Change Point Estimation of Location Parameter in Multistage Processes

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1 Proceedigs of e World Cogress o Egieerig 0 Vol I WCE 0 July Lodo U.K. Chage Poit Estimatio of Locatio Parameter i ultistage Processes Seyed Taghi Ahava Niai ehdi Davoodi Elaz Asghari Toramai Abstract owig e time of a process chage would simplify e search idetificatio ad removal of e special causes at disturbed e process. Sice i may real world maufacturig systems e productio of goods comprises several autocorrelated stages; i is paper e problem of e chage poit estimatio for such processes is addressed. A first order autoregressive model (AR()) is used to model a multistage process observatios where a X -chart is established for moitorig its mea. A step chage is assumed for e locatio parameter of e model. After receivig a outof-cotrol sigal i order to determie e stage ad e sample at caused e chage (hece fidig e time of e step chage) two maimum lielihood estimators are proposed. At e ed e applicability of e proposed estimators is demostrated by a umerical eample. Ide Terms Statistical Process Cotrol Chage Poit ultistage Quality Cotrol aimum Lielihood Fuctio AR(). I. INTRODUCTION AND LITERATURE REVIEW The defiitio of multistage processes ca be Tfoud i may research wors ( e.g. [] [] ad [3]). Accordig to [3] multistage maufacturig processes (Ps) are e processes i which multiple statios are set up to maufacture a specific product. Eamples of Ps are: () a automotive body assembly at has multiple parts assembled at multiple statios; () a egie head productio at ivolves multiple machiig operatios of a sigle part at multiple statios; (3) a trasfer or progressive stampig process at ivolves multiple die statios to form a part; ad (4) semicoductor maufacturig processes i which a silico wafer develops i several stages wi several layers to form a chip. Several meods have bee developed i literature to model e Ps. [4 5] proposed e regressio adjustmet approach to moitor multistage processes. [6] [7] [8] [9] [0] ad [] developed multistage egieerig models wi liear state space structure to describe quatitative data of a multistage process. Some oer research wors have used autocorrelated AR() model to detect shifts i various multistage processes (e.g. [] [] ad [3]). [4] ad [5] auscript received arch 0; revised April 3 0. S. T. A. Niai is e departmet of Idustrial Egieerig Sharif Uiversity of Techology Azadi Av. Tehra Ira ( iai@sharif.edu).. Davoodi is wi e departmet of Idustrial Egieerig Sharif Uiversity of Techology Azadi Av. Tehra Ira (ehdi Davoodi is e correspodig auor phoe: ; ( m.davoodi@yahoo.com). E. Asghari is wi e departmet of Idustrial Egieerig Amirabir Uiversity of Techology 44 Hafez Av. Tehra Ira ( asghari.eg_ie@yahoo.com). ISBN: ISSN: (Prit); ISSN: (Olie) employed a Bayesia approach to model short ru processes. Whe a out-of-cotrol sigal is received by a cotrol chart e eact time at which e process wet out of cotrol is usually ot e time of e sigal. Quality ad process egieers desire to have a good estimate of e eact time (e chage poit) to search for special causes at disturbed e process. This problem is so called e chage poit estimatio. So far few studies have bee coducted wi e aim of process moitorig ad chage poit detectio i multistage processes i which most of em have bee performed usig e liear state space model to describe multistage processes. I e curret wor a multistage productio process is first modeled by a first-order autoregressive model (AR()) ad e a maimum lielihood chage poit estimatio meod is proposed to determie e time of a step-chage i e locatio parameter by idetifyig bo e stages ad e samples of a out-of-cotrol process. The remaider of e paper is orgaized as follows. I e et sectio e required bacgroud of is research is briefly itroduced. I sectio III e autocorrelated multistage process is described. I additio e lielihood fuctio is derived ad maimized to determie e time of e step-chage i e locatio parameter. A umerical eample is used i sectio IV to illustrate e applicatios of e proposed estimators. Coclusio ad recommedatios for future study are preseted i sectio V. II. BACKGROUND Xiag ad Tsug [6] used a liear state space model give i () for moitorig a multistage process. y C A () j m ; N Where y is e quality characteristic of e j j product i stage of e process ~ N 0 ad ~ N 0. Furer A ad C are e data matrices of e process ad e model parameters at eed to be estimated are (). T N () Sice e lielihood fuctio is too comple to be maimized ey used e E algorim wi a Kalma filter. The ey proposed a group epoetial weighted movig average chart (GEWA) to moitor e mea of e process. Assumig ow process parameters [] used e WCE 0

