hotelling t2 for batch

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Applying Hotelling’s T2

Statistic to Batch Processes

ROBERT L. MASON

Southwest Research Institute, San Antonio, TX 78228-0510

YOUN-MIN CHOU

The University of Texas at San Antonio, San Antonio, TX 78249-0664

JOHN C. YOUNG

McNeese State University, Lake Charles, LA 70609-2340

In this paper we show the usefulness of Hotelling’s T2 statistic for monitoring batch processes in

both Phase I and Phase II operations. Discussions of necessary adaptations, such as in the formulas for

computing the statistic and its distribution, are included. In a Phase I operation, where the focus is on

detecting and removing outliers, consideration is given to batch processes where the batch observations are

taken from either a common multivariate normal distribution or a series of multivariate normal distributions

with different mean vectors. In a Phase II operation, where the monitoring of future observations is of

primary concern, emphasis is placed on the application of the T2 statistic using a known or estimated

in-control mean vector. A variety of data sets taken from different types of industrial batch processes are

used to illustrate these techniques.

Introduction

HOTELLING’S T 2 is a very versatile multivariatecontrol chart statistic. It can be used not only

in Phase I operations to identify outliers in the histor-ical data set (HDS), but also in Phase II operationsto detect process shifts using new incoming observa-tions. In either situation, orthogonally decomposingthe T 2 statistic using the MYT decomposition pro-cedure (Mason, Tracy, and Young (1995, 1997)) canhelp identify the variables causing the signal or out-lier. There are many other applications of the T 2

statistic in multivariate control charts. For example,procedures exist for enhancing the sensitivity of theT 2 statistic to the detection of small process shifts(Mason and Young (1999a)), and for adjusting the

Dr. Mason is a Staff Analyst in the Statistical Analysis

Section. He is a Fellow of ASQ.

Dr. Chou is a Professor in the Division of Mathematics

and Statistics. She is a Fellow of ASQ.

Dr. Young is a Professor in the Department of Mathemat-

ics, Computer Science, and Statistics.

T 2 statistic to accommodate observation vectors thatare correlated over time (Mason and Young (1999b)).A useful overview of the T 2, as a control statistic formultivariate processes, can be found in Mason andYoung (1998). In this paper we extend the use of theT 2 statistic to multivariate batch processes.

Multivariate statistical process control (SPC) forbatch processes is an extension of the correspondingunivariate methods. The most common of these uni-variate techniques involves the plotting of the batchsample means on an individuals chart with the con-trol limits computed using the average moving rangeof the sample means. These methods are reviewedin Woodall and Thomas (1995). Multivariate tech-niques for batch processes have only recently beendeveloped. An excellent summary of the literature onbatch processes is given in Nomikos and MacGregor(1995), and it also includes a method for monitoringbatch processes that is based on multiway principal-component analysis.

Most industrial processing units consist of threecomponents: input, processing, and output. Control

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APPLYING HOTELLING’S T 2 STATISTIC TO BATCH PROCESSES 467

procedures may be established on any or all of theprocess components. When the processing unit usesbatches as input, we label it a batch process (e.g.,see Fuchs and Kenett (1998)). For example, in theproduction of certain grades of silica specially pre-pared feedstock is required as input for a given pro-duction run. However, variation in feedstock prepa-ration produces different batches in production.

Batches also occur in processes where the entireprocessing unit is changed at specified time intervals.Many types of reactors are run in this manner. Atthe end of a run cycle the reactor is shut down andrefurbished, and production resumes with a newlyrebuilt reactor. Although the relationships amongthe process variables are maintained, variation in thestart position (due to refurbishing) produces differentruns.

Processes that may purposely be shifted to adifferent configuration to change production fromone product to another product inherently generatebatches. This is a common occurrence in the plasticsindustry and job shops where many different prod-ucts are made at different times. Feedstock remainsconstant in this type of process, but the process-ing component is changed to accommodate the newproduct grade being produced. The runs for eachspecified product comprise a batch process since theruns are separated by the production of other prod-ucts. The preliminary data set in a Phase I evalua-tion would be composed of the observations on thedifferent runs or batches that characterize the prod-uct being produced.

In this paper we demonstrate the usefulness of theT 2 as a control chart statistic for batch processesoccurring in both Phase I and Phase II operations.For Phase I settings we consider preliminary datasets where the batch observations are taken from ei-ther (1) a series of multivariate normal distributionshaving the same covariance matrix but possibly dif-ferent mean vectors; or (2) a common multivariatenormal distribution with the same mean vector andcovariance matrix. It is assumed that the batches arewell defined and that the mean vectors and covari-ance matrices are known or can be estimated. ForPhase II operations we consider process monitoringusing new observations where there is a known orestimated target mean vector. In either phase, dis-cussions of necessary adaptations, such as parameterestimation, are included. A secondary focus of thispaper is on using the T 2 statistic to identify outliersin multivariate batch processes.

