MONITORING THE COVARIANCE MATRIX OF MULTIVARIATE PROCESSES WITH SAMPLE RANGES

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1 MONITORING THE COVARIANCE MATRIX OF MULTIVARIATE PROCESSES WITH SAMPLE RANGES Atoio F. B. Costa Departameto de Produção, UNESP Guaratiguetá, 56-4, SP, Brasil Marcela A. G. Machado Departameto de Produção, UNESP Guaratiguetá, 56-4, SP, Brasil ABSTRACT For the uivariate case, the R chart ad the S chart are the most commo charts used for moitorig the process dispersio. With the usual sample size of 4 ad 5, the R chart is slightly iferior to the S chart i terms of efficiecy i detectig process shifts. I this article, we show that for the multivariate case, the chart based o the stadardized sample rages, we call the RMAX chart, is substatially iferior i terms of efficiecy i detectig shifts i the covariace matrix to the VMAX chart, which is based o the stadardized sample variaces. User s familiarity with sample rages is a poit i favor of the RMAX chart. A example is preseted to illustrate the applicatio of the proposed chart. KEYWORDS. Cotrol charts. Multivariate processes. Sample rage. RESUMO Para o caso uivariado, os gráficos de R e de S são os mais utilizados para o moitorameto da dispersão do processo. Com os tamahos de amostras usuais de 4 e 5, o gráfico de R é ligeiramete iferior ao gráfico de S em termos da eficiêcia em detectar desajustes o processo. Neste artigo, mostra-se que para o caso multivariado, o gráfico baseado as amplitudes amostrais padroizadas, deomiado gráfico de RMAX, é substacialmete iferior, em termos da eficiêcia em detectar desajustes a matriz de covariâcias, do que o gráfico de VMAX, o qual é baseado as variâcias amostrais padroizadas. O fato do usuário estar mais familiarizado com amplitudes amostrais é um poto a favor do gráfico de RMAX. Um exemplo é apresetado para ilustrar a aplicação do gráfico proposto. PALAVRAS CHAVE. Gráficos de cotrole. Processos multivariados. Amplitude amostral. XLI SBPO 9 - Pesquisa Operacioal a Gestão do Cohecimeto Pág. 883

2 . Itroductio Cotrol charts are ofte used to observe whether a process is i cotrol or ot. Whe there is oly oe quality characteristic Shewhart cotrol charts are usually applied to detect process shifts. The power of the Shewhart cotrol charts lies i their ability to separate the assigable causes of variatio from the ucotrollable or iheret causes of variatio. Shewhart cotrol charts are relatively easy to costruct ad to iterpret. As a result, they are readily implemeted i maufacturig eviromets. However, there are may situatios i which it is ecessary to cotrol two or more related quality characteristics simultaeously. Hotellig (947) provided the first solutio to this problem by suggestig the use of the T statistic for moitorig the mea vector of multivariate processes, where the umber of quality characteristics uder cotrol p is equal or more tha two. May iovatios have bee proposed to improve the performace of the T charts. Das ad Prakash (8) studied the iterpretatio of the out-of-cotrol sigal whe the T chart is i use. Costa ad Machado (8) ad Champ ad Aparisi (8) cosidered the use of the double samplig procedure with the chart proposed by Hotellig. The first multivariate cotrol chart for moitorig the covariace matrix Σ was based o the chartig statistic obtaied from the geeralized likelihood ratio test. For the case of two variables, Alt (985) proposed the geeralized variace statistic S to cotrol the covariace matrix Σ. Cotrol charts more efficiet tha the S chart have bee proposed. Recetly, Costa ad Machado (8a, 8b), Machado ad Costa (8) ad Machado et al. (8) cosidered the VMAX statistic to cotrol the covariace matrix of multivariate processes. The poits plotted o the VMAX chart correspod to the maximum of the sample variaces of the p quality characteristics. There are a few recet papers dealig with the joit cotrol of the mea vector ad the covariace matrix of multivariate processes (see Chou et al. (), Khoo (5), Che et al. (5), Zhag ad Chag (8)). For the uivariate case, the R chart is a basic tool for moitorig process dispersio. With the usual sample size of 4 ad 5, the R chart is slightly iferior to the S chart i terms of efficiecy i detectig process shifts. Accordig to the literature, a cotrol chart based o sample rages has ot yet bee proposed for moitorig the covariace matrix of multivariate processes. I this article we propose a chart based o sample rages for moitorig the covariace matrix of multivariate processes, amed as the RMAX chart. The performace of the proposed chart is ivestigated ad compared with the performace of the VMAX chart, which is based o sample variaces. We assume that the correlatio coefficiets are ot affected by the assigable cause. The paper is orgaized as follows. I sectio we compare the efficiecy of the R ad S charts. I Sectio 3 we describe the cotrol chart based o the RMAX statistic. We also ivestigate the performace of the proposed chart ad compare it with the VMAX chart. I Sectio 4, a example is preseted to illustrate the applicatio of the proposed chart. Coclusios are i Sectio 5.. Uivariate charts for moitorig process dispersio The most commo charts used for moitorig the process dispersio are the R chart ad the S chart. Both charts are briefly discussed i Sectios. ad.. I Sectio.3 the charts are compared i terms of their efficiecy i detectig process shifts. I this paper the average ru legth (ARL) measures the efficiecy of a cotrol chart i detectig a process chage. Durig the i-cotrol period the ARL=/ α ad is called ARL, ad durig the out-of-cotrol period the ARL= p, beig p = β the power of the cotrol w chart. The risks α ad β are, respectively, the well-kow Type I ad Type II errors. A chart with a larger i-cotrol ARL (ARL ) idicates lower false alarm rate tha other charts. A chart w XLI SBPO 9 - Pesquisa Operacioal a Gestão do Cohecimeto Pág. 884

