The design of the multivariate synthetic-T 2 control chart. · 1 The design of the multivariate synthetic-T2 control chart. Francisco Aparisi1 and Marco A. de Luna2. 1Departamento

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The design of the multivariate synthetic-T2

control chart. Francisco Aparisi1 and Marco A. de Luna2.

1Departamento de Estadística e Investigación Operativa Aplicadas y Calidad.

Universidad Politécnica de Valencia.

46022 Valencia. España. [email protected] 2Departamento de Ingeniería Industrial y Mecánica. ITESM

45140 Guadalajara, México. [email protected]

ABSTRACT

One of the objectives of the research done in statistical process control is to obtain

control charts that show few false alarms but, at the same time, are able to detect

quickly the shifts in the distribution of the quality variables employed to monitor a

productive process. In this paper the synthetic-T2 control chart is developed, which

consists of the simultaneous use of a CRL chart and a Hotelling’s T2 control chart. A

procedure of optimization has been developed to obtain the optimum parameters of the

synthetic-T2, given the values of in-control ARL and magnitude of shift which needs to

be detected rapidly. The comparison versus the charts MEWMA, T2 with variable

simple size and T2 with double sampling shows that the synthetic-T2 chart always

performs better than the standard T2 chart. The comparison with the remaining charts

reveals in which cases the performance of this new chart makes it interesting to employ

in real applications.

KEY WORDS: synthetic, multivariate, SPC, quality control, ARL

Correspondence author: Francisco Aparisi

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1.- INTRODUCTION

The statistical quality control has the objective of detecting shifts in the distribution of

the quality variable(s) that are monitored. In the majority of productive processes the

possible spontaneous changes in this distribution will worsen the quality of the product

manufactured. Therefore, the quick detection of these shifts is very important to

maintain quality. The statistical process control (SPC) has shown itself to be a powerful

tool to achieve this objective.

It is possible to employ two strategies when it is desired to control simultaneously p

variables from one piece or from a productive process: to use simultaneously p

univariate control charts or to employ only one multivariate control chart [Montgomery

(2005)]. During many years a lot of research has been made in order to design control

charts capable of detecting quickly small shifts in the process. The problem is that the

standard control charts ( X in the univariate case and Hotelling’s T2 in the multivariate

case) are not efficient in detecting small shifts. The efficacy of a quality control chart is

normally measured using the ARL (Average Run Length), the average number of points

in the chart until the first out-of-control signal appears. In the classical design of control

charts, a chart is considered efficient when shows a large value of ARL when the

process is in an in-control state, but having low ARL to detect a shift that produces an

out-of-control state.

Many strategies have been proposed with the aim of improving the ARL values when

the process is out of control, keeping a desired ARL value when the process is in

control. From our point of view, the more efficient strategies are: 1.- Cummulative

sums, in the univariate case the CUSUM charts [Luceno and Puig-Pey (2000), Jones,

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Champ and Rigdon (2004)] and in the multivariate case the MCUSUM charts [Woodall

and Ncube (1985) and Khoo and Quah (2002)]. 2.- The charts employ exponentially

weighted averages, the EWMA charts in the univariate case [Lucas and Saccucci (1990)

and Knoth (2005)] and MEWMA charts for the multivariate case [Lowry, Woodall,

Champ and Rigdon (1992), Prabhu and Runger (1997) and Aparisi and García-Díaz

(2004)]. 3.- The use of the variable sample size [VSS univariate charts: Costa (1994,

2001), Reynolds and Arnold (2001); VSS multivariate case: Aparisi (1996), Aparisi and

Haro (2003)]. 4.- The use of the double sampling [univariate DS charts: Daudin (1992),

Carot, Jabaloyes and Carot (2002), He and Grigoryan (2006); multivariate DS charts:

He and Grigoryan (2005), Champ and Aparisi (2006)]. 5.- The use of the synthetic

charts, only developed for the univariate case [Wu and Spedding (2000) and Chen and

Huang (2005a)].

The performance of the SPC has been improved employing simultaneously two control

charts. Examples of this strategy are: the Shewhart-CUSUM chart [Lucas (1982)], the

Shewhart-EWMA chart [Klein (1997)] and the synthetic chart, which is a Shewhart-

CRL (conforming run length) chart. The synthetic- X chart was introduced by Wu and

Spedding (2000) as a control chart useful for detecting small shifts in the mean of a

process. Calzada and Scariano (2001) and Scariano and Calzada (2003) studied the

robustness of this charts and developed a synthetic chart for quality variables with

exponential distribution.

All the previous referenced methodologies have been developed for univariate and

multivariate cases, except the synthetic chart. In this paper the multivariate synthetic-T2

chart will be developed, obtaining a procedure to find the parameters of this chart that

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minimises the ARL for a given size of shift, and given a desired in-control ARL.