Classification of Batch Processes

Certain types of batch processes generate productsimilar to that produced by a steady-state continu-ous process. Limited between-batch variation is ac-ceptable, but when variation becomes large the out-lying batches are rejected and must be reworked. Inthese processes, we assume that observations comefrom the same p-dimensional normal distribution,Np(µ,Σ), have a common mean vector, µ, and have acommon covariance matrix, Σ. We will label batchprocesses of this type as Category 1. An exampleof such a process arises in the production of certaintypes of pigments used in specialty grades of paintsand coatings. Although different batches of paint areproduced, customers seldom tolerate much variationin the color of the paint across the batches.

Since the batches from a Category 1 process areassumed to come from the same multivariate distri-bution, the resulting data should be centered as closeas possible about the common mean vector, µ. If µis known and attainable, it is designated as the in-control mean vector. An example of the in-controlproduction regions for a set of batches taken from aCategory 1 process containing two process variables,x1 and x2, is illustrated in Figure 1. The elliptical re-gions represent the in-control data from the separatebatches, and the darkened circles denote the corre-sponding batch means. Observe the closeness of thebatch means to one another, and to the overall popu-lation mean, which is represented by a shaded circle.

Another type of batch process, which we will labelas Category 2, produces runs with significant separa-tion between its batch mean vectors. The batch ob-servations in this type of process come from differentmultivariate normal distributions, Np(µi, Σ), whereµi, i = 1, 2, . . . , k, represents the population meanvector of the ith batch. Nevertheless, all batches pro-duced in a Category 2 process must be contained inan acceptable region defined by customer satisfac-tion. The differences among the batch means can bedue to known or unknown causes.

As an example, consider a bivariate Category 2batch process where the operation is chemical in na-ture and depends on the addition of a particular cat-alyst. A new “barrel” of catalyst is used on eachbatch produced. Although the catalyst concentra-tion for a given barrel is fixed, the concentration levelvaries within acceptable process limits between bar-rels. Thus, the output is dependent on the level ofthe catalyst.

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468 ROBERT L. MASON, YOUN-MIN CHOU AND JOHN C. YOUNG

FIGURE 1. In-Control Production Regions for Batches

from a Category 1 Process.

An example of the output from such a process isillustrated in Figure 2. The small ellipses representthe in-control production regions of the individualbatches while the large ellipse depicts the in-controlproduction region for the overall process. The verti-cal axis on the right side of the graph represents thelevel of the catalyst. Changing the level of the cat-alyst moves the mean of the distribution of the pro-cess variables for a given batch to a different position.The relationship between the two process variables,x1 and x2, is represented by the orientation of thesmall ellipses centered about the batch mean points.Observe the positive correlation of the two processvariables within a batch, although there is negativecorrelation between the batch mean components.

Since the batches from a Category 2 process areassumed to come from different distributions, no spe-cific target mean vector is specified. However, allbatches should produce data that are located withina fixed statistical distance from the overall mean ofall the batches when the process is stable. For exam-ple, the large elliptical region of Figure 2 defines thein-control region of batch production for the givenset of batches.

Control procedures for batch processes, whetherthe type is Category 1 or Category 2, must addresstwo issues. The first concerns the determination ofwhether the relationships between and among theprocess variables are being maintained relative tothose observed in the historical database. The sec-ond concerns the determination of whether the pro-cess is in statistical control. For a Category 1 batchprocess, where a single multivariate normal distribu-

FIGURE 2. In-Control Production Regions for Batches

from a Category 2 Process.

tion is applicable, the two determinations are equiva-lent. However, for a Category 2 batch process, wheredifferent multivariate normal distributions can exist,the determinations can be different. For example,several batches may be maintaining in-control rela-tionships between the process variables yet may beout-of-control relative to the overall mean vector.

Phase I Operation

The construction of an in-control data set ina Phase I multivariate control chart operation forbatch data includes the investigation of some of thesame potential problems as occur when construct-ing control charts for non-batch data. For example,an analyst must decide which variables to monitor,select the correct transformations for the chosen vari-ables, and determine if the observations are indepen-dent or correlated over time. Consideration also mustbe given to the effects of missing observations, andthe data set must be checked for the possibility ofsevere collinearities existing among the variables.

Some problems require different solutions withbatch processes. There are those that involve de-termining if any outliers are contained in the dataset, and some that require deciding on an estima-tion procedure for the common covariance matrix. Inaddition, the batch means must be examined to de-termine if they differ significantly from one another.Unfortunately, these problems are interrelated. Forexample, significant differences between batch meansmay be due to the presence of outliers. Outliers,in turn, can have a significant influence on parame-ter estimation. However, outliers cannot be detectedwithout proper estimates of the parameters.

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Batch Size = 1

Suppose we collect data on k batches taken fromeither a Category 1 or a Category 2 batch pro-cess. Assume the batch size is one (i.e., ni = 1,i = 1, . . . , k) so that a single observation is made onp variables from each batch. Thus, the total samplesize is N =

∑ki=1 ni = k. In this setting, individual

outliers can be detected by computing the T 2 valuefor each observation, X, and comparing the value toan upper control limit (UCL).

