A Study of Process Capability Analysis on Second-order Autoregressive Processes

Transcription

1 A Sudy of Process apabiliy Analysis on Second-order Auoregressive Processes Dja Shin Wang, Business Adminisraion, TransWorld Universiy, Taiwan. Szu hi Ho, Indusrial Engineering and Managemen, Naional Yunlin Universiy of Science and Technology, Taiwan. Tong Yuan Koo, Indusrial Engineering and Managemen, Naional Yunlin Universiy of Science and Technology, Taiwan. Absrac Process capabiliy analysis is conduced assuming ha he process under sudy is in saisical conrol and independen observaions are generaed over ime. However, in pracice i is very common o come across process which due o heir inheren naures, generae auo correlaed observaions. In he presen paper, we discuss he effec of auocorrelaion on he process capabiliy analysis when a se of observaions are esimaed by second-order auoregressive process, AR (). We propose an esimaion mehod for he process capabiliy analysis, when a se of observaions are produced by second order auoregressive model. We use he VBA sofware o simulaion he daa on he AR (). Then we use regression analysis o calculae process capabiliy inde a differen levels of auocorrelaion. Using he proposed mehod, we can find powerful decision rules o deermine he capabiliy of a process a given significance levels. Key words: Process apabiliy analysis, Saisical process conrol, Auocorrelaion, Auoregressive process. JEL lassificaion: 9, G, G 4

2 . Inroducion Process capabiliy analysis is conduced assuming ha he process under sudy is in saisical conrol and independen observaions are generaed over ime. Process capabiliy indices (PIs) are inroduced o give a clear indicaion of he capabiliy of a manufacuring process. In fac, PIs are organized o deermine wheher he process is capable of visiing specificaion limis on he qualiy feaures of ineres or no. Basic assumpions of PIs are he observaions are idenically, independen and normally disribuion. According o definiions and assumpions menioned. We can use he following well-known capabiliy indices: p USL LSL 6 LSL pl USL, pk min pl,, and Where LSL and USL are he lower and upper specificaion limis, respecively. Furhermore, is he process mean and is he sandard deviaion. However, in pracice i is very common o come across process which due o heir inheren naures and generae auo correlaed observaions. In recen years, many issues relaed he effecs of auocorrelaions on he process capabiliy analysis have been sudied. However, he direc impac of auocorrelaions on process is less known. Shore (997) is among he few researchers who have invesigaed he effec of auocorrelaions on process capabiliy analysis. He models he auocorrelaion srucure of a se of daa using an auoregressive model on AR (). He shows ha auo correlaed ime series may lead o a biased esimae of he rue capabiliy and ulimaely o wrong claims regarding he process performance. Auo correlaed observaions are common in indusry, especially when daa are sampled a a high frequency from processes wih ineria or carry-over effecs. Alhough no covering all siuaions, he AR () and AR () processes cover a relaiviy wide range of siuaions encounered in pracice. Wallgren. e al., (00) derives approimae confidence inervals for pk and pm when he daa can be modeled according o an AR()- or MA()-process wih unknown auocorrelaion funcion. He shows, hrough simulaions, ha if auocorrelaion is ignored when calculaing confidence inervals he empirical coverage rae differs considerably from he nominal one. Noorossana R. (00) combine procedure based on muliple regressions and ime series modeling, and proposed o remove he auocorrelaion paerns ha may be presen in he daa and also o esimae parameers effecively. He shows an eample ha auo correlaed daa could lead o biased esimaes of he rue process capabiliy indices and ulimaely o wrong decisions regarding he process performance. () ()