3 with a smaller out-of-cotrol ARL idicates a better ability of detectig process shifts tha other charts... The R chart Let X, X,, X be a radom sample of size from a ormally distributed process with mea µ ad stadard deviatio. The sample rage R is defied as max[ x, x,, x ] mi[ x, x,, x ], x i = ( X i µ ). As we oly cosider icreases i the stadard deviatio, the R chart sigals if R > CL, beig CL = k the cotrol limit for the R chart. The occurrece of the assigable cause chages the stadard deviatio from to a. The power of the R chart is give by R k p w = Pr[ R > k = = ] ; a = Pr w = > = ; = a a a k = Pr w > = = F ( w) a () where w = R a ad F beig y [ x, x,, ] ( w) = [ F ( y + w) F ( y) ] df ( y) = mi x, see more details i Pearso ad Hartley (94). The ARL for the R chart is give by : ( w), () ARL =. (3) F.. The S chart Let X, X,, X be a radom sample of size from a ormally distributed process with mea µ ad stadard deviatio. The sample variace S chart is defied as ( X X i ) ij j =. As we oly cosider icreases i the variace, the S chart sigals if S = S > CL, beig CL = χ, α the cotrol limit for the S chart. The occurrece of the assigable cause chages the variace from to = a. The power of the S chart is give by χ, α pw = Pr χ >. (4) a Cosequetly, the ARL for the S chart is give by : ARL = Pr χ χ, α > a. (5) XLI SBPO 9 - Pesquisa Operacioal a Gestão do Cohecimeto Pág

4 .3. Comparig the R chart with the S chart I this sectio we compare the R chart with the S chart. Table shows the ARL for both charts, cosiderig =5. The S chart is always faster tha the geeralized variace R chart i detectig shifts i the process dispersio. However, the ARL reductio provided by the S chart is small, less tha.%, see the Pv values i Table. We adopt a Type I error of five per oe thousad ( α =.5), that is, ARL =.. The study with > 5 led to larger values of Pv. Table. The ARL of the R ad S charts (=5). R S Pv (%) a CL Multivariate charts for moitorig process dispersio 3.. The RMAX chart I this sectio we propose a ew statistic based o the sample rages for moitorig the covariace matrix Σ of multivariate processes with p quality characteristics that follow a multivariate ormal distributio. This statistic is the maximum R i = max[ X i, X i,, X i ] mi[ X i, X i,, X i ], that is, RMAX = max{ R, R,, R p}, where X ij, i =,,, p, j =,,,, is the p x matrix of quality characteristic measuremets made o the items of the sample. The i-cotrol covariace matrix is give by p p Σ =, where ij, i = j =,,, p, are the p variaces of the quality p p pp characteristics ad ij, with i j =,,, p, are the covariaces betwee two quality characteristics. The cotrol chart based o the RMAX statistic is called the RMAX chart. If the RMAX statistic falls beyod the cotrol limit (CL), the cotrol chart sigals a out-ofcotrol coditio. Oce the RMAX chart sigals, the user ca immediately examie the sample rages to discover which variable was affected by the assigable cause, that is, the oe with the sample rage larger tha the cotrol limit. For safety, we might cosider that the assigable cause has also affected other variables whe their sample rages are close to the cotrol limit, eve if they do ot exceed it. XLI SBPO 9 - Pesquisa Operacioal a Gestão do Cohecimeto Pág

