MULTIVARIATE STATISTICAL PROCESS OF HOTELLING S T CONTROL CHARTS PROCEDURES WITH INDUSTRIAL APPLICATION

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1 Journal of Statistics: Advances in heory and Alications Volume 8, Number, 07, Pages -44 Available at htt://scientificadvances.co.in DOI: htt://dx.doi.org/0.864/jsata_ MULIVARIAE SAISICAL PROCESS OF HOELLING S CONROL CHARS PROCEDURES WIH INDUSRIAL APPLICAION M. S. HAMED Deartment of Administrative Sciences Faculty of Business Administration aibah University Kingdom of Saudi Arabia Deartment of Alied Statistics Faculty of Commerce Benha University Egyt moswilem@gmail.com Abstract A control chart is a tool that monitors quality characteristics of a rocess to ensure that rocess control is being maintained when monitoring multile characteristics that are correlated, it is imerative to use multivariate control chart. Hotelling s quality control chart is used to determine whether or not the rocess mean vector for two or more variables is in-control. It is allowing us to simultaneously monitor whether two or more related variables are in control, and it is shown that multivariate quality control chart do not indicate which variables cause the out-of control signal so that the interretation of the out-ofcontrol signal. Also, in this aer develos the multivariate quality control charts of the out of-control signal, this will be maintained by making and industrial alication on the fertilizers factory. It is imortant to know that this 00 Mathematics Subject Classification: 6P30. Keywords and hrases: quality control, multivariate analysis, average run lengths erformance, monitoring rocess. Received August, Scientific Advances Publishers

2 M. S. HAMED factory has a secial deartment for quality control, in order to maintain International Standardization Organization (ISO).. Introduction he statistical control chart is a well-known tool in today s industry, and it is one of the most owerful tools in quality control. First develoed in the 90 s by Walter Shewhart, the control chart found widesread use during World War II and has been emloyed, with various modifications ever since. he drawbacks to multivariate charting schemes is their inability to identify which variable was the source of the signal. With today s use of comuters, it is common to monitor several correlated quality characteristics simultaneously. Various tyes of multivariate control charts have been roosed to take advantage of the relationshis among the variables being monitored. Alt []; Jackson [7]; Lowry and Montgomery []; and Mason et al. [] discuss much of the literature on this toic. he formatter will need to create these comonents, incororating the alicable criteria that follow. he raid growth data acquisition technology and the uses of online comuters for rocess monitoring led to an increased interest in the simultaneous control of several related quality characteristics. hese techniques are often referred to multivariate statistical rocess control rocedures. he use of searate univariate control chart for each quality characteristic has roved to be inaroriate. his is because, it neglects the correlation between the multile quality characteristics; and this leads to incorrect results. he modern statistical rocess control took lace when Walter Shewhart [0] develoed the concet of a control chart based on the monitoring of the rocess mean level through samle mean ( chart ) and rocess disersion through samle range (R chart) or samle standard deviation chart. In the multivariate setting, Harold Hotelling [4] ublished what can be called the first major works in multivariate quality control. Hotelling develoed the statistic and the statistics

3 MULIVARIAE SAISICAL PROCESS OF 3 based on the samle variance-covariance matrix S rocedure, and its extensions to control charts to combine measurements taken on variables in several dimensions into a single measure of excellence. After Hotelling there was no significant work done in this field until the early sixties, when with the advances in comuters, interest in multivariate statistical quality control was revived. Since then, some authors have done some work in this area of multivariate quality control. Houshmand and Javaheri [5] resented two rocedures to control the covariance matrix in a multivariate setting. he advantages of these rocedures are that they allow the investigators to identify the sources of the out-of-control signal. hese rocedures are based on constructing tolerance regions to control the arameters of the correlation matrix. Linna et al. [0] resented a model for correlated quality variables with measurement error. he model determined the erformance of the multivariate control charting methods. he usual comarison of control chart erformance does not directly aly in the resence of measurement error. he most familiar multivariate rocess monitoring and control rocedure is the Hotelling s control chart for monitoring the mean vector of the rocess. It is a direct analog of the univariate Shewhart chart. Shewhart, a ioneer in the develoment of the statistical control chart (Shewhart charts), first recognized the need to consider quality control roblems as multivariate in character. Hotelling [4] did the original work in multivariate quality control. He alied his rocedure, which assumed that P-quality characteristics are jointly distributed as P-variate normally and random samles of size n are collected across time from the rocess. is sensitive to shifts in the means, as well as to shifts in the variance, but it cannot distinguish between location shifts and scale shifts. Multivariate charts are also useful for monitoring quality rofiles as discussed by Woodall et al. [4]. Alt [] defined two hases in

4 4 M. S. HAMED constructing multivariate control charts, with hase I divided into two stages. In the retrosective Stage of hase I, historical data (observations) are studied for determining whether the rocess was in control and to estimate the in-control arameters of the rocess. he Hotelling s control chart is utilized in this stage (Alt and Smith []; racy et al. []; and Wierda [3]). In hase II, control charts are used with future observations for detecting ossible dearture from the rocess arameters estimated in hase I. In hase II, one uses charts for detecting any dearture from the arameter estimates, which are considered in the in-control rocess arameters (Vargas []). An imortant asect of the Hotelling s control chart is how to determine the samle variance-covariance matrix used in the calculation of the chart statistics, the uer control limit (UCL) and the lower control limit (LCL). Onwuka and Hotelling [5] discussed the rincial comonent analysis and Hotelling s tests were used, with the 3-characteristics measured showing negligible low correlation with nearly all the correlation coefficients small.. Multivariate Quality Control Chart Multivariate quality control charts are a tye of variables control that how correlated, or deendent, variables jointly affect a rocess or outcome. he multivariate quality control charts are owerful and simle visual tools for determining whether the multivariate rocess is incontrol or out-of-control. In other words, control charts can hel us to determine whether the rocess average (center) and rocess variability (sread) are oerating at constant levels. Control charts hel us focus roblem solving efforts by distinguishing between common and assignable cause variation. Multivariate control chart lot statistical from more than one related measurement variable. he multivariate