2 Proceedigs of e World Cogress o Egieerig 0 Vol I WCE 0 July Lodo U.K. model of [6] to describe multistage processes. They developed a directioal multivariate epoetial weighted movig average chart (DEWA) to moitor e mea of a multistage process by e use of eteded maimum lieliess ratio. They showed e performace of eir proposed DEWA chart to be superior to GEWA chart. oreover ey developed maimum lielihood estimators for estimatig out-of-cotrol stage ad product (sample) whe a sigal was received from e proposed DEWA cotrol chart. III. AUTOCORRELATED ULTISTAGE PROCESSES Cosider a process icludig stages where i each stage a equal umber of characteristics are measured. Every characteristic is correlated wi its correspodig characteristic i previous stages. Such a process is called a autocorrelated multistage process (AP). Oe of e mai assumptios of AP is at e samples are idepedet. For each sample ere are p quality characteristics at must be measured ad cotrolled. Each sample passes all stages of productio process ad all characteristics i various steps are to be measured. Fig. shows e passig of oe sample rough all e stages of a AP [3]. j j p j Stage j j p j Stage Fig.. The Pass of a sample rough all e stages of a autocorrelated process Let i j be t i measured characteristic of j sample i t stage i which i p j ad t. Assumig to be e sample umber based o which e cotrol chart sigals a out-of-cotrol coditio e aim is to idetify bo e stage ad e sample at which e out-of-cotrol has bee iitiated. The owledge of e chage poit ca greatly aid process egieers i idetifyig special causes ad ruig suitable correctio actio. IV. PROCESS ODELING Stage [7] used AR() time series to model processes wi autocorrelated observatio. We use e same cocept to model e autocorrelated multistage process wi e differece at stages have replaced observatios ad despite e origial model samples or observatios at are tae from e process are idepedet from each oer. The Equatio (3) ca describe a AP. j t j t j t (3) Where is e locatio parameter j is e measured t qualitative characteristic of j sample i t stage is e autocorrelatio coefficiet ad are idepedet j t radom variables followig a ormal distributio wi mea zero ad variace i.e. N 0. Now if e samples pass rough Q stages to eit e process e probability j j p j desity fuctio of e j sample is give i (4) ([7 8]). f ( X ) f (... ) j j j j Q Q ( ) ( ) ( ) j ep Q j t. j t t Where Q stage. (4) X is e characteristic vector of j j sample at A. Chage i Locatio Parameter Assume a multistage process starts wi a iitial value of. Whe a step chage occurs i e process e value of 0 becomes 0 0 d. This chage happes i sample of stage r. The cotrol chart detects is chage i sample ad stage wi a delay. Therefore e poit of receivig sigal differs from e actual chage time. (see Fig. i which Si ; i... deotes e i stage). Sample S S S Sample Sample τ- Sample τ Sample τ+ Sample S S S chage S S.. happeed.... S S S S S. Sr.... S S S S... S Out-of-cotrol Sigal Fig.. Sigal time of cotrol chart differs from e actual chage time B. Lielihood Fuctio Derivatio The lielihood fuctio of e process is derived by multiplicatio of e probability desity fuctios (pdf) of all tae samples where e pdf of each sample is give i (4). I e lielihood fuctio derivatio process two poits should be tae ito accout. First oe otices at for each sample e probability desity fuctio i (4) comprises of two parts. The first part correspods to e first stage ad e secod part relates to e rest of e stages. However e locatio parameter eists i bo parts causig e form of pdf to deped o e stage i which e chage has happeed. I oer words e shape of pdf will be differet if e chage happes i e first stage compared to e coditio at e chage happes durig oe of e et stages. The secod importat poit is e time of receivig sigal ad e questio is wheer is time correspods to e stage i which e chage has happeed o i oer stages. If e cotrol chart sigals at e same stage e shape of lielihood fuctio will be differet from Fig.. Noeeless e derivatio of e lielihood fuctio depeds o ese coditios. If e chage happes i ay oer stages a e first ad e cotrol chart sigals i e et stage(s) e logarim of lielihood fuctio is epressed i(5). ISBN: ISSN: (Prit); ISSN: (Olie) WCE 0

