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1 0 Multivariate Cotrol Chart 3 Multivariate Normal Distributio 5 Estimatio of the Mea ad Covariace Matrix 6 Hotellig s Cotrol Chart 6 Hotellig s Square 8 Average Value of k Subgroups 0 Example 3 3 Value of Idividual Observatio 4 Example 4 3 wo-sample Hotellig s Square 6 Example 3 7 Example Cofidece Level of wo-sample Differece Mea 0 5 Pricipal Compoet Aalysis 6 Disadvatage of Usig Multivariate Cotrol Chart - -

2 Figure : A by p matrix 3 Figure : 5 by 3 measuremet matrix for parameters legth, width, ad height of a simple for five times 3 Figure 3: Variace-covariace matrix of results show i Fig 4 Figure 4: Calculatio of mea, variace, ad covariace of data show i Fig 5 Figure 5: (a) mea ad (b) d dispersio of multivariate cotrol charts 7 Figure 6: k subgroup of by p measuremet matrices Figure 7: Measuremet results of five samples each from (a) a old facility ad (b) a ew facility 7 - -

3 0 Multivariate Cotrol Chart Multivariate aalysis is a brach of statistics cocerig with the aalysis of multiple measuremets made o oe or several samples It is a array represetig measuremet i row o each of p parameter i colum Figure shows a by p matrix he matrix has row value from j = to ad row value from i = to p p p p Figure : A by p matrix ake for a example, measurig the legth, width, ad height of a box five times, the results i matrix show i Fig Colum oe is the legth measuremet, colum two is the width measuremet, ad colum three is the height measuremet Figure : 5 by 3 measuremet matrix for parameters legth, width, ad height of a simple for five times his set of five measuremets, measurig three parameters ca be described by its mea vector ad variace-covariace matrix S Each row vector j is has measuremet of the three parameters he mea vector cosists of the mea of each parameter Usig the example show i Fig, the mea is defied as i () j ij 4 hus, the mea vector matrix for three parameters is

4 he variace-covariace matrix cosists of the variaces of the parameters alog the mai diagoal ad the covariace betwee each pair of parameters i the other matrix positios Covariace betwee p parameter is defied as Covariace j i Y Yi ij ij () he variace of p parameter is defied Variace j i ij (3) he variace-covariace S matrix is defied as S j i i ij ij (4) Usig the results show i Fig, the ij i matrix is equal to ij i , while the traspose of the matrix is ij i variace-covariace matrix S is S hus, usig equatio (4), the ad the results are show i Fig S Figure 3: Variace-covariace matrix of results show i Fig

5 he calculatio of mea, variace, ad covariace are show i Fig 4 Parameter ij i i L W H L W H L-W L-H W-H Mea Covariace Variace Figure 4: Calculatio of mea, variace, ad covariace of data show i Fig Multivariate Normal Distributio ij i Multivariate ormal distributio is the commo model for multivariate data aalysis he multivariate ormal distributio model is a extesio of uivariate ormal distributio model to fit vector observatio A p-dimesioal vector of radom parameters is =,, 3,, p for - < i <, for i =,, 3,, p It is said to have a multivariate ormal distributio if its probability desity fuctio f() is i this form f ) f (,,, ) (5) ( p ρ/ / π Σ exp ' m) ( m) where m = (m, m,, m p ) is the vector of mea, is the correlatio betwee two parameters, ad is the variace-covariace matrix of the multivariate ormal distributio he shortcut otatio of multivariate ormal distributio is = N p (m, ) (6) Whe p = that is oe dimesioal vector, thus, =, whereby it has ormal distributio with mea m ad variace he distributio is f () exp ( m) /( ) for - < < Whe p =, = (, ), it has bivariate ormal distributio with two dimesioal vector meas m = (m, m ) ad ij - 5 -