5 The occurrece of the assigable cause chages the covariace matrix from Σ to a a a a a a p p a a a a a a p p = ij Σ. The correlatios ρ ij =, i, j =,,, p, i j a a p p a a p p a p a p pp with i j, are ot affected by the assigable cause. The power of the RMAX chart for the bivariate case is give by: p w = F ( w, w ) (6) beig w = R a ad w = R a. F ( w, w ) is the bivariate stadard ormal cumulative distributio fuctio. We obtaied a closed ARL expressio for the RMAX chart whe p = ad 3, respectively. We ca exted the study for the cases where p > 3. Tables ad 3 brig the ARL for the RMAX chart. I Table, p =, =5 ad ρ =,.3,.5,.7. I Table 3, p = 3, =5 ad ( ρ, ρ 3, ρ 3 ) = (.,.,,), (.5,.5,.5), (.7,.5,.). We adopt a Type I error of five per oe thousad ( α =.5), that is, ARL =.. The correlatio coefficiet ρ has some ifluece o the RMAX chart (see Tables ad 3). Whe p= ad the assigable cause chages oly oe variable, the ARL is practically the same for ay level of correlatio. Whe the assigable cause chages both variables, as ρ icreases from to.7, the ARL icreases very slightly (Table ). Whe p=3, the ARL always icreases as the correlatio becomes higher (Table 3). Table. The ifluece of the correlatio o the RMAX chart s performace (p=). ρ CL a a ARL Table 3. The ARL for the RMAX chart (p=3, =5). XLI SBPO 9 - Pesquisa Operacioal a Gestão do Cohecimeto Pág

6 ρ..5.7 ρ ρ a a a3 CL The VMAX chart I the ext sectio we compare the RMAX chart with the VMAX chart proposed by Costa ad Machado (8a). The VMAX statistic is the maximum of the values of the stadardized sample variaces, p max{ S, S,, S }. S xij j = i,,, p i = =, with ij ( X ij i ) i x = µ, that is, VMAX = A sigal is give if VMAX>CL, the cotrol limit for the VMAX chart. Accordig to Costa ad Machado (8a), the power of the VMAX chart for the bivariate case is give by: P S = CL a Pr χ < CL,( tρ ρ ) b ( ) Γ ( ) ρ e t ( ) t dt. (7) Recallig that the otatio χ,m represets a o-cetral chi-square distributio with degrees of freedom ad o-cetrality parameter give by m. The subroutie DCSNDF available o the Microsoft Fortra library (995) was used to compute the o-cetral chi-square distributio fuctio. Costa ad Machado (8b) obtaied the power of the VMAX chart for the multivariate case Comparig the RMAX ad the VMAX charts Tables 4 ad 5 show the ARL values for the RMAX ad VMAX charts. I Table 4, p = ad i Table 3, p = 3. These Tables were built fixig the correlatio coefficiets equal to.5. The VMAX chart is always faster tha the RMAX chart i detectig shifts i the covariace matrix. The ARL reductio provided by the VMAX chart is substatially large, more tha 7.%, see the Pv values i Tables 4 ad 5. Table 4. The ARL for the RMAX ad VMAX charts (p=, ρ =.5). XLI SBPO 9 - Pesquisa Operacioal a Gestão do Cohecimeto Pág

7 a 4 5 VMAX RMAX % VMAX RMAX P v (%) a CL Table 5. The ARL for the RMAX chart ad for the VMAX chart (p=3, ρ = ρ 3 = ρ 3 =.5). 4 5 VMAX RMAX % VMAX RMAX P v (%) a a a3 CL Illustrative example I this sectio we provide a example to illustrate the ability of the RMAX ad the VMAX charts i detectig shifts i the covariace matrix. To this ed, we cosidered a process with three quality characteristics that follow a trivariate ormal distributio. Whe the process is i-cotrol,..5.5 the mea vector ad the covariace matrix are give by μ =, Σ =.5..5,.5.5. respectively. We iitially geerate samples of size = 5 with the process i cotrol. The remaiig 3 samples were simulated cosiderig that the assigable cause icreased the variability of the first quality characteristic, that is, a =.. Table 6 presets the data for the illustrative example. Table 7 presets the values of the sample variaces ( S, S ad S 3 ) ad the sample rages ( R, R ad R 3 ) ad the statistics RMAX ad VMAX. The cotrol limits for the RMAX chart ad for the VMAX chart are 5.94 ad 3.85, respectively. They are based o a Type I error of five per oe thousad ( α =.5). Figures ad show the RMAX ad the VMAX cotrol charts. Accordig to Figure, the RMAX chart sigals the out-of-cotrol coditio at sample 3 (ru legth=3). The VMAX chart sigals at sample (ru legth=). Oce the RMAX or the VMAX chart sigals, we ca coclude from Table 7 that the first variable was the oe resposible for the out-of-cotrol sigal. 5. Coclusios I this article, we provided a evaluatio of the performace of a chart based o the stadardized sample rages (RMAX chart) for moitorig the covariace matrix of multivariate XLI SBPO 9 - Pesquisa Operacioal a Gestão do Cohecimeto Pág