3 Proceedigs of e World Cogress o Egieerig 0 Vol I WCE 0 July Lodo U.K. log L ( r X ) ( ) log( ) log( ) 0 ( ) j j 0. j t r t 0. t t (5) t. t tr ( ) j j. j t. t However if e chage happes i e first stage ad e cotrol chart sigals durig e et stage(s) e logarim lielihood fuctio becomes (6). log L ( r X ) ( ) log log 0 ( ) j j 0. j t (6) ( ) j j. j t. t oreover if e chage happes i a stage oer a e first stage ad e cotrol chart sigals durig e same stage e chage has happeed e logarim of lielihood fuctio ca be epressed as (7). log L3( r X ) ( ) log( ) log( ) 0 ( ) j j 0. j t r 0. t. tr Fially if e chage happes i e first stage ad e cotrol chart sigals i e same stage e logarim of lielihood fuctio ca be derived as (8). log L4 ( r X ) ( ) log log 0 ( ) j j 0. j t ( ). t C. The Proposed Chage Poit Estimators As previously metioed e form of e lielihood fuctio depeds bo o e stage i which e chage happes ad o e sigalig sample; implyig four separate estimatios. We ote at e stage ad e out-of-cotrol sample are discrete variables wi ow limits. Therefore i e estimatio process e partial deviatios of e logarim of e lielihood fuctio wi respect to e locatio parameter should be obtaied. The LE of ad r is e value of e estimated stage ad e out-of-cotrol sample at maimize e lielihood fuctio or equivaletly its logarim. ) Chage Poit Estimator for e Locatio Parameter The estimator of e locatio parameter is differet for various combiatios of ad r because of differet lielihood fuctios derived i sectio 3.. If e chage happes at a stage oer a e first stage ad a out-ofcotrol sigal is give i e samples after e chage e locatio parameter estimator is obtaied by (9). (7) (8) ISBN: ISSN: (Prit); ISSN: (Olie) WCE 0

4 Proceedigs of e World Cogress o Egieerig 0 Vol I WCE 0 July Lodo U.K. t. t ( ) j tr j. j t. (9) t ˆ r If e chage happes at e first stage ad out-of-cotrol sigal comes i e et stages e (0) epresses e estimatio of e locatio parameter. Note at e stage is ow i is situatio. ( ) j j.. (0) j t t ˆ If e chage occurs at a stage oer a e first ad outof-cotrol sigal comes o e same stage e locatio parameter estimatio is give i ().. ˆ tr () 3 r Fially if e chage happes at e first stage ad e receivig sigal is i e stage i which e chage has occurred e estimatio of e locatio parameter is stated i (). t. t ˆ t 4 () ) Estimatig e out-of-cotrol stage ad sample I order to estimate e out-of-cotrol stage ad e sample simultaeously e log lielihood fuctio for each combiatio of ad r wi correspodig estimated locatio parameter should be evaluated. The combiatio set of ad r at provides a maimum of e log lielihood fuctio is e estimator deoted by ˆ ad rˆ respectively. They are give i Error! Referece source ot foud.. L ˆ r L r ˆ ˆ rˆ arg ma L3 r r ˆ 3 L 4 r ˆ 4 V. NUERICAL EXAPLE (7) I is sectio e applicatio of e proposed LE estimators for e locatio parameter e out-of-cotrol sample ad e out-of-cotrol stage is illustrated. Cosider a AP wi a sigle quality characteristic at four stages. Fiftee samples each cotaiig 4 observatios are geerated for a process wi ad 0.5. The subsequet samples are observed from a similar AP wi The X chart is used for moitorig purposes. This chart sigals i sample 8 of stage 3. Table I shows e estimates of e locatio parameter e out-of-cotrol sample ad e out-of-cotrol stage. I additio e samples obtaied i each stage e log lielihood values ad e estimatio of e locatio parameter for every sample at each stage is show i is table. It ca be see at sample 6 i stage is e simultaeous estimate of e out-of-cotrol sample ad stage respectively. Table. Estimatig e locatio parameter e out-of-cotrol sample ad e out-of-cotrol stage sample Observed quality characteristic Estimated Locatio Parameter stage stage stage 3 stage 4 Log Lielihood Fuctios ˆ ˆ ˆ ˆ 3 4 log L log L log L3 log L ISBN: ISSN: (Prit); ISSN: (Olie) WCE 0