6 variace-covariace matrix radom parameters give by Σ ad the correlatio betwee the two Estimatio of the Mea ad Covariace Matrix Let,,, be p-dimesioal vectors of observatios that are sampled idepedetly from Np(m, Σ), where p < -, ad Σ is the variace-covariace matrix of he observed mea vector ad the sample dispersio matrix are show i equatio (7) Covar iace i ( )(Y Y) i i (7) hey are the ubiased estimators of m ad Σ respectively Hotellig s Cotrol Chart I 947 Harold Hotellig itroduced a statistic which uiquely leds itself to plottig multivariate observatios his statistic is amed as Hotellig s, which is scalar combiig iformatio from the dispersio ad mea of several parameters Owig to the fact that calculatio is laborious ad fairly complex coupled with requirig kowledge of matrix algebra, acceptace of multivariate cotrol chart by idustry low I this moder computer age, with the help of computer to calculate complex solutio, multivariate cotrol charts bega to attract attetio Ideed, the multivariate chart which displays Hotellig statistic became so popular that it is at time beig ame as Shewhart cotrol chart As for the uivariate case, whe data are grouped, the chart ca be paired with a chart that displays a measure of variability withi the subgroups for all the aalyzed characteristics he combied mea ad d dispersio cotrol charts are the multivariate couterpart of the uivariate ad S or ad R cotrol charts Example of a Hotellig mea ad show i Fig 5 d dispersio pair of cotrol charts are - 6 -

7 (a) Figure 5: (a) mea ad (b) (b) dispersio of multivariate cotrol charts d Each chart represets cosecutive subgroup measuremets o the meas of four parameters he chart for meas idicates a out-of-cotrol state for subgroup, 9 ad 0, ad he d dispersio chart idicates that subgroup 0 is also out of cotrol Based o the results, multivariate system is suspect However, to fid a assigable cause, oe has to resort to the idividual uivariate cotrol charts or some other uivariate procedure that should accompay this multivariate chart he Hotellig distace is a measure that accouts for the covariace structure of a multivariate ormal distributio It was proposed by Harold Hotellig i 947 ad is called Hotellig It may be thought of as the multivariate couterpart of the Studet s t-statistic he distace is a costat multiplied by a quadratic form his quadratic form is obtaied by multiplyig the three quatities, which are the vector of deviatios betwee the observatios ad the mea m that is expressed by ( - m), the iverse of the variace-covariace matrix S -, ad the vector of deviatio ( - m) For idepedet parameter, the covariace matrix is a diagoal matrix ad becomes proportioal to the sum of squared stadardized parameter I geeral, the larger the value, the more distat is the observatio from the mea he formula for calculatio the value is ( m) S ( m) (8) he costat is the sample size where the covariace matrix is estimated - 7 -

8 cotrol chart is the most popular, easiest to use ad iterpret method for hadlig multivariate process data, ad is begiig to be widely accepted by quality egieers ad operators but it is ot a woder Firstly, ulike the uivariate case, the scale of the value displayed o the cotrol chart is ot related to the scale of ay moitored parameter Secodly, whe statistic exceeds the upper cotrol limit UCL, the user does ot kow which particular parameter is causig it With respect to scalig, oe is strogly advised to ru idividual uivariate charts alog with the multivariate cotrol chart his will also help to idetify the parameter that has problem Idividual uivariate cotrol chart caot explai situatios that are the result of some problems i the covariace or correlatio betwee the parameters his is why a dispersio cotrol chart is also be used Aother way to aalyze the data is to use pricipal compoets For each multivariate measuremet or observatio, the pricipal compoets are liear combiatios of the stadardized p parameters It is doe by subtractig its respective targeted value ad dividig by its respective stadard deviatio he pricipal compoets have two importat advatages, which are the ew parameters are almost ucorrelated ad very ofte, a few or at time or pricipal compoets may capture most of the variabilities i the data so that oe does ot have to use all of the p pricipal compoets for cotrol he disadvatage is idetifyig the lost origial parameter However, i some cases the specific liear combiatios correspodig to the pricipal compoets with the largest eigevalues may yield meaigful measuremet uits What is beig used i cotrol chart is the pricipal factor, whereby a pricipal factor is the pricipal compoet divided by the square root of its eigevalue Hotellig s Square A multivariate method is the multivariate couterpart of Studet s t-distributio, which forms the basis for multivariate cotrol chart he two tail ( - α)00% cofidece limits for testig the mea of sample is defied as s t /, (9) he t-distributio statistic is - 8 -