5 MULIVARIAE SAISICAL PROCESS OF 5 control chart shows how several variables jointly influence a rocess or outcome. It is demonstrated that if the data include correlated variables the use of searate control chart is misleading because the variables jointly affect the rocess. If we use searate univariate control chart in a multivariate situation, tye I error and robability of a oint correctly lotting in-control are not equal to their exected values the distortion of those values increases with the number of measurement variables. It is shown that multivariate control chart has several advantages in comarison with multily univariate charts: he actual control region of the related variables is reresented. We can maintain secification tye I error. A signal control limit determines whether the rocess is in control. Multivariate control chart simultaneously monitors two or more correlated variables. o monitor more than one variable using univariate charts, we need to create a univariate chart for each variable. he scale on multivariate control charts unrelated to the scale of any of the variables. Out-of-control signals in multivariate charts do not reveal which variable or combination of variables cause the signal. A multivariate control chart consists of: Plotted oints, each for which reresents a rational subgrou of data samled from the rocess, such as a subgrou mean vector individual observation, or weighted statistic. A center line, which reresents the exected value of the quality characteristics for all subgrous. Uer and lower control limits (UCL and LCL), which are set a distance above and below the center line. hese control limits rovide a visual dislay for the exected amount for variation. he control limits are based on the actual behaviour of the rocess, not the desired

6 6 M. S. HAMED behaviour or secification limits. A rocess can be in control and yet not be caable of meeting requirements. 3. Construction of Hotelling s Control Chart he Hotelling multivariate control chart signals that a statistically significant shift in the mean has occurred as soon as: χ ( ) i µ o ( i µ o ). () If the samle covariance matrix and the samle mean vector µ 0 are known, but if and µ 0 are known, then the statistic is the aroriate statistic for the Hotelling multivariate control chart. In this case, the samle covariance matrix, S and samle mean vector, are used to estimate and µ 0, resectively. his statistic has the from: ( ) S ( ). i i () Suose that we have a random samle from a multivariate normal distribution Say,,,,, where the i -th samle vector, 3 n contains observations, i, i, i3,, i. Let the samle mean vector is: where ( ), i n L il ( i,,, ), and the samle covariance matrix is:

7 MULIVARIAE SAISICAL PROCESS OF 7 S S S S S S S S S, S (3) where s i is the variance of the i -th variable and ( ij )-th element of S-matrix is the estimated covariance between the variables i and j, S ij n ( il i )( jl j ). n L Note that we can show that the samle mean vector and the samle covariance matrix are unbiased estimators of the corresonding oulation quantities that is ( ) µ and E ( S ). E (4) Seber [8] gives the distribution roerties of this estimate as follows: (i) ~ N, µ. (5) n (ii) If distribution as in (i) then: n( ) µ ( µ ) ~ χ. (6) (iii) ( n ) S ~ W ( n, ), where W ( n, ) stands for the Wishart distribution. (iv) If Z and D are indeendent, random variables distributed, resectively as: Z ~ N ( 0, ) Z

8 8 M. S. HAMED ( n ) D ~ W ( n, ), Z then the quadratic form: Z D Z, (7) is distributed as: ( n ) ( n ) ~ F, n. (8) (v) If Z ~ N ( 0, ) and ( n ) D ~ W ( n, ), ( n ) >, where ( n )D can be decomosed as: Z ( n ) D ( n ) D + ZZ, Z where ( n ) D ~ W ( n, ), Z and Z is indeendent of D then the quadratic form: Z D Z, (9) is distributed as: ~ ( n ) β(, n ), where β(, n ) is the central Beta distribution. (vi) If the samle is comosed of k subgrous of size n with subgrou means j, j,, 3,, k and grand mean, i.e., k j j k k n j i ij kn, then

9 MULIVARIAE SAISICAL PROCESS OF 9 ( ) ~ ( 0, ). kn j N k (0) (vii) If the samle is comosed of K subgrous of n identically distributed multivariate normal observations and if S j is the samle covariance matrix from the j -th subgrou, j,, 3,, K ), then ( n ) S j ~ W( Kn, ), () these distributional roerties of multivariate quality control rocedures., S and are used in the Now, we resent two versions of Hotelling (a) Subgrou data chart: Suose that P-related quality characteristic,, 3,, are controlled jointly according to the P-multivariate normal distribution. he rocedure requires comuting the samle mean for each of the P- quality characteristics from a samle of size n. Let the set of quality characteristic means is reresented by the ( ) vector as:. hen the test statistic lotted on the Chi-square control chart for each samle is: where χ0 µ ( µ ) ( ), n ()

10 0 M. S. HAMED µ ' ( µ, µ,, µ ) is the ( ) vector of in-control means for each quality characteristic and is covariance matrix. Now, suose that m-subgrou are available. he samle means and variances are calculated from each subgrou as usual that is: n n jk ijk i, S jk n ijk jk i n ( ), j,,, ; k,,, m, (3) where the is the i -th observation on the j -th quality characteristic in ijk k -th subgrou. S jhk n ( ijk jk )( ihk hk ). n i k,,, n, j h, (4) reresents the covariance between quality characteristic j and quality characteristic h in the k -th subgrou. he statistics to obtain jk jk, S, S jhk are the averaged over all m-subgrous j m j, m k k j,,,, S j m S j m k k, j,,,, and