The procedure for purging outliers for batch pro-cesses where a single observation is made on eachbatch is equivalent to the procedure for continuousproduction processes. The corresponding T 2 statis-tic and its distribution for the purging procedure aregiven as

T 2 =(X− X

)′S−1

(X− X

)

∼[(N − 1)2

N

]

B[p/2,(N−p−1)/2], (1)

where X and S are the common estimators, respec-tively, of the overall sample mean and the over-all covariance matrix, and B(p/2,(N−p−1)/2) repre-sents the beta distribution with parameters (p/2)and ((N − p − 1)/2). The UCL is given as

UCL =[(N − 1)2

N

]

B[α,p/2,(N−p−1)/2], (2)

where B[α,p/2,(N−p−1)/2] is the upper αth quantile ofB[p/2,(N−p−1)/2].

It is useful to note that the T 2 statistics in aPhase I operation are not independent of one an-other. This occurs because the observation X is notindependent of X and S. However, in a Phase IIoperation future observations are independent of theestimated parameters obtained in Phase I, but theplotted T 2 statistics are dependent. More details onthese results can be found in Tracy, Young and Ma-son (1992).

Batch Size > 1

When the batch sizes in a Category 1 or Cate-gory 2 batch process exceed one the outlier purgingprocedures are more complicated. In this case, wecan have outliers among the batch means or amongthe individual batch values. For example, Category1 batch processes are checked for individual outliersusing the T 2 statistic in Equation (1) and the corre-sponding UCL in Equation (2). However, the sample

size N no longer reduces to k but has the general formN =

∑ki=1 ni.

Since observations may be taken from differ-ent distributions for the batches in a Category 2batch process, the definition and detection of out-liers changes from those given for a Category 1 batchprocess. With batches, the overall sample mean inEquation (1) is equivalent to the weighted average ofthe batch means; i.e.,

X=

k∑

i=1

ni∑

j=1

Xij

N=

k∑

i=1

ni Xi

N, (3)

where Xij is the jth observation in the ith batch, andXi represents the mean vector of the ith batch. Sim-ilarly, the covariance matrix estimator used in Equa-tion (1) measures not only process variation withinbatches but also process variation due to differencesbetween batches. The estimator S is the total varia-tion, SST , divided by (N − 1) and is written as

S =1

N − 1

k∑

i=1

ni∑

j=1

(Xij − X

) (Xij − X

)′

=SST

N − 1.

The total variation also can be written as

SST =k∑

i=1

ni∑

j=1

(Xij − Xi

) (Xij − Xi

)′

+k∑

i=1

ni

(Xi −X

) (Xi −X

)′

= Within-Batch Variation+ Between-Batch Variation

= SSW + SSB .

In a Category 1 batch process, we assume thatthe between-batch variation is minimal and is strictlydue to random fluctuations. Thus, the correspond-ing elements of SST and SSW are very similar insize. However, in a Category 2 batch process, thebetween-batch variation can be large and producea considerable difference between corresponding el-ements of SST and SSW . Further, the ratio of thesample size to the number of variables might be smallfor a given batch. This could lead to a within-groupcovariance matrix estimate that is near singular. In-dividual batches also may contain statistical outliers,which tend to greatly influence the estimate of thecovariance matrix for small sample sizes.


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FIGURE 3. Two Batches of Data With Outliers and

Similar Correlations.

As an illustration of the effects of between-batchvariation on a Category 2 process, consider the plotgiven in Figure 3 of a set of data taken from a bivari-ate process composed of two separate batches andhaving an outlier in each batch. The orientation ofthe data implies that x1 and x2 have the same cor-relation in each batch, but the batch separation im-plies the two batches have different mean vectors.If the batch classification is ignored, then the over-all sample covariance matrix, S, is based on devia-tions taken from the overall mean. However, giventhe large separation between the two sets of data,a better estimator of process variation could be ob-tained by taking a weighted average (weighted onthe degrees of freedom) of the two separate within-batch covariance matrix estimators. This estimator,in general, is given by

SW =

k∑

i=1

(ni − 1)Si

N − k=

SSW

N − k, (4)

where Si is the covariance matrix estimate for theith batch.

The problems noted above can be circumvented bysubtracting the corresponding estimated batch meanvectors from each set of batch data prior to dataanalysis. With this translation, the mean differencesamong the batches are removed and all data are cen-tered at the origin. Translating the batch data alsowill provide the appropriate estimate of the commoncovariance matrix.

Consider the scatter plot of observations presentedin Figure 4. Suppose the graph represents data takenfrom two separate independent batches from a Cate-gory 2 batch process involving two variables. Thereis an indication in the plot that both batches haveobservations that are potential outliers, since several

FIGURE 4. Bivariate Data for Two Separate Batches.

of the data points are separated from their respectivebatch means. Also, notice the similarity of the twobatches with respect to their orientation and vari-ation, even though their mean vectors are different.This is substantiated by examination of the summarystatistics presented in Table 1 for the two batches ofdata.

Translation to the origin is achieved by subtract-ing the estimates of the respective batch means fromthe corresponding batch data. For the Batch 1sample in Table 1 we compute (x1 − 2.367) and(x2−1.665), whereas for the Batch 2 sample we com-pute (x1 − 4.289) and (x2 − 4.714). This re-locatesboth batches at a common origin. As noted from thelast column of Table 1, the variation statistics for thecombined translated data are very similar to thosefor the individual batches. The correlation betweenthe translated x1 and x2 is 0.909, the variance of thetranslated x1 is 0.105, and the variance of the trans-lated x2 is 0.140. These values compare favorablywith the statistics of either batch, and they reinforcethe usefulness of the mean translation. The graph ofthe overall mean-translated data set is presented inFigure 5.