9 x t (0) s / If the hypothesis test for mea µ is equal to µ 0 the equatio (0) becomes x 0 t () s / hus, by squarig the test statistic, it becomes t where x x s 0 x x 0 Whe is geeralized to p parameter, it becomes μ S 0 μ0 t x 0 s / () x p ad p Equatio (8) is same as equatio () S - is the iverse of the sample variace-covariace matrix S, ad is the sample size for i variate, where i =,,, p he diagoal elemets of S matrix are the variaces ad the off-diagoal elemets are the covariaces for the p parameter Whe µ = µ 0, the has distributio fuctio equal to or ~ p( p ) F p, p (3) where F[p, -p] is F-distributed with p ad ( - p) degrees of freedom If µ = µ 0, it ca be tested by takig a sigle p-variate sample of size, calculatig value, p( ) ad comparig it with F p, p value at ( - α)00% cofidece level p Although the result applied to hypothesis testig, it does ot apply directly to multivariate Shewhart cotrol chart for which there is o μ 0 he result may be used as a approximatio whe large samples are used ad data are i subgroups, with the upper cotrol limit UCL of a cotrol chart based o the approximatio - 9 -

10 Whe a uivariate cotrol chart is used durig the aalysis of historical data phase ad subsequetly phase used for real-time process moitorig, the geeral form of the cotrol limit is the same for each phase, although it may ot ecessary be the case Specifically, three-sigma limit is used i the uivariate case, which skirts the relevat distributio theory for each phase hree-sigma uit is geerally ot used with multivariate cotrol chart However, it makes the selectio of differet cotrol limit for each phase based o the relevat distributio theory Average Value of k Subgroups If there are k subgroups of matrix measuremets as show i Fig 6, the calculatio of r average value of r th subgroup, the variace-covariace matrix Sp, ad other associated data like upper cotrol limit are described here Sice the value μ is geerally ukow, it is ecessary to estimate the mea μ aalogous to the way that μ is estimated whe a chart is used Specifically, whe there are ratioal subgroups from r = to k, μ is estimated by average if the mea of r th subgroup, with p Each i value of i parameter for i =,,, p is obtaied the same way as like a Shewhart cotrol chart by takig k subgroups of size ad calculatig usig equatio (4) where is k i ri (4) k r ri is deoted as the average for the r th subgroup of the i th parameter, which ri (5) rij j with rij deotig the j th observatio out of for the i th parameter i the r th subgroup - 0 -

11 p p p k k kp k k k p k k k kp Figure 6: k subgroup of by p measuremet matrices he variace ad covariace are similarly averaged over the k subgroups Specifically, S ij the elemets of the variace-covariace matrix S are obtaied as k S S (6) ij k r rij where S rij for i j deotig the sample covariace betwee i ad j for the r th subgroup ad S ij for i = j deotes the sample variace of i he S rij = S rii for r th subgroup with parameter i =,, 3,, p is calculated usig equatio (7) S rij i ri rij (7) Similarly, the covariace S rij betwee variable i ad j for a subgroup ca be calculated usig equatio (8) S rij i ri rj rij rij (8) Like a cotrol chart or ay other chart, r value of k subgroup would be tested with agaist the upper cotrol limit UCL If ay value falls above the upper cotrol limit UCL, the correspodig subgroup would be ivestigated Note that there is o lower cotrol limit hus, oe would plot, which is r Sp r r (9) - - j