8 processes. The RMAX chart is substatially iferior i terms of efficiecy i detectig shifts i the covariace matrix tha the VMAX chart, which is based o the stadardized sample variaces. This result differs from the uivariate case, where the R chart is slightly slower tha the S chart i sigalig. Ackowledgemets This work was supported by CNPq Natioal Coucil for Scietific ad Techological Developmet, Project 37744/6- - ad FAPESP - The State of São Paulo Research Foudatio, Project 6/49-. The authors are thakful to the referee for the comprehesive review that helped i improvig the mauscript. XLI SBPO 9 - Pesquisa Operacioal a Gestão do Cohecimeto Pág. 89 8

9 Table 6. The data for the illustrative example. Observatios Sample umber j = X j X j X 3 j XLI SBPO 9 - Pesquisa Operacioal a Gestão do Cohecimeto Pág. 89 9

10 Table 7. The S, S, S 3, R, R, R 3, VMAX ad RMAX values. Sample umber S S S3 VMAX R R R3 RMAX RMAX Sample umber (i ) Figure. The RMAX chart example. VMAX Sample umber (i ) Figure. The VMAX chart example. XLI SBPO 9 - Pesquisa Operacioal a Gestão do Cohecimeto Pág. 89

11 Refereces Alt, F. B. (985), Multivariate quality cotrol. I: Kotz, S., Johso, N. L., eds. Ecyclopedia of Statistical Sciece, 6, -. Champ, C. W. ad Aparisi, F. (8), Double samplig Hotellig s T charts, Quality ad Reliability Egieerig Iteratioal, 4, Che, G., Cheg, S. W. ad Xie, H. (5), A ew multivariate cotrol chart for moitorig both locatio ad dispersio, Commuicatios i Statistics-Simulatio ad Computatio, 34, 3-7. Chou, C. Y., Liu, H. R., Che, C. H. ad Huag, X.R. (), Ecoomic-statistical desig of multivariate cotrol charts usig quality loss fuctio, Iteratioal Joural of Advaced Maufacturig Techology,, Costa, A. F. B. ad Machado, M. A. G. (8), Bivariate Cotrol Charts with Double Samplig, Joural of Applied Statistics, 35, Costa, A. F. B. ad Machado, M. A. G. (8a), A ew multivariate cotrol chart for moitorig the covariace matrix of bivariate processes, Commuicatios i Statistics-Simulatio ad Computatio, 37, Costa, A. F. B. ad Machado, M. A. G. (8b), A ew chart based o sample variaces for moitorig the covariace matrix of multivariate processes, Iteratioal Joural of Advaced Maufacturig Techology. Das, N. ad Prakash, V. (8), Iterpretig the out-of-cotrol sigal i multivariate cotrol chart a comparative study, Iteratioal Joural of Advaced Maufacturig Techology, 37, Hotellig, H. (947), Multivariate quality cotrol, illustrated by the air testig of sample bombsights, Techiques of Statistical Aalysis, -84. Khoo. B. C. (5), A ew bivariate cotrol chart to moitor the multivariate process mea ad variace simultaeously, Quality Egieerig, 7, 9-8. Machado, M. A. G. ad Costa, A. F. B. (8), The double samplig ad the EWMA charts based o the sample variaces, Iteratioal Joural of Productio Ecoomics, 4, Machado, M. A. G. ad Costa, A. F. B. ad Rahim, M.A. (8), The sythetic cotrol chart based o two sample variaces for moitorig the covariace matrix, Quality ad Reliability Egieerig Iteratioal. Microsoft Fortra Power Statio 4.. Professioal Editio with Microsoft IMSL Mathematical ad Statistical Libraries, Microsoft Corporatio, Washigto, USA, 995. Pearso, E.S. ad Hartley, H.O. (94), The probability itegral of the rage i samples of observatios from a ormal populatio, Biometrika, 3, 3-3. Zhag, G. ad Chag, S. I. (8), Multivariate EWMA cotrol charts usig idividual observatios for process mea ad variace moitorig ad diagosis, Iteratioal Joural of Productio Research, 46, XLI SBPO 9 - Pesquisa Operacioal a Gestão do Cohecimeto Pág. 893

MONITORING BIVARIATE PROCESSES

MONITORING BIVARIATE PROCESSES Marcela A. G. Machado Departameto de Produção, UNESP Guaratiguetá, 1516-41, SP, Brasil marcela@feg.uesp.br Atoio F. B. Costa Departameto de Produção, UNESP Guaratiguetá,