3 . Time series models. Firs-order Auoregressive Process An approach ha has proved useful in dealing wih auo correlaed daa is o direcly model he correlaive srucure wih an appropriae ime series model, use ha model o remove he auocorrelaion from he daa, For eample, suppose ha we could model he qualiy characerisic as following: () Where are unknown consans, is auocorrelaion coefficien and is normally and independenly disribued wih mean zero and sandard deviaion. Equaion is called a firs-order auoregressive models, AR (). The firs-order auoregressive model used in he equaion is no he only possible model for ime-oriened daa ha ehibis correlaive srucure.. Second-order Auoregressive Process An obvious eension o equaion 4 is as follow: (4) This is a second-order auoregressive model, AR (). Where is an unknown consan, and were auocorrelaion coefficien and is normally and independenly disribued wih mean zero and sandard deviaion. The second-order auoregressive model, AR () saionary model,, and. This model ofen occurs in chemical and process indusries. (See Mongomery, Johnson, and Gardiner (990) and Bo, Jenkins, and Reinsel (994)).. Auoregressive moving average models, ARMA The auoregressive moving average model (p,q) i.e., ARMA(p,q) is Z Z Z... Z... p p p q (5) Where i he average auoregressive of ih order is, i is he moving average of ih order, is consan, is he random error, Z is he value of ime. If p=, q=., i.e., ARMA(,), A firs-order mied model is Z. Where Z is he observed error erm a ime and is an uncorrelaed residual wih mean zero and sandard deviaion. (Noorossana R., 00) using he auocorrelaion funcion (AF) and parial auocorrelaion funcion (PAF) of error erms as an ARIMA (, 0, ) model or equivalenly ARMA (, ). The variance of he observaions required o perform he rue process capabiliy analysis is he same as he variance of Z which is given as following equaion:

4 Z Var ( ), where Z is he variance of random error. A combined procedure based on muliple regressions and ime series modelling was proposed o remove he auocorrelaion paerns ha may be presen in he daa and also o esimae model parameers effecively. Hereinafer, Mohamadi (0) esimaed he process capabiliy inde (PI) of auoregressive model AR () is called p au shows as follow: p au bias And 0, where p au he is process capabiliy inde of auoregressive model AR (), is model coefficien and is auocorrelaion coefficien. Since AR() parameers impac on he bias in in p au comparison o he PI which is known for independen observaions.. Model Building and Eplanaion In his paper we propose o use hese parameers o diminish second order auocorrelaion, AR (), effecs on he PI esimaion. Here, we apply a mulivariae regression analysis model as shown in Eq. (6). I has been known ha and, where is a linear combinaion of,, denoe he PI in Eq. () based on he independence assumpion. The, and give are correlaion coefficien and model parameer, respecively. (6) 0 The 0,, and are esimaed from he observaions of he process by mulivariae regression. In fac, he effecs ha, and may have on he PIs are he main moivaion for using he proposed model in he presence of AR () auocorrelaion. We follow a wo-sep procedure in order o calculae 0,, and. The firs sep is generaing ses of daa which are auo correlaed and calculaing for each se of daa. Then, 0,, and for each se of daa are esimaed by he mulivariae of he model, a simulaion is also performed. The research procedure in his paper is show as figure. We consider an AR () process in Eq. (4) o generae ses of auo correlaed daa where is a random variable ha represens he amoun by which he h measuremen will differ from he mean due o he effec of common causes. Typically, are independen and idenically disribued random variables wih mean zero and sandard deviaion. 4

5 Figure : The research procedure of his paper reae Regression model Simulaion daa Finess AR () Auocorrelaion model Regression model Finess Resul Analysis Regression model s Validiy We rewrie AR () model Eq. (4) as following: (7) Therefore, we can ransfer he auocorrelaion daa o he independen daa, and we can find ou sandard deviaion. We can calculae simulaion daa mean and AR () model coefficien and auocorrelaion coefficien,. We define four level auocorrelaion coefficiens,, hen calculaed simulaion daa o conduc opimizaion regression model well fied AR () auoregressive model, respecively. Finally, we use he saisic sofware finess es o find ou he opimizaion regression model. Meanwhile, we consider a siuaion where = in his simulaion, and hen make simulaion for,, and a predeermined sample size (N=0000). Aferwards, his procedure mus be repeaed for 0 ieraions wih differen, and in each ieraion. Eq. (6) can be used o obain he upper specificaion limi of each se of 0000 observaions. usl onclusion and suggesion f ( ) d ( usl) Where f () and are he probabiliy densiy funcion and he cumulaive densiy funcion of, respecively. I is clear ha Eq. (8) can be rewrien as usl ( ( )). (8) In he presen paper, we use he VBA sofware o simulaion he daa on he AR () auoregressive process. I should be noed ha we use differen =0.5, 0.6, 0.7, 0.8, 0.85, 0.9, 0.95,.0,.05,.,.,.5,.,.5,.4,.5,.6,.7,.8 for each ime, 400 ses of observaions are creaed. Then we se four level on auocorrelaion coefficien and. 5

