Power of Mean Chart under Second Order Auto- Correlation with Known Coefficient of Variation

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1 Iteratioal Joural of Scietific a Research ublicatios Volume Issue December 0 ISSN 50-5 ower of Mea Chart uer Seco Orer Auto- Correlatio with Kow Coefficiet of Variatio Sigh D. a Sigh J.R Vikram uiversity Ujjai Iia Abstract- I this paper we ivestigate the effect of autocorrelatio o the power fuctio of mea chart with kow coefficiet of variatio. We sythesie the seco orer autocorrelatio process by its three ifferet roots. I particular the shift i the auto-correlatio structure from iepeet ata to a raom walk this is a special case of the structural shift occurrig i the process. For various values of roots the values of power fuctios are tabulate with kow coefficiet of variatio. Ie Terms- cotrol chart power fuctio auto-correlatio coefficiet of variatio. C I. INTRODUCTION otrol charts are effective for o-lie process moitorig a recet research i statistical process cotrol SC has focuse o the evelopmet of avace moitorig a cotrol techiques. Traitioally cotrol charts were evelope uer the assumptio that observatios of quality characteristics of iterest are statistically iepeet both withi a betwee samples. Furthermore successful implemetatio of traitioal cotrol charts relies o the assumptio of process stability. However these assumptios are usually violate i practice a ew approaches ee to be aopte. May processes show certai itrisic tre patter. Usually as log as this is withi a certai rage it is cosiere acceptable. A commo eample is the tool wear process which may ehibit a tre i the process mea level because of the physical ature. For such tree processes the absolute level is of great cocer a it shoul ot ecee a certai specificatio; otherwise they shoul be replace. However we also ee to moitor the process i a traitioal way i the sese that ay abormal chages ca be etecte quickly. I geeral the traitioal Shewhart cotrol charts are ot suitable for moitorig auto correlate process irectly as the iepeece a stability assumptios are violate. The purpose of the preset stuy is to evelop a suitable cotrol chartig a ecisio makig proceure to moitor auto correlate processes. I particular the shift i the autocorrelatio structure from iepeet ata to a raom Walk which is cosiere by Bo a Luceo 997 Bo et al. 994 AstrÖm a Wittemark 989 age 955 Roberts 958 Bissel 978 Wheeler 987 Champ a Wooall 987 Nelso 984 Ishikawa 976 ASQC 986 Davis a Wooall 988 Aere et al. 99 Koth et al a Motgomery 985. Detectio a elimiatio of parameter shifts are the typical task of statistical cotrol chartig. These techiques i be applie for the etectio of shifts from the i-cotrol to the out-of-cotrol state i the raom compoets. To etect shifts samples are take from the process. Sice the power fuctio will play mai role i hypothesis testig we use sample ata for repeate statistical test of simple hypothesis versus alterative hypothesis i case of autocorrelatio as follows : inull hypothesis Ho : process is i cotrol i our case : H o : = 0 = = 0 agaist the iialterative Hypothesis H : prouctio is out of cotrol i our case: H : 0 or a. The rejectio of Hypothesis H o is iterprete as a alarm or out-of- cotrol sigal : the process is stoppe for ivestigative a corrective actio. Acceptace of hypothesis H o o alarm meas that the process cotiuous without itervetio. The alterative H : 0 or a is ofte restricte to more specific alteratives by aitioal prior iformatio kowlege e.g. that the autoregressive parameters a is kow to be i cotrol kow particular alteratives 0. The ceter lie i cotrol charts are fie at 0 a power fuctios are calculate agaist alterative hypothesis. I this paper we ivestigate the effect of autocorrelatio o the power fuctio of X -chart is stuie with kow coefficiet of variatiocv. We sythesie the seco orer auto correlate process AR by its three ifferet roots. I particular the shift i the autocorrelatio structure from iepeet ata to a raom walk which is a special case of the structural shift occurrig i a AR process. For various values of roots the values of power fuctios are tabulate a compare with those the iepeet case with kow cv case. II. SECOND ORDER AUTOREGRESSIVE MODEL Cosier a maufacturig process where a quality characteristic is measure at equiistace time poits.... This situatio may occur i a iscrete maufacturig process which prouces iscrete time... with oe quality characteristic of iterest. It may also occur i a cotiuous maufacturig process where the quality characteristic of iterest is measure at iscrete equiistat time poits. We eote the behavior of the quality characteristic as.... It will assume that o EC cotrol actio ca be represete by some cotrollable variable or factor t such that t = + t