Section 2 shows a summary of multivariate SPC and the methodologies which enables

more efficient control charts to be obtained. In Section 3 the synthetic-T2 chart is

defined. The optimization of the parameters of this chart can be found in Section 4.

Section 5 shows the comparison between several multivariate control charts, in order to

find out in which cases the synthetic-T2 chart shows better performance. The

conclusions can be found in Section 6.

2.- MULTIVARIATE QUALITY CONTROL

As was commented in the Introduction, the multivariate quality control consists of

monitoring in one chart p variables of the same productive process or piece. The first

proposal to control possible shifts in the mean vector was Hotelling’s T2 control chart

[Hotelling (1947)] that consist of plotting the values of the statistic 2iT ,

( ) ( )120 00

Ti i iT n −= − −∑X Xμ μ (1)

where n is the sample size, iX is the mean vector of the p sample means, 0μ is the

mean vector when the process is in control and 0Σ is the in-control covariance matrix.

The distribution of the statistic 2iT , if the process is in an in-control state and 0μ and

0Σ are known, is a chi-square with p degrees of freedom. Therefore, it is easy to obtain

the control limit (CL) fixing the probability of the Type I error, α. Given the value of α,

the control limit is 2,pCL X α= .

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As the points plotted in the Hotelling’s T2 chart are independent, the in-control ARL is

given by 01ARLα

= . The performance of this chart to detect shifts in the mean vector

depends on the non-centrality parameter 2ndλ = , where d is the Mahalanobis’ distance,

( ) ( )11 0 1 00

Td −= − −∑μ μ μ μ , where 1μ is the out-of-control mean vector. The value

of ARL when d ≠ 0 is 1( 0)1

ARL dβ

≠ =−

(where β is the probability of the type II

error) is easily calculated, knowing the distribution of the 2iT statistic when the process

is out of control, a non-central chi-square, )( 222 ndXT p =≈ λ .

Commonly, the mean vector and the covariance matrix are estimated. Lowry and

Montgomery (1995) and Champ, Jones-Farmer and Rigdon (2005) present guides for

the design of the T2 control chart with estimated parameters. Vargas (2003) and

Williams, Woodall, Birch and Sullivan (2006) have studied alternatives to estimate the

covariance matrix for the case of individual multivariate observations.

As was commented, the T2 chart has a poor performance to detect small shifts in the

mean vector. In this article, the use of a synthetic-T2 control chart is proposed to reduce

the values of ARL to detect process shifts. The analysis is done employing the zero-

state approach [Champ (1992) and Zhang and Wu (2005)] and considering that 0Σ and

0μ are known. Subsequently there will be a description of the multivariate control charts

that are going to be used in the comparison of performance shown in Section 5.

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2.1 Variable Sample Size T2 control chart

Aparisi (1996) and Aparisi and Haro (2001, 2003) improve the sensibility of the T2

chart employing variable sample size (VSS). The strategy consists of determining the

value of a warning limit (w). The sample size at time i depends on the value of 21−iT . if

wTi ≤−21 a sample of size n1 at the time i is taken. Otherwise, if 2

1iw T CL−< < the sample

size to be used is n2. When 2,

2αpi XT > the process is considered to be out of control,

otherwise, it is decided that only random causes are present in the process. It is possible,

for a given in-control ARL, to find the values of CL, w, n1 and n2 that minimises the

out-of-control ARL for a specified size of magnitude d, considering the restrictions

1 2 maxn n n+ ≤ and a desired mean sample size, 0n , 0)( nnE = .

2.2 MEWMA control chart

The multivariate EWMA control chart, MEWMA, [Lowry, Woodall, Champ and

Rigdon (1992)] is a chart where the plotted statistic takes into account the information

from past samples. The MEWMA control chart shows an out-of-control signal when the

2iT statistic is larger that the control limit (h), selected to obtain a desired in-control

ARL. In order to compute the 2iT statistic is necessary to employ a smoothing

coefficient, r ( 10 ≤< r ). The MEWMA vector is obtained through

1(1 )i i ir r −= + −Z X Z and the statistic to be plotted is given by:

12 T

i i iiT −

= ∑zZ Z (2)

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being ∑iz

the covariance matrix of iZ , where 0 0=Z μ . The exact covariance matrix

of iZ is computed according to:

21 (1 )

2

i

i

r rr

⎡ ⎤− −⎣ ⎦=−∑ ∑z (3)

It is a common practice to use the asymptotic approximation of the matrix ∑ iz . When

1r = the resultant chart is the Hotelling’s control chart. If the value of r is reduced the

weight of the past samples is more important. Aparisi and García-Díaz (2004) have

developed software that employing genetic algorithms finds the parameters r and h (for

univariate and multivariate cases) that produces the minimum ARL for a given size of

shift and for a given in-control ARL.