Although the use of centering appears to be veryhelpful, a close examination of the data in Figure 5reveals the continued presence of some mean-vector

TABLE 1. Summary Statistics for Two Batches of Data

Batch 1 Batch 2 Translated

Sample Size 245 272 517x1 Mean 2.367 4.289 0.000x2 Mean 1.665 4.714 0.000x1 Variance 0.107 0.105 0.105x2 Variance 0.140 0.140 0.140Corr (x1, x2) 0.914 0.904 0.909


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FIGURE 5. Translation of Two Batches to Common

Origin.

differences between the two batches. Observe thatthe bulk of the translated observations of Batch 1are located in the first quadrant with a lesser amountlocated in the third quadrant; however, the oppositeoccurs with the translated observations of Batch 2.This mean-vector difference is unexpected and occursprimarily because the frequency distributions of thetwo sets of batch data are not symmetrical (as in anormal distribution), but skewed.

This disparity between the batches can be at-tributed to the effects of operator control on the pro-cess. The first variable (x1) is an operator-controlvariable and the second variable (x2) moves in re-sponse to changes in x1. In addition, there are nu-merous other “lurking” variables having great influ-ence on x2. Lead operators (on different shifts) runthe process differently, but obtain the same overallprocess results (e.g., the same production level or thesame product). Thus, although the correlation be-tween the two variables remains relatively constant,the mean vectors across the two batches differ due tooperator differences.

The above data example poses an interestingproblem. If we use a mean translation, all the trans-formed observations from a Category 2 batch processwill have a p-variate normal distribution, Np(0,Σ),with mean vector 0 and covariance matrix Σ. Thisresult is based on the assumption that the untrans-formed data have a multivariate normal distribution.However, as seen in the data in Figure 5, there maybe situations where the underlying distribution isnonnormal. Because this has occurred in this ex-ample, it is useful to briefly describe one possible so-lution to this assumption violation. To do so we willconsider the distribution, not of the original data,but of the T 2 values. If this distribution is similar tothe beta distribution given in (1), we will consider itappropriate to continue using the T 2 charting proce-dure.

The distribution of the T 2 statistic and its appli-cability to a particular situation can be readily es-tablished using a Q-Q plot (e.g., see Seber (1984)).For example, if one considers the transformed dataof Batch 1 in Figure 5, there are three obvious out-lying points. If these are removed during the datacleaning procedures of Phase I and a correspondingQ-Q plot is constructed using the beta distributionin (1), then the resulting plot is given in Figure 6.The trend in the plot lies along a 45-degree line andindicates that the beta distribution provides a goodfit to the T 2 values for Batch 1.

The Batch 2 data in Figure 5 appear to have sevenoutlying points that could be removed in the datacleaning effort. With their removal the resulting Q-Q plot, given in Figure 7, is similar to that obtainedfor the Batch 1 data. The trend is again highly lin-ear, indicating the validity of the beta distributionassumption for the T 2 values for Batch 2.

The Q-Q plot of the combined batches (with thecombined 10 outliers removed) is presented in Figure8. As expected, the trend is highly linear, and the re-sults indicate that, even with the disparity observeddue to operator differences, the departure from thespecified beta distribution used in describing the T 2

statistic for separate or combined batches is small.This small discrepancy will not affect overall conclu-sions drawn in using the T 2 as the control statistic forthis situation. Thus, we can identify individual out-liers by ignoring the batch differences, treating thetranslated data as a single group, and determiningsignaling points using the T 2 statistic in Equation(1) with the translated data. This is accomplishedby plotting the T 2 values on a control chart (e.g., seeFuchs and Kenett (1998) or Wierda (1994)). Any T 2

value exceeding the UCL of the chart is declared anoutlier.

Using an α = 0.01 and the T 2 statistic in Equation(1) with the translated data from Figure 5, we cannow identify the observations located a significantdistance from the center of the data (i.e., from theorigin). In the first pass through the data, threeobservations from Batch 1 and seven observationsfrom Batch 2 are designated as outliers (i.e., these arethe same 10 observations deleted in Figure 8). Thecovariance matrix is re-estimated from the remainingtranslated observations. One observation from eachbatch is removed on the second pass, but a third passdetects no remaining outlying observations. Thus,there are a total of 12 observations designated as


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FIGURE 6. Q-Q Plot of Batch 1 With Three Outliers

Removed.

outliers, four from Batch 1 and eight from Batch 2.The outlying observations are designated with sym-bols in Figure 9. With the removal of these outliers,an estimate of the covariance matrix can be obtainedfrom the remaining translated data.

