Multivariate EWMA Control Chart


 Frederick Roberts
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1 Multvarate EWMA Control Chart Summary The Multvarate EWMA Control Chart procedure creates control charts for two or more numerc varables. Examnng the varables n a multvarate sense s extremely mportant when the varables are hghly correlated, snce jont outofcontrol condtons can occur wthout any ndvdual varable volatng ts control lmts when plotted separately. The EWMA chart s smlar to the TSquared chart, except that the ponts plotted on the chart are a weghted average of current and past observatons. Sample StatFolo: mvewma.sgp Sample Data: The fle grt.sgd contans measurements made on n = 56 batches of grt, from Holmes and Mergen (1993). The data represent the percentages of large, medum, and small partcles n the grt. The table below shows a partal lst of the data n that fle: Large Medum Small Means Covarances , Snce the percentages n the frst three columns sum to 100% n each row, t s only necessary to create a chart based on the frst 2 columns. In addton to the data, the fle contans columns wth the standard means and covarances for the percentage of large and medum partcles, establshed when montorng of the process was frst begun by StatPont Technologes, Inc. Multvarate EWMA Control Chart  1
2 Data Input The data to be analyzed consst of p = 2 or more numerc columns contanng the varables of nterest, wth optonal entry of the varable means and varancecovarance matrx. Data: 2 or more numerc columns contanng the n samples, one sample per row. Subgroup numbers or sze: If the data were obtaned as ndvduals, leave ths feld blank or enter 1. If the data were collected n subgroups, each of sze m, enter the sngle value m. In such a case, each consecutve m rows n the fle wll be consdered to form a subgroup. If the subgroup szes are not equal, enter the name of an addtonal numerc or nonnumerc column contanng group dentfers. The program wll scan ths column and place all sequental rows wth dentcal codes nto the same group. Standard Means: For an ntal study or Phase I analyss where the data wll be used to determne the control lmts, leave ths feld blank. For a controltostandard or Phase II analyss, enter the name of a column contan p means. Standard Covarances: For an ntal study or Phase I analyss, leave ths feld blank. For a controltostandard or Phase II analyss, enter the name of a column contan the p 2 varances and covarances. In enterng the values n a covarance matrx, enter the values n the frst row of the matrx, then the values n the second row, and so forth. Note: f you select Save Results after performng a Phase I analyss, the covarance matrx wll be saved n ths exact format by StatPont Technologes, Inc. Multvarate EWMA Control Chart  2
3 Labels: optonal labels for each subgroup. The labels wll be appled n sequence to the subgroups when plottng the control charts. Select: subset selecton. Analyss Summary The Analyss Summary shows the number of observatons ncluded n the analyss and the locaton of the control lmts on the control charts. Multvarate EWMA Charts Data varables: Large Medum Number of observatons ncluded = 56 Number of observatons excluded = 0 Smoothng parameter lambda: 0.2 Intalzaton: centerlne Phase 2  covarance specfed based on standard Chart Alpha LCL UCL Beyond lmts TSquared Included n the table are: Smoothng parameter lambda: the value of the EWMA parameter, specfed on the Analyss Optons dalog box. The default value of s determned from the settngs on the Control Charts tab of the Preferences dalog box, accessble from the Edt menu. Phase: If Phase 1, the method for estmatng the covarance matrx s dsplayed. If Phase 2, the assumptons about the nput covarance matrx are shown. Chart: the type of chart. For ndvduals data, only a TSquared chart s created. For grouped data, a generalzed varance chart s ncluded f the subgroup sze exceeds the number of varables. Alpha: the false alarm probablty of the chart, specfed usng Analyss Optons. For standard 3sgma control charts, = LCL: the lower control lmt. UCL: the upper control lmt. Beyond lmts: the number of ponts on the control chart that are beyond the control lmts. In the example, the process generated 15 outofcontrol sgnals on the EWMA TSquared chart by StatPont Technologes, Inc. Multvarate EWMA Control Chart  3