12 for the r th subgroup r =,, 3,, k, with deotig a vector with p parameter cotaiig the subgroup s average for each p parameters for r th subgroup S p is the iverse matrix of the pooled variace-covariace matrix S p, which is obtaied by averagig the subgroup s variace-covariace matrix over k subgroups For every k s r value obtaied from equatio (9) would be compared with upper cotrol limit UCL show i equatio (0) for specified ( - )00% cofidece level p( k )( ) UCL Fα p, k k p (0) k k p he lower cotrol limit is geerally ot used i multivariate cotrol chart, although some cotrol chart method does utilize the lower cotrol limit LCL Although a small value for r may seem desirable but very small value would likely idicate a of r th problem of some types as oe would ot expect every parameter of r subgroup to be virtually equal to every parameter i If there is ay plotted poit above the upper cotrol limit, its cause ca be idetified, ad has bee corrected, the poit should be discarded ad the upper cotrol limit should be recalculated he remaiig r values would the be compared with the ew upper cotrol limit After discardig the out of cotrol kow cause r value, it is ecessary to recalculate S p ad values for subsequetly applicatio to future subgroups because differet distributio theory is ivolved sice future subgroups are assumed to be idepedet of the curret set of subgroups As the illustratio, oe assumes that a subgroups have bee discarded so that (k - a) subgroups left for recalculatig S p ad Let s let these two ews values be represeted by R Sp ad R to distiguish them from the origial values S p ad values before ay subgroups are beig discarded Future rr value to be plotted would be o the multivariate cotrol chart will obtai from R R ( future) S R ( future) ( future) rr p with deotig a arbitrary vector cotaiig the average for the p parameter from a sigle subgroup obtaied i the future Each of these future values would be plotted o the multivariate cotrol chart ad compared with upper cotrol limit show i equatio () - -

13 p( k - a )( -) UCL Fα p, ( k - a) k a p () ( k - a) k a p Notice that the equatio for upper cotrol limit show i equatio () does ot deduce to the equatio for the upper cotrol limit show i equatio () whe there is o subgroup is beig discarded ie a = 0 his is because a differet set of data is used i the calculatio Example Calculate the Hotellig s mea value usig the data show i Fig, which is if the expected value of legth is 40, width is 00, ad height is 060, ad test the sigificace of the measuremet versus the specificatios Solutio he matrix is traspose of he matrix is 0 0 matrix is hus, he variace-covariace matrix S is S , while the iverse of variace-covariace matrix is S Note the iverse variace-covariace ca be calculated usig Adjoit method, Gauss-Jorda elimiatio method, or ay other available methods he mea value is equal to =

14 p( ) p From distributio, at 95% cofidece level value is F p, p 3(5 ) 5 3 = 3, F 005 = 6x96 = 496 Sice the calculated mea value is less tha the statistical value, it cocludes that it is ot sigificace, which shall mea 95% cofidece that the box is made accordig specificatios 3 Value of Idividual Observatio If there are m historical multivariate data to be tested whether they are i cotrol, the procedure to establish the Hotellig s h value ca be costructed like the way to costruct uivariate idividual observatio h value is defied as m S m () h For each of the h value, it is compared with lower ad upper cotrol - )00% which are defied as ( m ) LCL m ( m ) UCL m p m p B ; ; p m p B ; ; (3) (4) where is B the Beta distributio with parameter p/ ad (m p - )/ Note that a b (a) (b) Beta distributio is defied as B a,b ) / for 0, (a b) Plottig multivariate cotrol chart usig the procedure stated above by maual calculatio is extremely impractical However, with Dataplot software should be able to reduce this tedious procedure Example he historical data take from two parameters are, 38,, 40, 9, ad 39 Calculate the Hotellig s value ad the cotrol 95% cofidece level - 4 -

15 Solutio he mea of the data is 98 ad the matrix traspose of matrix is S variace-covariace matrix is S ad the , is he variace-covariace matrix ad the iverse of hus, the h value is h = ( m ) p m p he lower cotrol limit is LCL B ; ; m (6 ) B ; ; 46B005;5; = 46 x 037 =