Category 1 Batch Mean Outliers

In Category 1 batch processes it is assumed thatthe batch mean vectors do not have significant differ-ences. Therefore, the specification, or estimation, ofthe means of the p process variables plays an impor-tant role in developing control charts for future obser-vations. When new observations are monitored, thetarget mean vector may be specified as µT . Other-wise, µT must be estimated using the overall samplemean estimate given in Equation (3). Unfortunately,with batch data, the outlying batch means can dis-tort the estimate of the in-control mean vector. Forexample, the plot given in Figure 10 represents a setof batch means with three obvious outliers locatedat points A, B, and C. The inclusion of Point B orPoint C will shift the overall batch mean to eitherthe left or right (respectively), while the inclusion ofPoint A will shift the overall mean upward and tothe right.

A basic assumption in using the T 2 statistic with

FIGURE 7. Q-Q Plot of Batch 2 With Seven Outliers

Removed.

FIGURE 8. Q-Q Plot of Combined Batches With Ten

Outliers Removed.

Category 1 batch data in a Phase I operation isthat the process is centered. This implies that themean vectors of the k individual batches are allequal. Translating the observations from the differ-ent batches to a common origin in order to identifyoutliers for individual observations will remove anypossible mean differences between the batches for thetranslated data. However, if the assumption of equalbatch means is invalid, batch mean differences amongthe untranslated data may go undetected, particu-larly if these differences appear after the purging ofthe individual outlier data. Given the importanceof this assumption, we recommend performing a testof hypotheses to determine the batch mean vectordifferences.

Assume that an observation from the ith batchis distributed as X ∼ Np(δi, Σ) and that an ob-servation from the jth batch is distributed as X ∼Np(δj ,Σ). In the form of a statistical hypothesis,the assumption of equal batch means is representedas

H0 : δ1 = δ2 = · · · = δk. (5)

The alternative hypothesis is represented as

HA : δi �= δj for some i �= j.

FIGURE 9. Translated Batch Data With Deleted Out-

liers.


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FIGURE 10. Plot of Batch Means With Outliers.

Under H0, each observation vector X is assumed tocome from a common multivariate normal distribu-tion; i.e., X ∼ Np(δ,Σ). Under HA, at least onebatch observation comes from a different multivari-ate normal distribution.

Before testing H0, the issue of batch sample sizemust be addressed. Even if the number of obser-vations was originally the same for each batch, thePhase I purging of individual outliers may have re-duced the sample size in several of the batches. Wesuggest using an average value for the batch samplesize; i.e., use

n =1k

k∑

i=1

ni, (6)

where ni, i = 1, 2, . . . , k, is the number of observa-tions in the ith batch, and n is rounded up to thenearest integer. Since most processing units take ob-servations electronically on a regular sampling basis,the discrepancies between the ni should be small.However, variation in the length of a run cycle canproduce a few extra observations per cycle.

Establishing the reasonableness of the hypothesisin Equation (5) is important. If there are no differ-ences between the batch means, the best estimator ofthe overall in-control mean vector is given in Equa-tion (3). If there are outliers, the outlying batchmeans will need to be excluded from the calculationsin Equation (3). Using the average value in Equa-tion (6) for sample size, we can test the validity ofthe hypothesis in Equation (5) with a T 2 statistic fordetecting batch mean outliers. The statistic is givenby

T 2 =(Xi − X

)′S−1

W

(Xi − X

), (7)

TABLE 2. Batch Means for Six Batches

Batch # x̄1 x̄2 x̄3 ni

1 10.633 9.808 2.425 122 8.816 16.875 2.058 123 3.745 9.800 1.027 114 5.900 7.033 3.408 125 9.563 2.809 0.463 116 7.358 9.325 1.683 12

Average 7.670 9.275 1.844 12

where SW is the sample covariance matrix computedusing Equation (4) and the translated data with theindividual outliers removed.

The distribution of the statistic in Equation (7),under the assumption of a true null hypothesis, isthat of an F variable. The distribution (i.e., seeWierda (1994)) is given as

T 2 ∼[

p(k − 1)(n − 1)n[k(n − 1) − p + 1]

]

F(p,nk−k−p+1),

where F(p,nk−k−p+1) represents the F distributionwith parameters (p) and (nk−k−p+1). For a givenα level, the UCL for the T 2 statistic is computed as

UCL =[

(k − 1)(n − 1)pn[nk − k − p + 1]

]

F(α,p,nk−k−p+1), (8)

where F(α,p,nk−k−p+1) is the upper αth quantile ofF(p,nk−k−p+1). The T 2 value in Equation (7) is com-puted for each of the k batch means and compared tothe UCL in Equation (8). Batch means with T 2 val-ues that exceed the UCL are declared outliers. Thecorresponding batches are removed and a new in-control mean vector is estimated for use in Phase IIoperations.

To demonstrate the procedure, consider a Cate-gory 1 batch process where 12 observations are col-lected on three process variables for each of k = 6batches. The estimated batch mean vectors, afterindividual outliers are removed, are given in Table2. The average sample size for the batches, usingEquation (6), is computed as 11.67. Thus, a valueof n = 12 is used in determining the UCL in Equa-tion (8). The correlation matrix for the three processvariables is given in Table 3.