11 MULIVARIAE SAISICAL PROCESS OF S jh m S jhk, m (5) k where j h and j are the i -th elements of the ( ) samle mean vector and ( ) average of samle covariance matrices S is formed as: S S S S S S. S (6) here are consider the unbiased estimate of µ and when the rocess is in control. If, we relace µ with and with S in (), the test statistic now becomes ( ) S ( ). n (7) Alt [] has ointed out that there are two distinct hases of control chart using. Phase I is the use of the chart for establishing control, that is, testing whether the rocess was in control when m-subgrous were drawn and the samle statistic and S comuted. he objective in hase I is to obtain an in-control set of observations, so that control limits can be established for hase II which is the monitoring of future roduction. In the hase I the control limits for the -control chart is given by: P( m )( n ) UCL F α, P, mn m + mn m +. LCL 0 (8)

12 M. S. HAMED In the hase II when the chart is used for monitoring future roduction, the control limits are as follows: P( m + )( n ) UCL F,, α P mn m + mn m +, LCL 0 (9) when the arameters µ and are estimated from a large number of subgrous, it is often to use χ α, UCL as the uer limit in both hases. Retrosective analysis of samles to test for statistical control and establish control limits also occurs in the univariate control chart setting. For the -chart, it is well-known that if use m 0 or 5, samles, the distribution between hase I and hase II limits is usually unnecessary, because the hase I and hase II limits will nearly coincide. However, with multivariate control charts, we must be careful. Lowry and Montgomery [] showed that in many situations a large number of samles would be required before the exact hase II control limits are well aroximate by the Chi-square. (b) Individual observations In some situation the subgrou size is naturally n. Suose that m samles each of size n are available and that is the number of quality characteristics observed in each samle. he Hotelling statistic becomes: ( ) S ( ). (0) Ryan [7] defined the hase II control limits for this statistic as: P( m + ) ( m ) UCL F,, α P m m( m ). LCL 0 () Jackson [7] suggested that for large m ( m > 00) then we can use an aroximate control limit, either

13 MULIVARIAE SAISICAL PROCESS OF 3 P( m ) UCL Fα, P, m, () ( m ) or UCL χ α,. (3) Equation (3) is only aroriate if the covariance matrix is known. Lowry and Montgomery [] suggested that if is large-say 0 then at least 50 samles must be taken ( m 50) before Chi-square uer control limit is a reasonable aroximation to the correct value. racy et al. [] oint out that if n, the hase I limits should be based on a beta distribution that is, the hase I limits defined as: UCL UCL β,, α P m ( m ) m 0. (4) he average run length erformance for a Hotelling s Mason et al. [] suggested that the average run length (ARL) for a control rocedure is defined as: ARL, where reresents the robability of being outside the control region. For a rocess that is in-control, this robability is equal to α, the robability of tye I error. he ARL has a number of uses in both univariate and multivariate control rocedures. hey suggested that it can be used to calculate the number of observations that one would exect to observe, on average, before a false alarm occurs. his given by: ARL. Another use of the ARL is to comute the number of observations one would exect to observe before detecting a given shift in the rocess. he

14 4 M. S. HAMED robability of a tye II error. he ARL for detecting the shift is given by: ARL. β Multivariate control charts using Hotelling s statistic is oular and easy to use. A major advantage of Hotelling s statistic is that it can be shown to be the otimal test statistic for detecting a general shift in the rocess mean vector for an individual multivariate observation. However, the technique has several ractical drawbacks. A major drawback is that when the statistic indicates that a rocess is out of control, it does not rovide information in which variable or set of variables is out of control. Further, it is difficult to distinguish location shifts from scale shifts since the statistic is sensitive to both tyes of rocess changes. he MY decomosition Mason, Young and racy (MY) extended that the interretation of signals from a chart to the setting where there is more than rocess variables MY decomosition. he MY decomosition is the rimary tool used in this effort, and they examined many interesting roerties associated with it. hey showed that the decomosition terms contained information on the residuals generated by all ossible linear regressions of one variable on any subset of the other variables. And they add that to being an excellent aid in locating the source of a signal in terms of individual variables or subsets of variables, this roerty has another major function. It can be used to increase the sensitivity of the statistic in the area of small rocess shifts. Mason et al. [4] resented decomosition rocedure. hey considered that, the statistic for a -dimensional observation vector (,,, ) can be reresented as ( ) S ( ), (5)

15 MULIVARIAE SAISICAL PROCESS OF 5 where and S are the common estimators of the mean vector and covariance matrix obtained from historical data set (HDS), they artitioned the vector ( ) as: where ( ) [( ( ) ( ) ), ( )], ( ) (,,, ) reresented the ( ) -dimensional variable vector excluding the -th variable and ( ) reresented the corresonding elements of the mean vector. hey also artition the matrix S so that where S sx, S (6) sx sxx S is the ( ) ( ) covariance matrix for the first ( ) variables, S is the variance of, and s x is a ( ) -dimensional vector the containing the covariances between and the remaining ( ) variables. he statistic in (5) can be artitioned into two indeendent arts (see Rencher [6]). hese comonents are given by he first term in (7), +,,,,. (7) ( ( ) ( ) ) S ( ( ) ( ) ), (8) uses the first ( ) variables and is itself a statistic. Mason et al. [4] roved that the last term in (7) was the -th comonent of the vector i adjusted by the estimates of the

16 6 M. S. HAMED mean and standard deviation of the conditional distribution (,, ). It is given by, given,,, (,,,., ),, (9) S,,,., where,,,., ( ( ) ( + B ) ), and B S sx, is the ( ) -dimensional vector estimate of the coefficients from the regression of on the ( ) variables,,,. It can be shown that the estimate of the conditional variance is given as S,,,, S sx S sx, since the first term of (7) is a statistic, it too can be searated into two orthogonal arts: +,,,,. he first term,, is a statistic, on the first ( ) comonents of the vector, and the second term,,,,, is the square of adjusted by the estimates of the standard deviation of the conditional distribution of given (,,, ). hey roosed one from of MY decomositions of a statistic. It is given by. +, + 3,, + +,,,, j+,,, j j +. (30) he term in (30) is the square of the univariate for the first variable of the vector and is given as