4 Analyss Optons Alpha: the false alarm probablty for postonng the control lmts. For the equvalent of a standard 3sgma control chart, set = 0.27%. Covarance Matrx: detals about the covarance matrx. There are 4 possbltes: 1. If the Standard Covarances feld was left blank on the data nput dalog box and the data are n subgroups, no entry s necessary snce the covarances wll be estmated from the pooled wthngroup varablty. 2. If the Standard Covarances feld was left blank on the data nput dalog box and the data are ndvduals, select Pooled estmator to estmate the covarance between varables j and k usng the usual estmator s jk 1 n 1 n 1 x x x k xk j j (1) Select Successve dfferences to estmate the covarance usng n 1 xj x 1, j x k x 1 k (2) s jk, 2( n 1) 1 The second estmator s a more local estmator, n the sense that t captures only shortterm varablty, n a smlar manner to the way n whch a movng range s used to estmate varablty for a standard ndvduals chart. 3. If an entry was made n the Standard Covarances feld and the estmates provded were obtaned from a prevous sample, enter the sze of that prevous sample (f grouped, the number of subgroups) n the Standard Sample Sze feld. 4. If an entry was made n the Standard Covarances feld and the covarances are assumed to be known, leave the Standard Sample Sze feld empty. Lambda: a value for the EWMA parameter 0 < < 1. The value of controls the amount of weght gven to the past hstory of the process. The smaller the value, the more weght gven to older observatons or subgroups. Ths also mpacts the average run length of the chart by StatPont Technologes, Inc. Multvarate EWMA Control Chart  4
5 EWMA Chart The EWMA Chart shows the exponentally weghted value of T 2 for each data value or subgroup: TSquared Multvarate EWMA Control Chart UCL = 11.83, lambda = Observaton The EWMA procedure begns by smoothng the observed data vector at tme by EWMA x (3) ( 1 ) EWMA 1 for ndvduals data and by EWMA x (4) ( 1 ) EWMA 1 for grouped data, wth EWMA 0 set equal to the mean vector or x. The th value of Tsquared s then calculated from 1 EWMA EWMA 2 T (5) Z where s the covarance matrx of the nput data, and 2 Z (6) For the sample data, the chart starts out well below the control lmt but then rses durng the latter secton. 15 outofcontrol sgnals are generated by the chart by StatPont Technologes, Inc. Multvarate EWMA Control Chart  5
6 Pane Optons STATGRAPHICS Rev. 7/24/2009 Pont Symbols: select Beyond Lmts to draw specal pont symbols only for ponts fallng above the control lmt. Select Largest Contrbutor to dentfy the varable that contrbutes most to each value of T 2. Decmal Places for Lmts: number of decmal places for dsplayng the control lmt. Color Zones: check ths box to dsplay green and red zones. Example: Identfyng Largest Contrbutor If Largest Contrbutor s selected, the chart wll take the followng form: TSquared Multvarate EWMA Control Chart UCL = 11.83, lambda = 0.2 Largest Large Medum Observaton Each pont on the chart s coded accordng to the varable that contrbutes the most to the value of T 2. In the plot above, the bggest contrbutor to the frst rse appears to be the percentage of Medum partcles, whle the bggest contrbutor to the second rse appears to be the percentage of Large partcles by StatPont Technologes, Inc. Multvarate EWMA Control Chart  6
7 Multvarate Control Chart Report Ths pane tabulates the ponts on the control chart: Multvarate Chart Report Observaton TSquared Large Medum 27 * * * * * * * * * * * * * * * An astersk ndcates any value beyond the control lmts. Any ponts excluded from the analyss usng the Exclude button are ndcated wth an X. Pane Optons Dsplay All Subgroups f checked, all observatons or subgroups wll be tabulated. Otherwse, only ponts beyond the control lmts wll be ncluded n the table. Dsplay Orgnal Data f checked, the values of each varable wll be dsplayed. Otherwse, only the control chart values wll be tabulated by StatPont Technologes, Inc. Multvarate EWMA Control Chart  7