16 ( m ) p m p he upper cotrol limit is UCL B ; ; = m (6 ) B ; ; = 4 6B 0975 ;5; = 46 x 48 = 66 6 he result shows that the historical multivariate data is out of cotrol limits 3 wo-sample Hotellig s Square wo samples from radom vector ad are take ad the test hypothesis is testig the ull hypothesis that the sample mea of vector ad vector are equal ie H 0 : = versus alterative hypothesis H : It is equivalet to oe sample Hotellig s statistical test o the radom vector Z = - equal to zero he calculatio of Hotellig s value ivolves calculatio the differece i the sample mea vectors of two multivariate samples It also ivolves the calculatio of the pooled variace-covariace matrix ad mea differece matrix he equatio for calculatig value is show equatio (5) S (5) P For large samples, this test statistic will be approximately Chi-square distributed with p degree of freedom However, this approximatio does ot take ito accout the variatio due to estimatig the variace-covariace matrix hus, it eeds to trasform Hotellig's p( ), which is ~ Fp, p ito F-statistic usig p equatio (6), which is a fuctio of the sample size ad of ad each havig p parameters p F ~ F, ( ) p p (6) p hus, the testig statistic is testig of ull hypothesis H 0 : = havig F-statistic with p ad ( + - p - ) degrees of freedom he ull hypothesis H 0 would be p rejected if the calculated F-value, which is F, is larger tha the p( ) table F-value with p ad ( + - p - ) degrees of defied value, F p p which is, - 6 -

17 F F p p (7), If the ull hypothesis is rejected, oe may wat to kow which parameter(s) is/are cotributig to the reject I this circumstace oe may resolve to oe variate Studet s t-statistic to fid the parameter he t-value is Studet t-distributed with ( + - ) degree of freedom, which is show i equatio (8) x x t ~ t( ) (8) sp x ad x are the scalar mea of a parameter scalar samples s p is the pooled variace of the two If the calculated t-value is larger tha the table t-value with ( + - ) degrees of defied value, which is t /,( ) the the test parameter is cotributig to the differece x x t t/,( ) (9) sp Example 3 he microelectroic compay has built a ew facility he egieer would like to perform a test to check if there is ay differece betwee the old facility ad ew facility i terms of fabricatio the device he egieer takes five devices fabricated i old facility ad five devices fabricated i ew facility hree parameters are measured he results are show i Fig (a) (b) Figure 7: Measuremet results of five samples each from (a) a old facility ad (b) a ew facility - 7 -

18 Solutio he ad matrices are respectively equal to 6 ad he mea differece matrix is ad 004 he variace-covariace matrix of devices fabricated i old facility is S j j j ad j j S = matrix of = 4 hus, the variace-covariace he variace-covariace matrix of devices fabricated i ew facility is S j j j ad j

19 j S = matrix of = 4 hus, the variace-covariace he pooled variace-covariace matrix S p S P = = S P = = p 0 3 he calculated F- value is F x5 9= 8 p( ) 3x8 F-value from F-table at = 005 is 3,6 F = Sice the calculated F-value is smaller tha F-table value, the coclusio is o differet betwee two facilities Example 4 Based o the experimet results show i Fig 7, calculate the t-value ad compare it with the t-table value at α = 005 for the three parameters - 9 -

20 Solutio Sample size is = = 5 for both samples Parameter he mea value differece is = he pooled variace is 0045 x x 006 he Studet t-value t = 387 sp t-value from table is t 005/,( 8) = 306 Sice t-value is larger tha t-value from α = 005, the coclusio is that there is differece betwee two facilities this parameter Parameter he mea value differece is = -44 he pooled variace is 4685 x x 44 he Studet t-value t = 550 sp t-value from table is t 005/,( 8) = 306 Sice t-value is smaller tha t-value from α = 005, the coclusio is that there is o differece betwee two facilities for this parameter Parameter 3 he mea value differece is 3= he pooled variace is 0055 x3 x he Studet t-value t = 05 sp t-value from table is t 005/,( 8) = 306 Sice t-value is smaller tha t-value from α = 005, the coclusio is that there is o differece betwee two facilities for this parameter 4 Cofidece Level of wo-sample Differece Mea I order to assess which parameter differs i two-sample vector, oe eeds to calculate ( - α)00% cofidece level iterval for the differece i the mea vectors of the two samples For the i th idividual parameter, oe is lookig at the differece betwee the sample meas for i th parameter plus or mius the radical - 0 -

