Multivariate Statistical Process Control Charts and the Problem of Interpretation: A Short Overview and Some Applications in Industry

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1 Multvarate Statstcal Process Control Charts and the Problem of Interpretaton: A Short Overvew and Some Applcatons n Industry S. Bersms 1 J. Panaretos and S. Psaraks Abstract- Woodall and Montgomery [35] n a dscusson paper, state that multvarate process control s one of the most rapdly developng sectons of statstcal process control. Nowadays, n ndustry, there are many stuatons n whch the smultaneous montorng or control, of two or more related qualty - process characterstcs s necessary. Process montorng problems n whch several related varables are of nterest are collectvely known as Multvarate Statstcal Process Control (MSPC).Ths artcle has three parts. In the frst part, we dscuss n bref the basc procedures for the mplementaton of multvarate statstcal process control va control chartng. In the second part we present the most useful procedures for nterpretng the out-of-control varable when a control chartng procedure gves an out-of-control sgnal n a multvarate process. Fnally, n the thrd part, we present applcatons of multvarate statstcal process control n the area of ndustral process control, nformatcs, and busness. Index Terms- Qualty Control; Process Control; Multvarate Statstcal Process Control; Hotellng's T²; CUSUM; EWMA; PCA; PLS; Identfcaton; Interpretaton. I. INTRODUCTION Most Statstcal Process Control (SPC) approaches are based upon the control chartng of a small number of varables, usually the fnal produce qualty, and examnng them one at a tme. Ths s napproprate for most process ndustry applcatons. It totally gnores the nformaton collected on the process varables possbly hundreds. The practtoner can not really study more than two or three charts to mantan process or product qualty. It s very helpful that n practce, only a few events are drvng a process at any one tme; dfferent combnatons of these measurements are smply reflectons of the same underlyng events. Multvarate SPC refers to a set of advanced technques for the montorng and control of the operatng performance of batch and contnuous processes. More specfcally, multvarate SPC technques reduce the nformaton contaned wthn all of the process varables down to two or three composte metrcs through the applcaton of 1 Unversty of Praeus, Department of Statstcs and Insurance Scence, Praeus, Greece Athens Unversty of Economc and Busness, Department of Statstcs, Athens, Greece statstcal modelng. These composte metrcs can then be easly montored n real tme n order to benchmark process performance and hghlght potental problems, thereupon provdng a framework for contnuous mprovements of the process operaton. Woodall and Montgomery [35] n a dscusson paper, state that multvarate process control s one of the most rapdly developng sectons of statstcal process control. Harold Hotellng establshed multvarate process control technques n hs 1947 poneerng paper. Hotellng [11] appled multvarate process control methods n a bombsghts problem. Jackson [1] stated that any multvarate process control procedure should fulfll four condtons: a) an answer to the queston: "Is the process n control?" must be avalable; b) an overall probablty for the event "Procedure dagnoses an out-of-control state erroneously" must be specfed; c) the relatonshps among the varables - attrbutes should be taken nto account; d) an answer to the queston: "If the process s out-of-control, what s the problem?" should be avalable. The Jackson s fourth condton s the most challengng problem at ths tme n the MSPC area, an appealng subject for many researchers n the last years, and the man topc under consderaton n ths artcle. In Secton, we dscuss n bref the basc procedures for the mplementaton of multvarate statstcal process control va control chartng. In Secton 3, we descrbe the most sgnfcant methods for the nterpretaton of an out-ofcontrol sgnal. Furthermore, n Secton 4, we present an extended set of applcaton of multvarate statstcal process control n the area of ndustral process control. Fnally, n Secton 5 some concludng remarks are gven wth some ponts for further research. II. CONTROLLING AND MONITORING MULTIVARIATE PROCESSES USING CONTROL CHARTS As we already stated, statstcal process control technques are wdely used n ndustry. The most common process control technque s control chartng. There are two dstnct phases of control chartng, Phase I and Phase II. In Phase I, charts are used for retrospectvely testng whether the process was n control when the frst subgroups were beng drawn. In ths phase, the charts are used as ads to the practtoner, n brngng a process nto a state of