6 There are 0 ses daa in every level and auocorrelaion coefficien and, and every se daa have 0000sample simulaion. Therefore, we seing, and using he, and o generae simulaed daa, hen using simulaed daa o eplore he regression models beween he difference of variables and he process capabiliy inde. 4. Resuls and Analysis Le us recall ha. Before assuming 0,,, 0 as he model coefficiens, i is vial o deermine wheher,, and are relaed or no. Since, we can assume as a response variable of he mulivariae regression, and 0,,, can be also assumed as he model coefficiens ha are esimaed from he observaions of process by he mulivariae regression. The regression equaions of are esimaed for differen inervals of and. In his regard, i is necessary o es he null hypohesis, he T- Saisic are used a he significance level. We should invesigae wheher is a significan difference he capabiliy inde and esimaed capabiliy inde a a given significance level. As a consequence, i seems reasonable o rejec he null hypohesis in ha confidence inerval is A= [-.96,.96] a 5% significance level. This means ha here is no significan difference beween Ĉ and in his siuaion. According o his procedure, i seems reasonable o conclude ha here is no significan difference beween Ĉ and for his value of and. i. The regression equaion of when 0 0., Table show as he, and, and he regression models coefficien beween he difference of variables and he process capabiliy inde, and he regression models as. 0 R-sq(adj)=0.6, i shows ha 6.% resoluion of he esimaed regression equaion is Table : The regression coefficien of, and Pred. Regression coefficien SSE T-Saisic onsan * * * * R-sq 0.67 R-sq(adj) 0.6 6

7 F 76.66* Noe: * saisic significan, R-sq(adj)=0.6 ii. The regression equaion of when 0 0., Table show as he, and, and he regression models coefficien beween he difference of variables and he process capabiliy inde, and he regression models as. 0 Table : The regression coefficien of, and Pred. Regression coefficien SSE T-Saisic onsan * * * * R-sq R-sq(adj) Noe: * saisic significan, R-sq(adj)= I shows ha 46.4% resoluion of he esimaed regression equaion is: iii. The esimaed when daa are auo correlaed The equaions which are usually used o esimae are obained by using he mulivariae regression and displayed on he classificaion of Figure. Figure : The esimaed when daa are auo correlaed wih 0 0., 0 0., , , Daa are auo correlaed ˆ ˆ ˆ ˆ

8 5. onclusions and Recommendaions In he presen paper we propose an esimaion mehod for he process capabiliy analysis when a se of observaions are auo correlaed and produced by an auoregressive model of order wo. The qualiy of he ou from an auo correlaed process can be easily managed by using his classificaion o monior he difference beween cusomer requiremens and he acual performance of an auo correlaed process. This paper is based on subracing consecuive observaions from each oher in order o obain samples wih independen observaions, and hen using regression analysis o calculae process capabiliy inde a differen levels of auocorrelaion. We can find ou powerful decision rules o deermine he capabiliy of process on given significance levels. A simulaion was also employed o evaluae he provided resuls. Acknowledgemen We would like o acknowledge his work was financially suppored by he Minisry of Science and Technology of he Reblic of hina (Taiwan). References Bo, G. E. P., G. M. Jenkins, and G.. Reinsel, 994, Time Series Analysis, Forecasing, and onrol, rd ediion, Prenice-Hall, Englewood liffs, NJ. Mongomery, D.., L. A. Johnson, and J. S. Gardiner, 990, Forecasing and Time Series Analysis, nd ed., McGraw-Hill, New York. Mohsen Mohamadi, Mehdi Foumani, Babak Abbasi, 0, Process apabiliy Analysis in he Presence of Auocorrelaion. Journal of Opimizaion in Indusrial Engineering 9, 5-0. Noorossana R.,00, Shor communicaion process capabiliy analysis in he presence of auo correlaion. Qualiy and Reliabiliy Engineering Inernaional 8, Shore H., 997, Process capabiliy analysis when daa are auo correlaed. Qualiy Engineering, 9(4), Wallgren E., 00, onfidence limis for he process capabiliy inde pk for auo correlaed qualiy characerisics. Froniers in Saisical Qualiy onrol 6, -. 8