miielu, An Analical Approach o EWMA Conrol Char for Trend Saionar Exponenial AR (1)Processes, Proceedings of he World Congress on Engineering, 01, Vol. IJul 4 6 [7] Karaoglan, A.D and G.M. Bahan., ARL performance of residual conrol chars for rend AR (1) process: A case sud on peroxide values of sored vegeable oil. Scienific Research and Essas, 01, Vol. 7(13), pp , [8] Deligonul and Mergen, Dependence bias in convenional p-chars and is correcion wih an appropriae lo quali disribuion, Jrnl. Of Applied Saisics, (1), [9] Bha, U.N. and Lal, R., Aribue Conrol Chars for Markov Dependen Producion Processes, IIE Transacions, (), 1990, [] Harve, A.C. and Fernandes, C., Time Series Modeling for coun or Correlaed Observaions, Jrnl. Of Business and economic Saisics, 1989,7(4), 7-4. [11] Wisnowski, J. W. and Keas, J. B., Monioring he availabili of asses wih binomial and correlaed observaions, Quali Engineering, 1999,11(3), [1] Simson, W.A. and Masrangelo, C.M., Monioring Serialldependen Processes wih Aribue Daa JQT,1996, 8(3), [13] Lai, C.D., Govindaraju, K. and ie, M., Effecs of Correlaion on Fracion non conforming Saisical Process Conrol Procedures, Jrnl. Of Applied Saisics, 1998, 5(4), [14] Lai, C.D., Govindaraju, K. and ie, M., Sud of Markov Model for a high quali dependen process, Jrnl. Of Applied Saisics, 000, 7(4), [15] Nembhard D.A., A Demeris Conrol Char for Auocorrelaed daa Quali Engineering, 000, 13(), [16] Tang, L.C. and Cheong, W.-T., A conrol scheme for high-ield correlaed Producion under group inspecion, Journal of Quali Technolog, 006, 38(1), [17] Shepherd, D.K., Champ, C.W., Rigdon, S.E. and Fuller, H.T. (006):Aribue Chars for monioring a dependen process, Quali and Reliabili Engineering Inernaional [18] Robers, S.W., Conrol Char Tes based on Geomeric Moving Averages. Technomerices, 1959, 1. [19] Huner, J.S., The Exponeniall Weighed Moving Average, Journal of Quali Technolog,1998, 18 38