8 Generalzed Varance Chart A TSquared chart s desgned to montor the means of p varables. To montor the varance, a Generalzed Varance Chart by sometmes be dsplayed: Generalzed Varance Chart Gen. Varance UCL = CL = LCL = Subgroup Ths chart s created only for data arranged n subgroups, and only f the subgroup sze s at least p + 1. The generalzed varance S for the th subgroup s defned as the determnant of ts varancecovarance matrx. The above chart shows the grt data grouped n subgroups of 4 consecutve observatons each. Any pont beyond the upper control lmt would ndcate an unusually large amount of varablty wthn that subgroup. In ths case, no such ponts are present. Control Ellpse If outofcontrol sgnals are shown on the control chart, t s useful to examne those values n detal. A good chart to use n the case of p = 2 varables s the Control Ellpse: Control Ellpse Medum Large The upper control lmt on the TSquared chart corresponds to an ellptcal regon n the space of any two of the varables, wth the other varables held at a fxed value. For p = 2, any outofcontrol sgnals wll show up as ponts outsde the ellpse by StatPont Technologes, Inc. Multvarate EWMA Control Chart  8
9 In the sample data, t wll be notced that some outofcontrol sngles correspond to a hgh percentage of Large partcles whle others correspond to a low percentage of Medum partcles. Pane Optons Select 2 varables: select any 2 varables to defne the control ellpse. The ellpse wll be plotted assumng that all other varables at held at ther mean levels. Care should be taken n nterpretng the plot f p > 2, snce the true ellptcal control regon for each pont depends on the value of the varables that are not plotted by StatPont Technologes, Inc. Multvarate EWMA Control Chart  9
10 EWMA Decomposton The T 2 statstc can be decomposed nto components attrbutable to each of the varables. One method for measurng the contrbuton of the jth varable to an outofcontrol T 2 value s by lookng at how much smaller T 2 would be f the jth varable was not ncluded n the analyss. The TSquared Decomposton pane does such a decomposton for each outofcontrol sgnal on the TSquared chart: TSquared Decomposton Observaton Large Medum Followng Runger, Alt and Montgomery (1996), the table shows d j 2 2 T T( j) (7) where 2 T ( j) s the value of the statstc usng all varables except the jth. For the current data, Medum appears to be the domnant varable for the early outofcontrol sgnals, whle Large appears to be the domnant varable for the later sgnals. 3D Control Chart When the data consst of p = 3 varables, a 3D control chart can be very helpful, snce the control regon s then an ellpsod by StatPont Technologes, Inc. Multvarate EWMA Control Chart  10
11 Control Ellpsod X X2 X1 The above plot shows the outlne of a typcal control ellpsod for 3 varables. Note: to explore ths plot, t s very helpful to use the dynamc rotaton button on the analyss toolbar. Pane Optons Select 3 varables: select any 3 varables to defne the control ellpsod. The ellpsod wll be plotted assumng that all other varables at held at ther mean levels. Care should be taken n nterpretng the plot f p > 3, snce the true control regon for each pont depends on the value of the varables that are not plotted by StatPont Technologes, Inc. Multvarate EWMA Control Chart  11
12 Save Results The followng results can be saved to the datasheet: 1. TSquared the calculated T 2 values for each observaton or subgroup. 2. Means the p varable means. 3. Covarances the p 2 varances and covarances n rowwse order. 4. Labels the labels correspondng to each value of T Generalzed Varance  f calculated, the values of S. Calculatons TSquared Control Lmt f Covarances are Known UCL X (8) 2, p TSquared Control Lmt f Covarances are Estmated from k Prevous Samples UCL p( k 1)( k 1) F (9) k( k p), p, k p TSquared Control Lmt f Covarances are Estmated from Current Data 2 ( n 1) UCL Beta, p / 2,( n p1) / 2 (10) n Generalzed Varance Control Lmts where UCL b 1 3 b 2 (11) CL b 1 (12) LCL b 1 3 b 2 (13) b 1 p ( n 1 p ( n 1) ) 1 p p p 1 b2 ( n ) ( n j 2) ( n j) 2 p ( n 1) 1 j1 j1 (14) (15) If s not known, t s replaced by the estmate S /b by StatPont Technologes, Inc. Multvarate EWMA Control Chart  12
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