2 Iteratioal Joural of Scietific a Research ublicatios Volume Issue December 0 ISSN 50-5 where is a costat a t is a statioary time series with ero mea a staar eviatio. A Durbi a Watso 950 statistic ca be use to etect the presece or absece of serial correlatio. The problem however is that to o oce the suspicio of epeece via the serial correlatio test is cofirme. If serial correlatio eist we use ietificatio techiques to efie the ature of t. Whe ietificatio is complete the likelihoo fuctio ca provie maimum likelihoo estimate of the parameters of the ietifie moel. Suppose that a correlatio test reveale the presece of ata epeece a ietificatio techique suggeste autoregressive moel of orer two AR say the we ca epress t of equatio as where t t t k t... ~ 0 i t N ii cov t 0 t t The class of statioary moels that assume the process to remai i equilibrium about a costat mea level. The variace of AR process is give by:. Followig Keall a Stuart 976 it ca be show that for statioarity the roots of the characteristic equatio of the process i equatio B B B must lies outsie the uit circle which implies that the parameters a must satisfy the followig coitios : 6 G Now If a G are the roots of the characteristic equatio of the process give by equatio 5 the G G For statioary we require that Thus three situatios ca theoretically arise: i. i Roots G a G are real a istict i. e. 4 0 ii Roots G a G are real a equal i. e. 4 0 iii Roots G a G are comple cojugate i. e Whe the serial correlatio is preset i the ata we have for the istributio of the sample mea its mea a variace is give by E Var ap 9 ap where epes o the ature of the roots G a G a for ifferet situatios is give as follows : i If G a G are real a istict G G G i G G G ap G G G GG G G GG Where ii ap iii ap = r 0 G G G G G If G a G are real a equal G G G G G G = re If G a G a comple cojugate G G G G G G u W u u cc

3 Iteratioal Joural of Scietific a Research ublicatios Volume Issue December 0 ISSN 50-5 Where W 4 cos u cos u u u 4 4 si u si u si u cos u si u Z u si u si u si u cos u si u = - u cos a. The t eote the chage i the level of the compesatig variable moel at the time t i.e. the ajustmet mae at the time poit t. The t. is Gaussia white oise with variace Throughout we suppose that the oise variace is kow. I practice this is justifie if reliable estimates of are available from the evaluatio of a large umber of previous values of the process e.g. urig the setup phase. The real - value parameters a the autoregressive parameters etermies the ifluece of the preceig time poit t - a t - o the preset time poit t. We assume a i-cotrol value = = O for the autoregressio parameters. It is possible that the autoregressio parameters may shift to a out-of-cotrol value 0. III. OWER OF THE CHART UNDER SECOND ORDER AUTOCORRELATION WITH KNOWN COEFFICIENT OF VARIATIONCV From a raom sample of observatios... a estimator has bee costructe by Searls 964 where w j j E where w is a scalar a is chose so that the MSE is miimum. Searls 964 has show that w v a hece MSE / 4 The istributio of sample mea uer seco orer autocorrelatio with kow coefficiet of variatio is give by f / where 5 e / I this evelop it is assume that the process has a ormal istributio with mea a MSE /. It is further assume that at the time of etermiig the cotrol limits the process is i a state of statistical cotrol a the same evice is use as will be employe for kow cv. The istributioal behavior of epes oly o v a. Thus the ata use for establishig the limits o the cotrol charts comes from a process N / that is. Whe the process shifts the ata is N / assume to come from a populatio. If samples of sie are take from the populatio N / a the value of is plotte with cotrol limits of the power of etectig the chage of process is give by the followig formula. X r{ } pr{ 6 uer seco orer autocorrelatio with kow power fuctio is r{ } pr{ } X 7 } c we have

4 Iteratioal Joural of Scietific a Research ublicatios Volume Issue December 0 4 ISSN 50-5 / / / / / / / r r 8 9 r r Where / a 0 r r Where u u ep.

5 Iteratioal Joural of Scietific a Research ublicatios Volume Issue December 0 5 ISSN 50-5 IV. ILLUSTRATIVE NUMERICAL COMARISON The values of power fuctio for some chose values of v a three ifferet roots alog with iepeece case have bee worke out usig equatio a give i Table a Table for = 5 a 7.To give a visual compariso the power curves have bee raw for =5 i figure to figure 4 attach. A compariso of various curves for AR process with classical cotrol chart havig iepeet observatios shows that comple cojugate roots have the teecy to brig the power curve for auto correlate ata very close to the power curve for iepeet observatio. However there is marke ifferece i the power curves for the other two situatios e.g. whe the roots are i real a equal a ii real a istict. I both the situatios there is a large eviatio i the power curves from the iepeet observatio with kow cv. I practical situatios however the case of real a equal roots harly arise. Moreover this is too simple a case for calculatios a iclue oly for the sake of completio. It is see from the table that possibility of shifts ot oly the mea parameters but also i the auto auto regressio parameters may result from assigable causes occurrig over prouctio time but ofte also from misietificatio of the auto regressio moel. e.g. a biase recomme to use of power curves for -chart uer the process moel where the shifts i the mea a the auto regressio parameters are possible. Whe cv icreases the values of power fuctio i all situatios close to each other.