To test the hypothesis of equal batch means, theT 2 statistic in Equation (7) is computed using the es-timated covariance matrix SW employed in comput-ing the correlation matrix. The T 2 values for the six


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TABLE 3. Correlation Matrix for Six-Batch HDS

x1 x2 x3

x1 1.000 −0.528 −0.043x2 −0.528 1.000 0.519x3 −0.043 0.519 1.000

batch means are presented in Table 4. For α = 0.01,n = 12, k = 6, and p = 3 the UCL in Equation (8) iscomputed as 0.882. Since the T 2 value for Batch 2exceeds this UCL, the corresponding batch mean isdeclared to be significantly different from the otherbatch means. An examination of Table 2 reveals thereason. The mean of the variable x2 for Batch 2is 16.875, which somewhat exceeds the other batchmeans on this variable. Thus, the analyst shoulddelete Batch 2 before computing the overall batchmean vector for use in the Phase II operations onthis process.

Category 2 Batch Mean Outliers

For Category 2 batch processes, it is known thatthe batch means differ. However, in a Phase II op-eration, it is necessary to define an in-control regionso that one can determine if a current batch givesevidence of process stability. The true p-dimensionalcenter of this region is given as

µ =(

1k

) k∑

i=1

µi,

where µi represents the mean vector of the ith batch,and the best estimate of µ is given by X in Equation(3). Assuming the batch data can be described by amultivariate normal distribution, an in-control regionbased on a T 2 statistic can be developed. The formand distribution of the statistic is given as

T 2 =(Xi − X

)′S−1

B

(Xi − X

)

∼ (k − 1)2

kB(α,p/2,(k−p−1)/2). (9)

where SB = SSB/(k−1). Similarly, the UCL is givenby

UCL =(k − 1)2

kB(α,p/2,(k−p−1)/2). (10)

The T 2 statistic in Equation (9) is based on theestimated between-batch variation, SB . Note that,in the special case of a univariate batch process, thisestimated between-batch variance is based on using

TABLE 4. T 2 Values for Six-Batch HDS

Batch # T 2 Value

1 0.0682 1.0623 0.0634 0.2205 0.4736 0.001

the sample variance of the pooled batch means (e.g.,see Woodall and Thomas (1995)). Thus, SB de-scribes the relationships between the components ofthe batch mean vectors rather than of the individualobservations. For example, in the Category 2 batchprocess example presented in Figure 2, the correla-tion between the two process variables is positive,while the correlation between the batch mean vectorcomponents is negative. The downward shifting ofthe overall process could be due to an unknown “lurk-ing” variable, or to a known uncontrollable variablesuch as the catalyst concentration of the previousbatch.

To demonstrate how to develop an in-control re-gion for later use in a Phase II operation, consider aCategory 2 batch process where 12 observations arecollected on three process variables for each of k = 16batches. The estimated means of the batches, afteroutliers are removed, are given in Table 5. The cor-

TABLE 5. Batch Means for 16 Batches

Batch # x̄1 x̄2 x̄3 ni

1 161.6091 207.0545 33.42727 112 152.6545 205.4182 15.11909 113 143.9455 206.6545 15.9 114 149.7 200.15 16.3 125 158.6333 194.025 12.425 126 178.3727 195.2636 13.57273 117 133.3636 200.7909 0.000909 118 140.2 202.4727 0.581818 119 159.2 197.1 24.33636 1110 145.2182 193.9273 9.8 1111 150.6333 194.8083 2.925 1212 148.8167 201.875 2.558333 1213 143.7455 194.8 1.527273 1114 145.9 192.0333 3.908333 1215 149.5636 187.8091 0.963636 1116 147.3583 194.325 2.183333 12


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TABLE 6. Correlation Matrix for 16-Batch HDS

x1 x2 x3

x1 1.000 −0.776 −0.571x2 −0.776 1.000 0.484x3 −0.571 0.484 1.000

relation matrix based on the between-batch variationis given in Table 6.

The T 2 chart for the 16 batch means is presentedin Figure 11. For α = 0.05, k = 16, and p = 3the UCL in Equation (10) is computed as 10.263.Since no batch mean is out of statistical control, allthe data can be used in computing the overall meanestimate given by Equation (3).

Phase II Operation: Category 1 Process

Before beginning a Phase II operation for a Cate-gory 1 batch process we must have an appropriate es-timator of the covariance matrix. We have availablethe estimator SW , which is based on the translatedobservations with individual outliers removed and isthe appropriate estimator to use in testing for meandifferences. However, with the removal of individ-ual outliers and the exclusion of observations fromthe outlying batches that exhibit a significant meandifference, one could use the common estimator Sobtained from the remaining observations. For ex-ample, consider the data in Table 2. Because Batch2 is an outlier, we would exclude it from the HDS andestimate the covariance matrix using S and the datafrom the remaining five batches. We use S ratherthan SW because the between-batch variation, SB ,is assumed to be entirely due to inherent process vari-ation (e.g., see Alt (1982) and Wierda (1994)).

The estimator SW excludes between-batch varia-tion, and thus it is not influenced by batch mean dif-ferences. However, if mean differences do not existamong the batches, the estimator S is a more efficientestimator of Σ than SW (e.g., see Wierda (1994)).Since both covariance estimators have merit depend-ing on the circumstances, we will consider Phase IIoperations for monitoring future process observationsusing both S and SW . We also present control pro-cedures for monitoring a Category 1 batch processfor the four cases of interest presented below.