17 MULIVARIAE SAISICAL PROCESS OF 7 ( ). (3) S his term is not a conditional term, as its value does not deend on a conditional distribution. In contrast, all other terms of the exansion in (30) are conditional terms, since they reresent the value of a variable by the mean and standard deviation from the aroriate conditional distribution. Comuting the decomosition terms hey considered that the first ( ) terms of (30) corresond to the, value of the sub vector (,, ); i.e., (,, ), +, + 3,, + +,,,,, similarly, the first ( ) terms of this exansion corresond to the sub vector (,,, ); i.e., (,, ), +, + 3,, + +,,,, 3, continuing in this fashion, they comute the values for all sub vectors of the original vector. he last sub vector, consisting of the first comonent () ( ) is used to comute the unconditional given in (3); i.e.,. term All the values, comuted using the general formula ( ), ( ),,,,,, ( ),, are (,,, j ) ( j ( ) ( j) ) ( ( j) ( j S ) jj ), (3) ( j ) reresents the aroriate sub vector, where corresonding sub vector mean, and ( j ) is the S jj denotes the corresonding

18 8 M. S. HAMED covariance sub matrix obtained from the overall S matrix given in (6) by deleting all unused rows and columns. he terms of the MY decomosition can be comuted as follows:,,,,,,,, ( ),,, (,,, ), ( ),,, (,,, ),, ( ),, ( ). (33) s Proerties of the MY decomosition Many roerties are associated with the MY decomosition. Consider they dimensional vector defined as (,,, ) they interchange the first two comonents to form another vector (,,, ) so that the only difference between the two vectors is the two vectors such that the first two comonents have been ermuted the value of the two vectors is the same; i.e., ( ),, (,,, )., his occurs because values cannot be changed by ermuting the comonents of the observation vector. his invariance roerty of ermuting the comonents that each ordering of an observation vector will roduce the same overall value. Since there are! ( )( )( ) ( )( ) ermutations of the comonents of the

19 MULIVARIAE SAISICAL PROCESS OF 9 vector (,,, ) this imlies that we can artition a value in! different ways. o illustrate his result, suose 3. here are 3! ( 3)( )( ) 6 decomositions of the value for an individual observation vector. hese are listed below: +, + 3,, + 3, +,, 3 + 3, +,, 3 +, + 3,, 3 +, 3 +,, 3 3 +, 3 +,, 3. (34) Each row of (34) corresonds to a different ermutation of the comonents of the observation vector. For examle, the first row corresonds to the vector written in its original form as (,, 3 ), whereas the last row reresents ( 3,, ). Note that all six ossible ermutations of the original vector comonents are included. he imortance of this result is that it allows one to examine the statistic from many different ersectives. he terms in any articular decomosition are indeendent of one another, although the terms across the decomositions are not necessarily indeendent. With artition and! artitions, there are! ossible terms to evaluate in a total MY decomosition of a articular artition, as certain terms occur more than once. In general, there are ( ) distinct terms among the ossible decomositions. hese unique terms are the ones that need to be examined for ossible contribution to a signal, when is large. Comuting all these terms can be cumbersome.

20 0 M. S. HAMED hey considered the MY decomosition given in (30) and suose dominates the overall value of the statistic. his indicates that the observation on the variable is contributing to the signal. However, to determine if the remaining variables in this observation contribute to the signal, we must examine the value associated with the sub vector (, 3,, ) which excludes the comonent. Small values of the statistic for this sub vector imly that no signal is resent. hey also indicate that one need not examine any term of the total decomosition involving these ( ) variables. hey considered another imortant roerty of statistic is the fact that the ( ( ) ) unique conditional terms of a MY decomosition contain the residuals from all ossible linear regressions of each variable on all subsets of the other variables. his roerty of the statistic rovides a rocedure for increasing the sensitivity of the sensitivity of the Locating signaling variables statistic to rocess shifts. hey seek to relate a signal and its interretation to the comonents of the MY decomosition. hey consider signaling observation vector: (,,, ) such that (,,, ) > UCL. hey roosed two methods one for locating the variables contributing to the signal is to develo a forward iterative scheme. his was accomlished by finding the subset of variables that do not contribute to signal from (7) and (9) such that a statistic can be constructed on any subset of the variables,,,. Construct the for each individual variable, j,,,, so that j statistic

21 MULIVARIAE SAISICAL PROCESS OF j ( ) j S j j, where j and S j are the corresonding mean and variance estimates as determined from the HDS. Comare these individual UCL, where j values to their UCL ( ) j ( n + ) ( n ) F ( ) ( α, n n n ), n + F( α,, n ), n (35) is comuted for an aroriate α level and for a value of. Exclude from the original set of variables all j for which j > UCL ( ), j since observations on this subset of variables are definitely contributing to the signal. From the set of variables not contributing to the signal. Comute the statistic for all ossible airs of variables. For examle, for all ( i, j ) with i j comute uer control limit, ( ), and comare these values to the i, j ( ) ( ) ( ) ( ). UCL n + n (, ) F i j α,, n n n Exclude from this grou all airs of variable for which (, ) > UCL ( i, j ). i j he excluded airs of variables in addition to exceeded single variable comrise the grou of variables contributing to the overall signal continue to iterate in this fashion so as to exclude from the remaining grou all variables of signaling grous of three variables four variables