2 statstcal n-control. Once ths s accomplshed, the control chart s used to defne what s meant by statstcal ncontrol. In Phase II, control charts are used for testng whether the process remans n control when future subgroups are drawn. There are multvarate extensons for all knds of unvarate control charts (see e.g. Fgure 1), such as multvarate Shewhart type control charts, multvarate CUSUM control charts, and multvarate EWMA control charts. In addton, there are unque procedures for the constructon of multvarate control charts, based on multvarate statstcal. Shewhart type control charts for controllng the mean of an ndustral process are usually based on the well known Mahalanobs dstance statstc. The alternatve forms of ths statstc (dstance) for the Phase I may be summarzed as χ n µ, for = 1,,..., m ratonal subgroups, where n s the sample sze of each ratonal subgroup (wth n=1 for ndvdual observatons), µ s the vector of known means, Σ s the known covarance matrx and fnally s the vector of samples means for 1 followng: a) = ( µ ) Σ ( ) t the th t 1 ratonal subgroup, b) T ( ) ( ) = n S for = 1,,...,m, where s the pooled vector of sample means calculated usng the n observed sample mean vectors, and S s the pooled sample covarance matrx. The T, and χ statstc represent the weghted dstance of any pont from the target (process mean under stable condtons). Under the assumpton that the m samples are ndependent and the jont dstrbuton of the p varables s the multvarate Normal, the follows a ch-square dstrbuton wth p degrees of freedom and the χ T, follows p( m 1)( n 1) tmes an F dstrbuton wth p, mn-m-p+1 mn m p + 1 degrees of freedom. Thus, the approprate probablty lmts may be obtaned usng the known dstrbutons of the correspondng statstc. In Fgure, a control chart for a bvarate Normal process based on T statstc s gven. Moreover, n the specal case of a bvarate Normal process a control ellpse may be used. The ellpsod presented n Fgure 3, represents the 95% probablty area of the bvarate Normal process. Shewhart type control charts for controllng the varance of an ndustral process are usually based ether on the determnant of the covarance matrx S whch s called the generalzed varance, or on the trace of the covarance matrx, trs, whch s the sum of the varances of the varables. For specfc applcatons of these charts, as well as applcatons of other multvarate methods n qualty mprovement, the nterested reader may consult Alt [1], Werda [34], Lowry and Montgomery [15], Ryan [8], Bersms [], Bersms et al. [3], Koutras et al. [14] or the more recent book by Mason and Young [17] Chart for Length UCL = 19,94 CTR = 1,4 LCL = 7,54 Fgure 1: An unvarate Shewhart Type Control Chart (p=1) T-Squared Multvarate Control Chart UCL = 11,5 Fgure : A multvarate Shewhart Type Control Chart (p=) Shape Control Ellpse Length Fgure 3: A Control Ellpse(p=) Multvarate Shewhart type control charts use the nformaton only from the current sample and they are relatve nsenstve to small and moderate shfts n the mean vector. Multvarate Cumulatve Sum (MCUSUM) and Multvarate Exponentally Weghted Movng Average (MEWMA) control charts are developed to overcome ths problem. The multvarate CUSUM control charts are dstngushed n two major categores. In the frst case, the drecton of the shft (or shfts) s consdered to be known (drecton specfc schemes) whereas n the second the drecton of the shft s consdered to be unknown (drectonally nvarant schemes). Here we may note that the Shewhart type s always drectonally nvarant and the EWMA type control charts at the most of the cases. Multvarate CUSUM schemes have been gven by Woodall and Ncube [3] (the Multple Unvarate CUSUM Scheme, by Healy [1] (the CUSUM Based on the SPRT), by Croser [5] (the CCV Scheme), as well as by Pgnatello