(i) Consider a single future observation vector, X,that is taken from a Category 1 batch pro-cess with an unknown mean target vector. Us-ing the covariance matrix estimator S, the T 2

FIGURE 11. T 2 Control Chart of HDS for a 16-Batch

Process.

statistic for monitoring this observation is givenby

T 2 =(X− X

)′S−1

(X− X

),

where the covariance matrix, S, and the targetmean vector estimate, X, are obtained usingthe HDS of size N = nk.

(ii) When process control is based on the moreconservative covariance estimator SW , the T 2

statistic is given by

T 2 =(X− X

)′S−1

W

(X− X

).

(iii) For the situation where the target mean vector,µT , is specified and the estimator S is used, theT 2 control statistic is given as

T 2 = (X − µT )′ S−1 (X − µT ) , (11)

where S is again obtained from the HDS.

(iv) When the in-control mean vector is specifiedbut the estimator SW is used, the T 2 statisticis given as

T 2 = (X − µT )′ S−1W (X − µT ) ,

The T 2 formulas, distributions, and UCLs for theabove four cases are presented in Table 7. Whenprocess control is based on the monitoring of thesubgroup mean, XS , of a future sample of m obser-vations taken on a given batch, slight modificationsmust be made to these formulas. These results arealso contained in Table 7.

Phase II Operation: Category 2 Process

For a Phase II operation for Category 2 batch pro-cesses, control procedures can be based on observingeither a single future observation or the sample meanvector of a subgroup of future observations for the


mss # ms238.tex; AP art. # 9; 33(4)


TABLE 7. Phase II Formulas for Category 1 Batch Process

Subgroup Target Cov. T 2 T 2

size: m Mean: µT Est. Statistic Distribution∗

S (X − µT )′ S−1 (X − µT )[

p(N−1)N−p

]F(p,N−p)

KnownSW (X − µT )′ S−1

W (X − µT )[

p(N−k)N−k−p+1

]F(p,N−k−p+1)

1S

(X− X

)′S−1

(X− X

) [p(N+1)(N−1)

N(N−p)

]F(p,N−p)

UnknownSW

(X− X

)′S−1

W

(X− X

) [p(N−k)(N+1)N(N−k−p+1)

]F(p,N−k−p+1)

S(XS −µT

)′S−1

(XS −µT

) [p(N−1)m(N−p)

]F(p,N−p)

KnownSW

(XS −µT

)′S−1

W

(XS −µT

) [p(N−k)

m(N−k−p+1)

]F(p,N−k−p+1)

> 1S

(XS − X

)′S−1

(XS − X

) [p(m+N)(N−1)

mN(N−p)

]F(p,N−p)

UnknownSW

(XS − X

)′S−1

W

(XS − X

) [p(m+N)(N−k)mN(N−k−p+1)

]F(p,N−k−p+1)

∗ The T 2 UCL is the upper αth quantile of the T 2 distribution given above.

batch being produced. Recall that the objective inthis situation is to determine if the present batchis in-control (i.e., the T 2 value corresponding to thebatch data is within the T 2 control region). Whenthe target mean is unknown, the historical data setprovides the estimates X and S. Note that, sincethe HDS is composed of single observations from kbatches, S = SB and N = k. Therefore, the formulasfor a Category 2 batch process are the same as thosegiven in Table 7 for a Category 1 batch process exceptthat we replace S with SB , and we let N = k. Also,none of the formulas based on SW are applicable.

FIGURE 12. T 2 Control Chart for HDS for Polymer

Process.

Phase II Data Example

To demonstrate a Phase II operation, consider abatch process for producing a specialty plastic poly-mer. A detailed chemical analysis is performed oneach batch to assure that the composition of sevenmeasured components adhere to a rigid chemical for-mulation. The rigid formulation is necessary for moldrelease when the plastic is transformed to a usableproduct.

A preliminary data set consisting of 52 batchesacceptable to the customer is used to construct a

FIGURE 13. T 2 Control Chart for New Observations on

Polymer Process.


mss # ms238.tex; AP art. # 9; 33(4)


TABLE 8. Summary Statistics for HDS for Polymer Process

x1 x2 x3 x4 x5 x6 x7

N 52 52 52 52 52 52 52Mean 87.10 7.18 3.31 0.35 0.09 0.86 1.16Minimum 86.60 6.80 3.00 0.30 0.00 0.80 0.98Maximum 87.50 7.70 3.60 0.42 0.18 0.97 1.40Std. Dev. 0.248 0.175 0.125 0.020 0.028 0.045 0.103

TABLE 9. Correlation Matrix for HDS for Polymer Process

x1 x2 x3 x4 x5 x6 x7

x1 1.000 −0.519 −0.659 −0.436 −0.092 −0.418 −0.426x2 −0.519 1.000 0.040 −0.021 −0.054 −0.186 −0.211x3 −0.659 0.040 1.000 0.634 0.308 0.136 0.125x4 −0.436 −0.021 0.634 1.000 0.399 0.020 0.018x5 −0.092 −0.054 0.308 0.399 1.000 −0.265 −0.220x6 −0.418 −0.186 0.136 0.020 −0.265 1.000 0.693x7 −0.426 −0.211 0.125 0.018 −0.220 0.693 1.000

historical data set. Each batch has only one obser-vation vector. This example suffices for either a Cat-egory 1 or a Category 2 batch process since controlis based on a single observation vector. A T 2 chart(with α = 0.001) of the data, using the statistic inEquation (1) with N = 52, p = 7, and α = 0.001,is presented in Figure 12. The UCL of 20.418 is ob-tained from Equation (2). Although observations #3and #7 produce signals on the chart, both were re-tained in the HDS.