22 M. S. HAMED etc. he rocedure roduces a set of variables that contribute to the signal. And another method of locating the vector comonents contributing to a sing al is to examine the individual terms of the MY decomosition of a signaling observation vector and to determine which are large in value. his method was accomlished by comaring each term to its corresonding critical value. Mason et al. [4] roosed that the distribution governing the comonents of the MY decomosition for the situation where there are no signals is F distribution. For the case of variables, these are given by for unconditional terms, and by ~ n + j F(, n ), (36) n ( ) ( ) ~ n + n j,,,, j F ( ) (, n k ), (37) n n k for conditional terms. Where k equals the number of conditioned variables. For k 0, the distribution in (37) reduces to the distribution in (36). Using these distributions, critical values for a secified α level and HDS samle of size n for both conditional and unconditional terms are obtained as follows: unconditional terms : n + CV F( α,, n ), n ( ) ( ) conditiona l terms : n + n CV F ( ) ( α,, n k k ). (38) n n hey add that we can comare each individual term of the decomosition to its critical value and make the aroriate decision. Interretation of a signal on a comonent hey (Mason et al. []) considered one of the ossible unconditional, terms resulting from the decomosition of the statistic associated with a signaling observation. As stated earlier, the term

23 MULIVARIAE SAISICAL PROCESS OF 3 ( j j ) j, j,,, is square of a univariate t statistic for sj the observed value of the j -th variable of an observation vector. For control to be maintained, this comonent must be less than its critical value, i.e., n + j < F( α,, n ), since t n ( α ) F, n ( α,, n ) they re-exressed this condition as j being in the following interval: n + n + ( ) t ( α ) < < ( ) ( ),, j t n α n n, n (39) or as n + n + ( ) t ( α ) < < + ( ) ( ),, n j t α (40) n n, n where t ( α, n ) is, the aroriate value from a t-distribution with n degrees of freedom. his is equivalent to using a univariate Shewhart control chart for the j -th variable. And they considered the form of a general conditional term given as j,,, ( j j,,,, j ), j, (4) s j,,,, j if the value in (4) is to be less than its control limit, ( n + ) ( n ) F ( ) ( α,, n ). n n k j,,,, j < k Its numerator must be small, as the denominator of these terms is fixed by the historical data this imlies that comonent j from the observation vector (,,, j,, ) is contained in the conditional distribution of j given,,, j and falls in the ellitical control region.

24 4 M. S. HAMED A signal occurs on the term in (4) when j is not contained in conditional distribution of j given,,, j, i.e., when Regression ersective ( n + ) ( n ) F ( ) ( α,, n ). n n k j,,,, j > k Mason et al. [3, 4] roosed that, in general, j,,,, j is a standardized observation on the j -th variable adjusted by the estimated of the mean and variance form the conditional distribution associated with ( j,,, j ). he general from of this term was given in (4). hey considered the estimated mean of j adjusted for,,, j, and estimated this mean by using the rediction equation. ( j ( ) ( j ) j,,,, j j + B j ), (4) where j is the samle mean of he sub vector j ) and j obtained from the historical data. ( j ) is comosed of the observation on (,,, ( j ) is the corresonding estimated mean vector, S jj, is the covariance matrix of the first j comonents of the vector. o obtain S jj artition S as follows: S S jj S j( j). S ( ) S j j ( j)( j) Further, artition the matrix S jj as S jj S( j )( j ) S j( j ). S j( j ) S j hen

25 MULIVARIAE SAISICAL PROCESS OF 5 B j S ( )( ) S j j j( j ), since the left-hand side of (4) contains, which is the j,,,, j redicted value of j, the numerator of (4) is a regression residual reresented by r ( j j,,,, ), j,,,, j j rewriting the conditional variance as S j,,,, j S j j,,,, j ( R ). (see, e.g., Rencher [6] and substituting r j,,,, j for ( j j,,,, j ), they exressed j,,, ( rj,,,, j ), j s ( R ). (43) j j,,,, j he above results indicate a signal may occur if something goes astray with the relationshis between subsets of various variables. his situation can be determined by examination of the conditional terms. A signaling value indicates that a contradiction with historical relationshi between the variables has occurred either () due to a standardized comonent value that is significantly larger or smaller than that redicted by a subset of the remaining variables or () due to a standardized comonent value that is marginally smaller or larger than that redicted by a subset of the remaining variables when there is a very severe collinearity (i.e., a large R value) among the variables. hus, a signal results when an observation on a articular variable or set of variables, is out of control and/or when observations on a set of variables are counter to the relationshi established by the historical data.

26 6 M. S. HAMED Imroving the sensitivity of the statistic Mason et al. [, 3] used the decomosition for imroving the sensitivity of the in signal detection. hey showed that the statistic to be a function of all ossible regressions existing among a set of rocess variables. Furthermore, they showed that the residuals of the estimated regression models are contained in the conditional terms of the MY decomosition. Large residuals roduce large comonents for the conditional terms and are interreted as indicators of counter relationshis among the variables. However, a large residual also could imly an incorrectly secified model. his result suggests that it may be ossible to imrove the erformance of the statistic by more carefully describing the functional relationshis existing among the rocess variables. Minimizing the effects of model missecification on the signaling ability of the should imrove its erformance in detecting abrut rocess shifts. hey showed when comared to other multivariate control rocedures, the lacks the sensitivity of detecting small rocess shifts. hey showed that this roblem can be overcome by monitoring the error residuals of the regressions contained in the conditional terms of the MY decomosition of statistic. Furthermore, they showed that such monitoring can be helful in certain tyes of on-line exerimentation within a rocessing unit. hey roosed an alternative form of condition terms, they considered the conditional term of the MY decomosition in (5). his is the squares of the j -th variable of the observation vector which adjusted by the estimates of the mean and variance of the conditional distribution of j given,,, j. hey showed that (5) could be written as j,,, rj,,,, j, j ( ), (44) s j,,,, j