5 Proceedigs of e World Cogress o Egieerig 0 Vol I WCE 0 July Lodo U.K. VI. CONCLUSION I is study a autocorrelated multistage process was first itroduced. For such a process ere are usually some quality characteristics at each stage at should be moitored. The quality characteristics of ay stage are autocorrelated wi e correspodig oes of e previous stage(s). The we cosidered a autocorrelated multistage process of a sigle quality characteristic at each stage. Net e process was modeled ad for a out-of-cotrol sigal obtaied by a X-bar cotrol chart e lielihood fuctio to estimate e locatio parameter e out-of-cotrol sample ad e out-of-cotrol stage was derived. Fially a umerical eample was give to demostrate e applicability of e proposed estimators. REFERENCES []. Agrawal R. J.F. Lawless ad R.J. ackay Aalysis of Variatio Trasmissio i aufacturig Processes-part II. Joural of Quality Techology vol. 3 o. pp []. Zhou S. Q. Huag ad J. Shi State Space odelig of Dimesioal Variatio Propagatio i ultistage achiig Process Usig Differetial otio Vectors. Robotics ad Automatio IEEE Trasactios o vol. 9 o. pp [3]. Shi J. Stream of Variatio odelig ad Aalysis for ultistage aufacturig Processes CRC Press Taylor & Fracis Group 007. [4]. Hawis D.. ultivariate quality cotrol based o regressio adjusted variables. Techometrics vol. 33 o. pp [5]. Hawis D.. Regressio adjustmet for variables i multivariate quality cotrol. Joural of Quality Techology vol. 5 o. pp [6]. Ji J. ad J. Shi State Space odelig of Sheet etal Assembly for Dimesioal Cotrol. Joural of aufacturig Sciece ad Egieerig vol. o. 4 pp [7]. Dig Y. J. Shi ad D. Ceglare Diagosability Aalysis of ulti- Statio aufacturig Processes. Joural of Dyamic Systems easuremet ad Cotrol vol. 4 o. pp [8]. Huag Q. S. Zhou ad J. Shi Diagosis of ulti-operatioal achiig Processes Through Variatio Propagatio Aalysis. Robotics ad CI Joural vol. 8 o. pp [9]. Djurdjaovic D. ad J. Ni Liear State Space odelig of Dimesioal achiig Errors. Trasactios of NARI/SE vol. o. pp [0]. Zhou S. Y. Che Y. Dig ad J. Shi Diagosability Study of ultistage aufacturig Processes Based o Liear ied-effects odels. Techometrics vol. 45 o. 4 pp []. Zou C. ad F. Tsug Directioal EWA Schemes for ultistage Process oitorig ad Diagosis. Joural of Quality Techology vol. 40 o. 4 pp []. Lawless J.F. R.J. ackay ad J.A. Robiso Aalysis of Variatio Trasmissio i aufacturig processes part I. Joural of Quality Techology vol. 3 o. pp [3]. Niai S.T.A. ad. Davoodi Desigig a ultivariate-ultistage Quality Cotrol System Usig Artificial Neural Networs. Iteratioal Joural of Productio Research vol. 47 o. pp [4]. Tsiamyrtzis P. A Bayesia Approach to Quality Cotrol Problems Faculty of e Graduate School Uiversity of iesota 000 [5]. So Y.S. ad S.W. Kim Bayesia Sigle Chage Poit Detectio i a Sequece of ultivariate Normal Observatios. Statistics vol. 39 o. 5 pp [6]. Xiag L. ad F. Tsug Statistical oitorig of ulti-stage Processes Based o Egieerig odels. IIE Trasactios vol. 40 o. 0 pp. 957(4) 008 [7]. Timmer D.H. J. Pigatiello ad. Logecer The developmet ad evaluatio of CUSU-based cotrol charts for a AR() process. IIE Trasactios vol. 30 o. 6 pp [8]. Hamilto J.D. Time series aalysis Priceto NJ Priceto Uiv. Press 994. ISBN: ISSN: (Prit); ISSN: (Olie) WCE 0

Estimating the Change Point of Bivariate Binomial Processes Experiencing Step Changes in Their Mean

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