21 multiply the stadard error of the differece betwee the sample meas for the i th parameter which ivolves the iverses of the sample sizes ad the pooled variace for parameter i hus, the ( - α)00% cofidece iterval for mea differece is where x i p( ) p xi F p, p s (30) spi is the pooled variace of parameter i Example 5 Based o the experimet results show i Fig 7, calculate the 95% iterval for the three parameters Solutio Sample size is = = 5 for both samples Parameter he mea value differece is = he pooled variace is % cofidece iterval for this parameter is x p( ) x F p = 06 4xF 3,6 04x p, p s Pi = 006 4x476 04x0045 = = (- 049, 036) Parameter he mea value differece is = -44 he pooled variace is % cofidece iterval for this parameter is x p( ) x F p = 44 4xF 3,6 04x p, p s = 44 4x476 04x4685 = = (- 4783, 903) Parameter 3 he mea value differece is 3 3= he pooled variace is % cofidece iterval for this parameter is x p( ) 3 x3 F p p, p s Pi Pi Pi - -

22 = 04 4xF 3,6 04x = 004 4x476 04x0055 = = (- 067, 059) Based o the results of 95% cofidece iterval, there is o distiguishable differece for the three parameters 5 Pricipal Compoet Aalysis A multivariate aalysis problem could start with a substatial umber of correlated parameters Pricipal compoet aalysis is a dimesio reductio tool that ca be used advatageously i such sceario Pricipal compoet aalysis is techique aims at reducig a large set of parameters to small set parameters ad yet still cotais most of the iformatio i the large set parameters he techique of pricipal compoet aalysis eables oe to create ad use a reduced set of parameters, which are called pricipal factors Reduced set parameters are much easier to aalyze ad iterpret o study a data set that has more tha 0 parameters is extremely difficult but if oe ca reduce them to five without losig iformatio of more tha 0 parameters, it would save the egieer a lot of time At this momet, the techique to derive the reductio will ot be discussed 6 Disadvatage of Usig Multivariate Cotrol Chart he disadvatages of usig multivariate cotrol chart may have bee metioed i the earlier text Nevertheless, the summary of it provided here Oe may ask why multivariate cotrol chart is ot popular ad ot frequetly used i the idustry he obvious reaso is that the associated calculatios are laborious Oe eeds to calculate the subgroup meas for each parameter,,, p, the subgroup s variace, ad the subgroup s covariace Beside these calculatios, oe eeds to calculate the mea of the subgroup s mea, the mea of subgroup s variace, the mea of the subgroup s covariace, the sample variace-covariace matrix, the mea vector, value, ad the cotrol limits Fially value eeds to be compared to the cotrol limits to determie if ay poit is out of cotrol However, livig i this iformatio age, such laborious work ca be replaced usig computer program Whe there is a out of cotrol situatio, it is difficult to idetify the parameter that causig the problem Oe eeds to resolve to idividual Shewhart cotrol chart or Studet paired t-statistic to idetify it Oe may eed to calculate the t-statistic value of every parameter ad compare it with the critical value of t- - -

23 statistic at the specified ( - α)00% cofidece level to idetify which parameter or parameters causig out of cotrol situatio - 3 -

24 A Adjoit method 3 B Beta distributio 4 C Chi-square 6 F F-distributio 9 G Gauss-Jorda elimiatio method 3 H Hotellig dispersio chart 6 Hotellig mea chart 6 Hotellig, Harold 7 Hotellig s cotrol chart 6 I Iverse matrix, 3 L Lower cotrol limit M Mea vector 3, 6 Multivariate cotrol chart 3 Multivariate ormal distributio 5 P Pooled variace-covariace 9 S Shewhart cotrol chart 6, 9, Studet s t-distributio 8 Studet s t-statistic 7, 7 average value of k subgroups 0 U Upper cotrol limit 9,, 3 V Variace-covariace matrix 3, 7, 9, - 4 -