27 MULIVARIAE SAISICAL PROCESS OF 7 this was achieved by noting that j,,, j can be obtained from the, regression of j on,,, j : i.e., (45) j,,,, j b0 + b + b + + bj j, where b j are the estimated regression coefficients. Since j,,, j is, the redicted value of j. he numerator of (4) is the raw regression residual, given in (44)., j ( j j,,,, j ) (46) r j,,, Another form of the conditional term in (4) is obtained by substituting the following quantity for the conditional variance contained in (44), i.e., by substituting S j,,,, j S j ( Rj,,,, j ), where j,,,, j R is the squared multile correlation between j and this yields:,,, j j,,, rj,,,, j, j s ( R ). (47) j j,,,, j Much information is contained in the conditional terms of the MY decomosition. Since these terms are, in fact, squared residuals from regression equations they can be helful in imroving the sensitivity of the statistic in detecting both abrut rocess changes and gradual shifts in the rocess. hey considered the better the fit of a model, the more sensitive the control rocedure will be to deartures from the model. his suggests that more effort should be taken in hase I oerations, during

28 8 M. S. HAMED construction of the historical database, to insure roer functional forms are chosen for the rocess variables and useful models are created. Princial comonents rocedure Jakson [6, 8] recommended to use the rincial comonents rocedure to aid in the interretation of an out-of-control signal. he rincial comonent analysis (PCA) technique decomoses the statistic into a sum of indeendent rincial comonents, which are linear combinations of the original variables, and using these comonents to hel for solving this identification roblem. he PCA used to reduce to dimensionality of a data set which consists of a large number of interrelated variables. While retaining as much as ossible of the variation resent in the data set. his goal is achieved by transforming the original variables to a new set of uncorrelated variables, which are called the rincial comonents. his transformation is a rincial axis rotation of the variance and covariance matrix of the data set, and the elements of the characteristic vectors or the eigenvectors of the covariance matrix are direction cosines of the new axes related to the old. he transformed new uncorrected variable or the rincial comonents are normally numbered in descending order according to the amount at the variation. he use of the method of PCA in the field of multivariate quality control was first introduced by Jackson and Morris [9]. hey identified a large number () of correlated variables that account for the quality of the rocess. hey notice that the use of Hotelling s may involve comutational roblems since the determinant of the variance and covariance matrix is near zero. he solution is to transform the original variables to lesser k rincial comonents. Jackson [6] roosed using PCA to interretation an out-of-control single by decomosing into indeendent comonent. Jackson roosed that the starting oint of the statistical alication of the

29 MULIVARIAE SAISICAL PROCESS OF 9 method of rincial comonents is the samle covariance matrix S, for a -variate roblem in (3). If the covariances are not equal to zero, it indicates a relationshi existing between those two variables, the strength of that relationshi (if Sij it is linear) being reresented by the correlation r ij ( S S ). A rincial axis transformation will transform correlated variables,,, into new uncorrelated variables Z, Z,, Z the coordinate axes of these new variables being described by the vectors u i which make u the matrix U of direction cosines used in the following transformation: i j Z U ( ), (48) the transformed variables are called the rincial comonents of. he j -th rincial comonent would be zi ui ( ). (49) If one wishes to transform a set of variables by a linear transformation Z U ( ), whether U is orthogonal or not, the covariance matrix of the new variables S z can be determined directly from the covariance matrix of the original variables S by the relationshi: U Sz SU. (50) However, the fact that U is orthonormal is not a sufficient condition for the new variables to be indeendent. Only a transformation such as the rincial axis transformation will roduce the diagonal element of the matrix L, where S z

30 30 M. S. HAMED L S S S he fact that S z is diagonal elements of L means that rincial comonents are uncorrected. He added that we can also determine the correlation of each rincial comonent with each of the original variables; this is useful for diagnostic uroses. he correlation of the i -th rincial comonent S z and determined as j -th original variable j can be r zi j uij Li. (5) s j Another interesting roerty of rincial comonents is the fact that the Equation (48) can be inverted to + U z, (5) by virtue of the fact that U is orthonormal so that U U. his means that if we know the values of the rincial comonents, we can determine what the original data were. Princial comonent form of Jackson [8] roosed that the can be exressed as a function of the rincial comonents of the estimated covariance matrix. He gives an alternative form of the statistic as: z i λi i ( ) S ( ), (53)

31 MULIVARIAE SAISICAL PROCESS OF 3 where λ > λ > > λ are the eigenvalues of the estimated covariance matrix S and the, i,,,, are the corresonding rincial comonents. z i Each of these comonent is obtained by multilying the vector quantity ( ) by the transose of the normalized eigenvector U i of S corresonding to λ i: i.e., zi Ui ( ). Each z i is a scalar quantity and the statistic is exressed in terms of these values. he reresentation in (53) is derived from the fact that the estimated covariance matrix S is a ositive definite symmetric matrix. hus, its singular value decomosition is given as S UAU, where U is a orthogonal matrix whose columns are the normalized eigenvectors U i of S, and A is a diagonal matrix whose elements are the corresonding eigenvalues, U λ λ ( U, U,, U ) and A. 0 0 λ Note that S UA U. (54) Substituting this quantity into the statistic of (53), we have ( ) UA U ( ) i i Z A Z

32 M. S. HAMED 3, i i i Z λ (55) where ( ) U z i i and.,,, z z z Z A Hotelling s statistic for a single observation also can be written as ( ) ( ), Y R Y S where R is the estimated correlation matrix and Y is the standardized observation vector of x, i.e., ( ), r r r r r r r r r r R where ( ) j r ij, corr and ( ) ( ) ( ) [ ].,,,,,, y y y S S S he matrix R (obtained from S) is a ositive definite symmetric matrix and can be reresented in terms of its eigenvalues and eigenvectors. Using a transformation similar to (64), the above can be written as, i i i W γ (56) where γ > > γ > γ are the eigenvalues of the correlation matrix R, and w w w,,, are the corresonding rincial comonents of the