3 and Runger [] (the Mean Estmatng CUSUM). The multvarate EWMA control chart proposed by Lowry et al. [1]. A problem wth utlzng tradtonal multvarate Shewhart charts or multvarate CUSUM and EWMA schemes s that they may be mpractcal for hgh-dmensonal systems wth collneartes. A common procedure for reducng the dmensonalty of the varable space s the use of projecton methods lke Prncpal Components Analyss (PCA) and Partal Least Squares (PLS). These two methods are based on buldng a model from a hstorcal data set, whch s assumed to be ncontrol. After the model s bult, the future observaton s checked to see whether t fts well n the model. These multvarate methods have the advantage that they can handle process varables and product qualty varables. Technques such as PCA and PLS are used prmary n the area of chemometrcs (see eg Kourt[13], Wasterhus et. al [33]) but they seem to be very promsng n any knd of multvarate process. III. IDENTIFYING THE OUT-OF-CONTROL VARIABLE In case that a unvarate control chart gves an out-ofcontrol sgnal, the practtoner may easly conclude what the problem s and gve a soluton snce a unvarate chart s related to a sngle varable. In a multvarate control chart the soluton to ths specfc problem s not straghtforward snce any chart s related to a number, greater than one, of varables and also correlatons exst among them. In ths secton we present methods for detectng, whch of the p varables s out of control. A frst approach to ths problem was proposed by Alt [1] who suggested the use of Bonferron lmts. Hayter and Tsu [9] extended the dea of Bonferron-type control lmts by gvng a procedure for exact smultaneous control ntervals for each of the varable means, usng smulaton. A smlar control chart s the Smulated MnMax control chart presented by Sepuldveda and Nachlas [9]. Alt [1] and Jackson [1] dscussed the use of an ellptcal control regon. However, ths process has the dsadvantage that t can be appled only n the specal case of two qualty characterstcs. An extenson of the ellptcal control regon as a soluton to the nterpretaton problem s gven by Chua and Montgomery [4]. Today the use of T² decomposton proposed by Mason et al. [18] s consdered as the most valuable. The man dea of ths method s to decompose the T² statstc nto ndependent parts, each of whch reflects the contrbuton of an ndvdual varable. The problem wth ths method s that the decomposton of the T² statstc nto p ndependent T² components s not unque. Thus, Mason et al. [19] gve an approprate computng scheme that can greatly reduce the computatonal effort. Mason et al. [] presented an alternatve control procedure for montorng a step process, whch s based on a double decomposton of Hotellng's T² statstc. Mason and Young [1] showed that by mprovng the model specfcaton at the tme that the hstorcal data set s constructed, t may be possble to ncrease the senstvty of the T² statstc to sgnal detecton. The methodologes of Murphy [4], Doganaksoy et al. [], Tmm [31] and Runger et al. [7], are specal cases of Mason's et al. [18] parttonng of T². Jackson [1] proposed the use of prncpal components for montorng a multvarate process. Snce the prncpal components are uncorrelated, they may provde some nsght nto the nature of the out of control condton and then lead to the examnaton of partcular orgnal observatons. Tracy et al. [3] expanded the prevous work and provded an nterestng bvarate settng n whch the prncpal components have meanngful nterpretatons. Prncpal components can be used to nvestgate whch of the p varables are responsble for an out-of-control sgnal. Untl nowadays, wrters have proposed varous methods whch use prncpal components for nterpretng an out-ofcontrol sgnal. The most common practce s to use the frst k most sgnfcant prncpal components, n the case that a T² control charts gves an out-of-control sgnal. The prncpal components control charts, whch were analyzed n the correspondng secton, can be used. The basc dea s that the frst k prncpal components can be physcally nterpreted, and named. Therefore, f the T² chart gves an out-of-control sgnal and for example the chart for the second prncpal component gves also an out-of-control sgnal, then from the nterpretaton of ths component, a drecton can be taken for whch varables are the suspect to be out-of-control. The practce just mentoned transforms the varables nto a set of attrbutes. The dscovery of the assgnable cause that led to the problem, wth ths method, demands a further knowledge of the process tself from the practtoner. The basc problem of ths method s that the prncpal components have not always a physcal nterpretaton. Accordng to Jackson [1], the procedure for montorng a multvarate process usng PCA can be summarzed as follows: For each observaton vector, obtan the z-scores of the prncpal components and from these compute T². If ths s n control, contnue processng. If t s out-of-control, examne the z-scores. As the prncpal components are uncorrelated, they may provde some nsght nto the nature of the out-of-control condton and may then lead to the examnaton of partcular orgnal observatons. Kourt and MacGregor [13], provde a dfferent approach based on prncpal components analyss. The T² s expressed n terms of normalzed prncpal components scores of the multnormal varables. When an out-of-control sgnal s receved, the normalzed score wth hgh values are detected, and contrbuton plots are used to fnd the varables responsble for the sgnal. A contrbuton plot ndcates how each varable nvolved n the calculaton of that score contrbutes to t. Computng varable contrbutons elmnates much of the crtcsm that prncpal components lack of physcal nterpretaton. Ths approach s partcularly applcable to large ll condtoned data sets due to the use of prncpal components. Contrbuton plots are also explored by Wasterhus et al. [33]. Maravelaks et al. [] proposed a new method based on prncpal components analyss. Theoretcal control lmts