6 Iteratioal Joural of Scietific a Research ublicatios Volume Issue December 0 6 ISSN 50-5 TABLE: ower Fuctio of X -Chart uer AR rocess with Kow cv =5 Iepeet Observatios Roots are Real a Distict α =0.α =0.6 Roots are Real a Equal α =0.8α =-0.6 Roots are Comple Cojugate α =0.8α =-0.6 v=0 v= v=4 v=6 v=0 v= v=4 v=6 v=0 v= v=4 v=6 v=0 v= v=4 v=

7 Iteratioal Joural of Scietific a Research ublicatios Volume Issue December 0 7 ISSN TABLE: ower Fuctio of X -Chart uer AR rocess with Kow cv =7. Iepeet Observatios Roots are Real a Distict α =0.α =0.6 Roots are Real a Equal α =0.8α =-0.6 Roots are Comple Cojugate α =0.8α =-0.6 v=0 v= v=4 v=6 v=0 v= v=4 v=6 v=0 v= v=4 v=6 v=0 v= v=4 v=

8 Iteratioal Joural of Scietific a Research ublicatios Volume Issue December 0 8 ISSN

9 Iteratioal Joural of Scietific a Research ublicatios Volume Issue December 0 9 ISSN 50-5 Fig.: ower Curve of X chart uer AR rocess with cv = 0. Fig.: ower Curve of X chart uer AR rocess with cv =.

10 Iteratioal Joural of Scietific a Research ublicatios Volume Issue December 0 0 ISSN 50-5 Fig.: ower Curve of X chart uer AR rocess with cv = 4. Fig.4: ower Curve of X chart uer AR rocess with cv = 6.

11 Iteratioal Joural of Scietific a Research ublicatios Volume Issue December 0 ISSN 50-5 REFERENCES [] Aere L. A. Champ C. W. a Rigo S. E. 99. Evaluatio of cotrol charts uer liear tre Commuicatios i Statistics Theory a Methos 00 pp [] ASQC986. Automotive Divisio. Statistical rocess Cotrol Maual America Society for Quality Cotrol Milwaukee. [] Astrom K. J. a Wittemark B Aaptive Cotrol Aiso- Wesley Reaig MA. [4] Bissel A. F 978. A attempt to uify the theory of quality cotrol proceures Bulleti i Applie Statistics 5 pp. -8. [5] Bo G. E..Jekis G. M. a Reisel G. C Time Series Aalysis Wiley & SosNew York. [6] BoG. E.. a Luceo A Statistical Cotrol by Moitorig a Feeback Ajustmet Wiley & Sos New York. [7] Champ C. W. a Wooall W. H Eact results for shewhart cotrol charts with supplemetary rus rules Techometrics 94 pp [8] Davis R. B. a Wooall W. H erformace of the cotrol chart tre rule uer liear shift" Joural of Quality Techology 04 pp [9] Durbi J. a Watso G. S Testig for serial correlatio i least squares regressio Biometrika 7. [0] Ishikawa K Guie to Quality Cotrol Asia rouctivity Orgaiatio Tokyo. [] Koth S. Schmi W. a Schoe A Simultaeous Shewhart type charts for the mea a variace of a time series Arbeitsbericht 07 of Europa-Uiversity at Frakfurt Oer. [] Motgomery D. C Itrouctio to Statistical Quality Cotrol Wiley & Sos New York. [] Nelso L. S The Shewhart cotrol chart { tests for special causes Joural of Quality Techology 6 pp [4] age E. S Cotrol charts with warig lies Biometrics 4 pp [5] Roberts S. W roperties of cotrol chart oe tests The Bell System Techical Joural 7 pp [6] Searls D. T "The utiliatio of kow coefficiet of variatio i the estimatio proceure" Joural of America Statistical Associatio [7] Wheeler D. J Detectig a shift i process average: tables of the power fuctio for X bar charts Joural of Quality Techology 54 pp AUTHORS First Author Sigh D. Vikram uiversity Ujjai Iia chouhaharm@reiffmail.com Seco Author Sigh J.R Vikram uiversity Ujjai Iia

Statistical Fundamentals and Control Charts

Statistical Fundamentals and Control Charts Statistical Fudametals ad Cotrol Charts 1. Statistical Process Cotrol Basics Chace causes of variatio uavoidable causes of variatios Assigable causes of variatio large variatios related to machies, materials,