33 matrix MULIVARIAE SAISICAL PROCESS OF 33 R, wi can be determined by the following transformation: ( ), i,,,, wi vi where the v, v,, v, are the corresonding normalized eigenvectors of R Equation (56) is not to be confused with (53). he first equation is written in terms of the eigenvalues and eigenvectors of the covariance matrix, and the second is in terms of the eigenvalues and eigenvectors of the estimated correlation matrix. hese are two different forms of the same Hotelling s mathematical transformations are not equivalent. as the he rincial comonent reresentation of the lays a number of roles in multivariate statistical rocess control (SPC). he control region can be defined by the UCL. he observations contained in the values less than the UCL, i.e., for each i, i < UCL. hus, by (56) w w γ γ w he control region is defined by the equality γ < UCL. w γ w w γ γ UCL, which is the equation of a hyer ellisoid in a -dimensional sace rovided the eigenvalues γ > γ > > γ are all ositive. he fact that the estimated correlation matrix R is a ositive definite matrix guarantees that all the γ s are ositive. i Note that in secial case of the rincial comonent sace of the estimated correlation matrix, can be reduced to

34 34 M. S. HAMED w w + UCL, (57) γ γ which gives the equation of the control ellise. he length of the major axis of the ellise in (57) is given by γ and the length of the minor axis is given by γ. he axes of this sace are the rincial comonents, w and w. he absence of a roduct term in this reresentation indicates the indeendency between w and w. his is a characteristic of rincial comonents, since they are transformed to be indeendent. Assuming that the estimated correlation r is ositive it can be shown that γ ( + r) and γ ( r). For negative correlations, the γ i values are reversed. One can also show that the rincial comonents can be exressed as w ( y + y ), w ( y ). From these y equations, one can obtain the rincial comonents as functions of the original variables. 4. Alication of Multivariate in Industrial Delta Fertilizers and Chemical Industries is considered one of the leading comanies in the field of fertilizers roduction in Egyt. About 4500 emloyees are working for it, on the various managerial levels. Urea roduction is one of the major roducts of the comany. he roduction of urea occurs through three stages, summarized as follows: A. High ressure stage In this stage, urea is roduced through two reactions; the first reaction occurs by condensation of Ammonia gas and Carbon dioxide under high ressure and temerature for the sake of the roduction of intermediate material, known as Carbamate. he second reaction haens by searating the water from the Carbamate in order to a chive urea. In this, stage the condensation of urea aroximately 56%.

35 MULIVARIAE SAISICAL PROCESS OF 35 It contains 6 variables, these are: E-0Outlet emerature Outlet cold NH3 from E CO to rain 4 4 CO ressure to synthesis 5 5 CO after E- 6 6 R emerature in reactor R emerature in reactor R emerature in reactor R emerature in reactor R-0 Strier level Liquid leaving the Strier 3 3 Stream from E-04 to j Conditioned water to scrubber E Conditioned water from scrubber E Stream from j-03 able analysis of laboratory in this stage: t. NH3 Reactor outlet t. CO Reactor outlet 3 t.3 UR Reactor outlet 4 t.4 B Reactor outlet 5 t.5 HO Reactor outlet 6 t. NH3 Strier outlet 7 t. CO Strier outlet 8 t.3 UR Strier outlet 9 t.4 B Strier outlet 0 t.5 HO Strier outlet

36 36 M. S. HAMED B. Low ressure stage In this stage, the condensation of urea liquid rises from 56% to 7%. his haens through the decomosition of the remaining Carbamate and the elimination of water under low ressure. It contains seven variables, these are: y Urea solution from strier E-0 y Steam to E-05 3 y3 Urea carbonate solution from strier -0 to E-05 4 y4 Gas leaving -0 5 y5 Level in K-0 6 y6 P-03 7 y7 Urea solution in K-0 able analysis of laboratory in this stage: t3. NH3 D 0 Outlet t3. CO D 0 Outlet 3 t3.3 UR D 0 Outlet 4 t3.4 B D 0 Outlet 5 t3.5 HO D 0 Outlet 6 t4. NH3 In K 0 7 t4. CO In K 0 8 t4.3 UR In K 0 9 t4.4 B In K 0 0 t4.5 HO In K 0 t5. NH3 In PI 30 t5. CO In PI 30 3 t5.3 UR In PI 30 C. Evaoration and rilling stage his stage occurs by two stages:

37 MULIVARIAE SAISICAL PROCESS OF 37 (i) Evaoration stage In this stage, the condensation of urea rises from 7% to 98.7% aroximately and the urea liquid trams forms to urea melt. his haens under high ressure and temerature. (ii) Prilling stage In this stage, the urea melt is through formed into rilling in the rilling tower. It contains four variables, these are: Z Urea solution from D-04 to E-09 Z D-05 Vacuum 3 Z3 Urea to rilling tower -0 4 Z4 E- Vacuum able analysis of laboratory in this stage: t6. B t6. HO 3 t6.3 Pills > t6.4 Pills 3.35:.4 5 t6.5 Pills.4:.4 6 t6.6 Pills.4:.0 7 t6.7 Pills <.0 8 t6.8 UR Data descrition: For the alication of multivariate quality control, chart data originate from urea roduction rocess, which consists of the three stages and the analysis of laboratory, which discussed above. he number of the samle is 73 observations taken er hour. he advantages of this samle that, it has several variables and several stages of the roduction. his advantage of the roduction is the