4 were derved and a detaled nvestgaton of the propertes and the lmtatons of the new method were gven. Furthermore, a graphcal technque whch can be appled n these lmtng stuatons was provded. Fuchs and Benjamn [7], presented a method for smultaneously controllng a process and nterpretng a outof-control sgnal. Ths s a new chart (graphcal dsplay) that emphaszes the need for fast nterpretaton of an out-ofcontrol sgnal. The multvarate profle chart (MP chart) s a symbolc scatterplot. Summares of data for ndvdual varables are dsplayed by a symbol, and global nformaton about the group s dsplayed by the locaton of the symbol on the scatterplot. A symbol s constructed for each group of observatons. The symbol s an adopton of a profle plot that encodes vsually the sze and the sgn of each varable from ts reference value. Fuchs and Kenett [8], developed a Mntab macro for creatng MP charts. Sparks et al. [3], presented a method for montorng multvarate process data based on the Gabrel bplot. They llustrated the use of the bplot on an example of ndustral data. Nottngham et al. [5], developed radal plots as SASbased data vsualzaton tool that can mprove process control practtoner's ablty to montor, analyze, and control a process. Fnally, Maravelaks and Bersms [3] presented an algorthm usng the well known Andrews curves for solvng the problem of nterpretng an out-of-control sgnal. the 1 th tme pont s n-control. Usng the fact that the process s n-control we may use the estmated parameters n the Phase II analyss. SQUARED Quantle-Quantle Plot Ch-Square dstrbuton Fgure 4: Quantle-Quantle Plot for the Values of Length Shape Weght T IV. APPLICATIONS OF MULTIVARIATE SPC TECHNIQUES IN THE INDUSTRIAL ENVIRONMENT In ths Secton, we wll dscuss n bref an applcaton of the multvarate SPC technques n ndustry. Specfcally, we wll analyze a three-varable real case relatve to the qualty of a chemcal process. In the begnnng, we proceed wth a Phase I analyss. In ths Phase, our nterest s to estmate the parameters (the mean vector and the covarance matrx), to check for the exstence of dependence among the varables and fnally, to check the valdty of the assumpton of multvarate normalty. As a frst step n the analyss, we must test the assumpton of multvarate normalty of the three varables. If the three varables come from a 3-dmensonal Normal dstrbuton the T values must follow a ch-square dstrbuton wth 3 degrees of freedom. As we may observe n Fgure 4 the hypothess that the T values follow a ch-square dstrbuton wth 3 degrees of freedom can not be rejected. The second step n the analyss s the calculaton of the correlaton matrx. The applcaton of a multvarate control chart s needed only f the three varables are strongly related. As we may easly see n Fgure 5 the three varables are strongly related wth correlaton coeffcents equal to r (, ) =.599, r(, ) =.787, r(, ) = Snce there s a strong correlaton among the three varables, we must use a multvarate procedure for controllng the mean level of the process. Thus, we may apply a Shewhart type control chart and evaluate the state of the process. In Fgure we may see that the process tll Fgure 5: Matrx Scatter Plot for the three varables T-Squared Multvarate Control Chart UCL = 14,1 Fgure : A multvarate Shewhart Type Control Chart (p=3) In Phase II, the multvarate Shewhart type control chart s used for testng whether the process remans n control from the 11 st tme pont and after. At the 11 st pont as we may easly observe n Fgure 7 the process moved to an out-ofcontrol state. At ths tme pont the practtoner must fnd the varable that contrbuted n the out-of-control sgnal. As we presented n Secton 3 there are too many optons for dentfyng the varable responsble for the out-ofcontrol message. In ths applcaton, we wll use the methods proposed by Maravelaks and Bersms [3] and Maravelaks et al. []. Both of these procedures are classfed to the graphcal technques for nterpretng the out-of-control sgnal.