Summary statistics for the seven measured vari-ables obtained from the HDS are presented in Table8. Examination of the ranges and standard devia-tions reveals tight control on the operational rangesof the individual variables. However, these statisticsare not as critical to the customer as the relationshipsbetween and among the variables. The correlationmatrix for the seven variables is presented in Table9. This matrix indicates the structure or relation-ships that must be maintained among the variables.Any deviation can produce serious problems for thecustomer.

Suppose that a target mean vector for this batchprocess is specified as

µT = (87.15, 7.16, 3.29, 0.34, 0.08, 0.85, 1.14)′.

The T 2 value of a future observation, X, on theprocess will be computed using Equation (11). For

α = 0.001, N = 52 (i.e., n = 1 and k = 52), andp = 7 the UCL = 34.238. Twenty six new batchesare produced and a T 2 control chart (using the abovetarget mean vector) is presented in Figure 13. Sig-nals are observed on the observations on Batches #5,#7, #9, #14, and #21.

Discussion

With the assumption that X is distributed as a p-variate normal, Np(µ,Σ), it can be shown that thecommon covariance matrix estimator S is a functionof the maximum likelihood estimator of Σ, and itis an unbiased, least-squares estimator. Due to theseproperties, one would expect superior performance ofthe T 2 statistic in detecting individual outliers. Thisis indeed the case. The work by Wierda (1994) con-cludes that the T 2 statistic, when based on the com-mon estimator, S, is more powerful than one basedon SW (as presented in Equation (4)) in detecting in-dividual outliers in a single batch when there are nomean differences among the batches. Studies com-paring the power of the T 2 in detecting outliers tothe power of T 2 statistics using alternative covari-ance matrix estimators have been performed (e.g.see Chou, Mason, and Young; 1999). With the ba-sic assumption of a single multivariate normal dis-tribution, the T 2 approach again produces superiorresults.


mss # ms238.tex; AP art. # 9; 33(4)


We have recommended translation of the batchdata to a common origin by subtracting individualbatch mean vectors from the corresponding batch ob-servations. This translation produces an overall setof data satisfying the basic assumption of a singlemultivariate normal distribution. Since the differentbatches have a common covariance structure, it ispossible to use the powerful T 2 statistic when seek-ing to remove individual outliers. The translationalso produces a more precise covariance estimate.

We believe the translation to a common origin tobe an important step in developing a control pro-cedure for batch processes with batch sizes greaterthan one. Clearly, the translation cannot be appliedto batches of size n = 1. The control procedure forthis situation reduces to the T 2 methodology usedfor monitoring individual observations. In this case,we use the common sample covariance estimator S inthe formula for the T 2 statistic. The exact distribu-tion of this T 2 and the corresponding UCL are wellknown (e.g., see Mason, Tracy, and Young (1997)).

Some researchers have shown that, in the abovemonitoring procedure, the covariance estimator Scan be affected by shifts in the mean vector. For ex-ample, Wheeler (1994) discusses the univariate situa-tion, and Sullivan and Woodall (1996) use simulationtechniques to examine different multivariate cases,such as an alternative covariance estimator based onthe differences of successive observations. AlthoughSullivan and Woodall (1996) indicate that small ran-dom shifts (equivalent to different batches means)have an effect on the covariance matrix estimator,translation removes the possibility of contaminatingthe estimate with between-batch variation.

In analyzing batch data in a process an extra com-ponent of variability has been added for considera-tion, namely batch-to-batch variation. Other sourcesof variation also could be considered. An applicationto multivariate SPC is given by Linna, Woodall, andBusby (2001) who examine the performance of multi-variate control charts in the presence of measurementerror variation.

Summary

The creation of a historical data set is an impor-tant aspect in implementing a multivariate controlprocedure for batch processes. We have emphasizedthree important, interrelated areas in this develop-ment: outlier removal, parameter estimation, andthe location of significant batch mean differences.

When the batch data are collected from the samemultivariate normal distribution, the T 2 statistic isan excellent choice for use in purging outliers. Whenthe batch data are collected from multivariate nor-mal distributions with different mean vectors, thetranslation of the different batches to a common ori-gin again allows one to use the overall T 2 statisticto identify outliers. The approach yields a powerfulprocedure.

Acknowledgments

The authors wish to thank the editor and the ref-erees for their helpful comments and suggestions onearlier versions of this paper.

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Key Words: Multivariate Quality Control, Outliers,Phase I, Phase II .

∼


hotelling t2 for batch

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