38 38 M. S. HAMED basic reason for choosing this roduction to allow us to study the multivariate quality control charts. In this alication, we shall introduce the most common using technique of multivariate quality control charts; MEWMA control chart & Hotelling s control chart. A Hotelling chart consists of: Plotted oints, each of which reresents statistic for each observation. A center line (green), which is the median of the theoretical distribution of statistic. Control limits (red), which rovide a visual means for assessing whether the rocess is in-control. he control limits reresent the exected variation. MINIAB marks oints outside of the control limits with a red symbol. MINIAB indicates which oints is out-of-control by using decomosition of statistic, along with the P-value for each significant variable. 4.. Hotelling chart of,, 6 and t.,, t. 5 est results for squared chart of,, 6 and t.,, t.5. ES. One oint beyond control limits. est Failed at oints: (Less than LCL)

39 MULIVARIAE SAISICAL PROCESS OF 39 est Failed at oints: (Greater than UCL) Figure. chart of,, 6 and t.,, t.5. he Hotelling chart of,, 6 and t.,, t. 5 can be summarized as follows: he lower and uer control limits are 9.7 and 5, resectively. herefore, we exect the Statistics to fall between 9.7 and 5. he center line or median, is 5.3. est results indicate that 5 oints less than LCL, for examle, oint 4 exceeds the lower control limit. est results indicate that 40 oints greater than UCL, for examle, the test results indicate that oint 3 exceeds the uer control limit.

40 40 M. S. HAMED est results indicate 9 oint through beyond the control limits. hen the out-of-control rate.6% and the in-control rate 87.4%. 4.. Hotelling chart of y,, y7 and t 3.,, t4. 5 est results for squared chart of y,, y7 and t 3.,, t4.5. ES. One oint beyond control limits. est Failed at oints: (Greater than UCL) Figure. chart of y,, y7 and t 3.,, t4.5. he Hotelling chart of y,, y7 and t 3.,, t4. 5 can be summarized as follows: he lower and uer control limits are 0. and 43.6, resectively. herefore, we exect the statistics to fall between 0. and he center line, or median, is 9.3.

41 MULIVARIAE SAISICAL PROCESS OF 4 est results indicate that 9 oint greater than UCL, for examle test results indicate that oint 9 exceeds the uer control limit. est results indicate 9 oint that are beyond the control limit. hen the out of control rate.59% and the in-control rate 97.4% Hotelling chart of Z,, Z4 and t 6.,, t6. 8 est results for squared chart of Z,, Z4 and t 6.,, t6. 8 ES. One oint beyond control limits. est Failed at oints: (Greater than UCL) Figure 3. chart of Z,, Z4 and t 6.,, t6. 8 he Hotelling chart of Z,, Z4 and t 6.,, t6. 8 can be summarized as follows:

42 4 M. S. HAMED he lower and uer control limits are.37 and 3.63, resectively. herefore, we exect the statistics to fall between.37 and he center line, or median, is.35. est results indicate that oint greater than UCL, for examle test results indicate that oint 45 exceeds the uer control limit. est results indicate oint that are beyond the control limit. hen the out of control rate.87% and the in-control rate 97.3%. 5. est Results of Alication he alication is shown that in high rocess stage, test results of Hotelling chart indicate that the out-of-control ercentage 87.4% and the in-control ercentage.6%, and it shown that in low rocess stage, test results of Hotelling chart indicate that the out-of-control ercentage.59% and the in-control ercentage 97.4%. It is shown that in the evaoration and rilling stage, test results of Hotelling chart indicates that the out-of-control ercentage.87% and the in-control ercentage 97.3%. 6. Conclusions he results allow us to determine whether the joint rocess variability is in control or out-of-control. Hotelling s charts used to determine whether or not the rocess mean vector (a vector of the rocess means that accounts for the mean of each charted variable) for two or more variables is in-control. An in-control rocess exhibits only random variation with the control limits. An out-of-control rocess exhibits unusually variation, which may be due to the rocess of assignable causes (unusual occurrences that are not normally art of the rocess). charts allow us to simultaneously monitor whether two or more related variables are in control.

43 MULIVARIAE SAISICAL PROCESS OF 43 Finally he comany should use multivariate Hotelling s quality control chart to monitor the quality of the urea roduction. oo, the comany should use the Hotelling s variables which causes the out-of-control signals. chart to determine Acknowledgement he author is grateful to the associate editor and referees for careful reading and the Editor for very useful comments and suggestions. References [] F. B. Alt, Multivariate Quality Control, Encycloedia of the Statistical Sciences, John Wiley & Sons, New York, 985. [] F. B. Alt and N. D. Smith, Multivariate Process Control, Handbook of Statistics, P. R. Krishnaiah and C. R. Rao, Eds., North-Holland, Amsterdam, Elsevier Science, B. 7 (988), [3] M. S. Hamed, MEWMA control chart rocedure: Average run length erformance with alication, Journal of Statistics: Advances in heory and Alications 5() (06), [4] H. Hotelling, Multivariate Quality Control in echniques of Statistical Analysis, McGraw Hill, New York, 947, -84. [5] A. Houshmand and A. Javaheri, Multivariate Ridge Residual Charts, Ph.D. Dissertation, University of Cincinnati, 998. [6] J. E. Jackson, Princial comonents and factor analysis: Part : Princial comonents, Journal of Quality echnology (980), 0-3. [7] J. E. Jackson, Multivariate quality control, Communications in Statistics-heory and Methods (988), [8] J. E. Jackson, A User s Guide to Princile Comonent, John Wiley & Sons, New York, 99. [9] J. E. Jackson and R. H. Morris, An alication of multivariate quality control to hotograhic rocessing, Journal of the American Statistical Association 5 (957), [0] W. Linna, H. Woodal and L. BusBy, he erformance of multivariate control charts in the resence of measurement error, journal of Quality echnology 33 (00),