5 The rato F 13, whch connects the thrd varable and the frst prncpal component, s charted n Fgure 8. In Fgure 9, the other two ratos F 1, F 11, are presented. From ths fgures t s clear that the thrd varable s responsble for the out-of-control sgnal at the 11 st samplng pont. varable s responsble for the out-of-control sgnal. In concluson, the two methods gave us the same result, thus the practtoner has to check for possble assgnable causes at the mechansms related to the thrd varable. Multvarate Control Chart V. COMMENTS T-Squared UCL = 14,1 Interestng areas for further research n the doman of multvarate SPC are robust desgn of control charts and nonparametrc control charts. The research for multvarate attrbutes control charts s also a promsng task. The problem of nterpretng an out-of-control sgnal s an open area whch needs further nvestgaton Fgure 7: A multvarate Shewhart Type Control Chart (p=3) 1,18,1,14,1 - -1,,9,,3,,17,14 Chart for F UCL =,7 CTR =,1 LCL =,15 Fgure 8: Identfyng the out-of-control varable (Procedure based on []) Chart for F1, UCL =,18 CTR =,15 LCL =,11,73,7,7,4,1 Chart for F, Fgure 9: Identfyng the out-of-control varable (Procedure based on []) Fgure 1: Identfyng the out-of-control varable (Procedure based on [3]) UCL =,71 CTR =,5 LCL =,59 In Fgure 1, the graphcal dsplay of the procedure based on Maravelaks and Bersms [3] s presented. Accordng to ths procedure each of the three varables corresponds to specfc ntervals of the [ π, π]. The nterval from 1 to 5 corresponds to the thrd varable mplyng that the thrd REFERENCES [1] Alt FB. Multvarate Qualty Control. The Encyclopeda of Statstcal Scences, Kotz S., Johnson, NL, Read CR (eds), New York: John Wley, 1985; pp [] Bersms, S. Multvarate Statstcal Process Control, M.Sc. Thess, Department of Statstcs, Athens Unversty of Economcs and Busness, 1, ISBN [3] Bersms S, Panaretos J, and Psaraks S. Multvarate Statstcal Process Control Charts: An Overvew, (under revson). [4] Chua M-K, Montgomery DC. Investgaton and characterzaton of a control scheme for multvarate qualty control. Qualty and Relablty Engneerng Internatonal, 199, 8: [5] Croser RB. Multvarate generalzatons of cumulatve sum qualty-control schemes. Technometrcs, : [] Doganaksoy N, Faltn FW, Tucker WT. Identfcaton of out-of-control multvarate characterstc n a multvarable manufacturng envronment. Communcatons n Statstcs - Theory and Methods, 1991, : [7] Fuchs C, Benjamn Y. Multvarate profle charts for statstcal process control. Technometrcs, 1994, 3: [8] Fuchs C, Kenett RS. Multvarate Qualty Control; Marcel Dekker INC: New York, 1998 [9] Hayter AJ, Tsu K-L. Identfcaton and quantfcaton n multvarate qualty control problems. Journal of Qualty Technology, 1994, : [1] Healy JD. A note on multvarate CUSUM procedures. Technometrcs, : [11] Hotellng H. Multvarate qualty control - Illustrated by the ar testng of sample bombsghts. Technques of Statstcal Analyss, Esenhart, C., Hastay, M.W., Walls, W.A. (eds), New York: MacGraw-Hll, 1947; pp [1] Jackson JE. A user gude to prncpal components; John Wley: New York [13] Kourt T, MacGregor JF. Multvarate SPC methods for process and product montorng. Journal of Qualty Technology, 199, 8:

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