2.3 Double Sampling T2 control chart

The strategy of double sampling (DS) consists of taking taken two samples of sizes n1

and n2 from the process at the same time. With the statistical information obtained in the

first sample it is determined whether the process is in control, out of control or whether

it is required to analyze the second sample, and combine it with the first one in order to

make a final analysis. Champ and Aparisi (2006) developed an analytical method to

obtain the ARL of the DS-T2 chart and, employing genetic algorithms, the parameters of

the optimum DS-T2 chart to detect a shift of given magnitude are found.

There are five parameters to define the DS-T2 chart: h, h1, w1, n1 and n2. The 21,iT

statistic is computed only considering the first sample of size n1, employing

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( ) ( )2 1,1 1 ,1 ,1

Ti i o i oT n −= − −X Xμ μΣ . If 2

,1 1iT h> the process is deemed to be out of

control; if 2,1 1iT w< it is accepted that the process is in an in-control state. However, if

121,1 hTw i <≤ it is not possible to take a decision and the second sample of size n2 is

studied. The two samples are combined employing 1 ,1 2 ,21 2

1 ( )i i in nn n

= ++

X X X . Hence,

2iT is calculated using the expression ( ) ( )2 1

1 2 0 0( )T

i i iT n n μ μ−= + − −X XΣ . If 2iT h>

it is assumed that the process is out-of-control, otherwise the process is deemed to be in

an in-control state.

3.- The Synthetic-T2 control chart

The univariate synthetic chart was introduced by Wu and Spedding (2000) as an

alternative to improve the performance of the Shewhart control chart to detect process

shifts. It is the result of combining a Shewhart chart and a CRL chart (a chart originally

designed to detect increments in the percentage of defective units). The synthetic-

X chart shows better ARL values to detect process shifts, for any shift magnitude, than

the X control chart. In some cases, specially for moderate and large shifts, the

synthetic- X chart has better performance than the EWMA control chart [Wu and

Spedding (2000)].

The synthetic chart has been also applied to monitor the variability of a process [Chen

and Huang (2005a, 2005b)] and the percentage of defective units of a process [Wu, Yeo

and Spedding (2001)]. With the objective of improving the performance of the synthetic

control charts, Chen and Huang (2005b) apply variable sampling intervals (VSI) and

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Davis and Woodall (2002) improves the performance of the synthetic- X control chart

adding the concept of side-sensitive. However, a multivariate synthetic control has not

been developed.

The synthetic T2 control chart (synthetic-T2) that is going to be defined in this paper

takes benefit of the advantages of the synthetic charts. It is a chart for monitoring

simultaneously two or more quality characteristics. It consists of two sub-charts, a T2

sub-chart and a CRL sub-chart. Figure 1 shows the concept of the synthetic-T2 chart.

The T2 sub-chart has a unique control limit, LCsynt. The CRL sub-chart has a low

control limit, L, 1L ≥ . The value of LCsynt is the criteria to classify a sample as

conforming or non-conforming. The value of L is the criteria to decide if the process is

in control or out of control.

[INSERT FIGURE 1 OVER HERE]

The CRL chart was proposed by Bourke (1991). In the synthetic-T2 chart the value of

CRL is defined as the number of inspected samples between two samples classified as

non-conforming, including the last non-conforming sample. In figure 2 the white circles

represent conforming samples and the black circles show non-conforming samples. In

this Figure four values of CRL are shown, CRL1 = 5, CRL2 = 4, CRL3 = 7, CRL4 = 6,

assuming that the sampling starts at t = 0.


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The CRL concept assumes that in t = 0 there is a point above the LCsynt limit (a non-

conforming sample in t = 0). This characteristic, called head start, is very important for

the performance of the synthetic charts. When this assumption is ruled out, the

performance of synthetic charts worsens [Davis and Woodall (2002)]. The routine of the

synthetic-T2 chart follows:

1. a sample of size n is taken from the process at time i and the sample mean vector

is computed, 1 2 3( , , ,...., )Ti pX X X X=X . The 2

iT statistic is calculated following

(1).

2. The value of the 2iT statistic is plotted in the T2 sub-chart. If 2

iT LCsynt≤ the

sample is classified as conforming and we move back to point 1. Otherwise, if

2iT LCsynt> , the sample is classified as non-conforming and we continue to the

next point.

3. It is counted the number of samples between this non-conforming sample and the

last one. This number is called CRL sample and it is plotted in the CRL sub-chart.

4. If CRL L> the conclusion is that the process is in an in-control state, and the

control routine begins again in point 1. If CRL L≤ the process is deemed as out of

control.

5. The out of control signal is investigated. If no assignable cause is founded the

signal is considered as a false alarm and we continue to point 1. Otherwise, the

assignable cause must be eliminated.

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For a synthetic-T2 chart, unlike the T2 control chart, 2iT LCsynt> does not mean that an

out-of-control state has to be assumed, but the inspected sample must be classified as

non-conforming.

Like the T2 control chart, the synthetic-T2 chart shows an out-of-control signal,

indicating the probably there is a shift in the process. However this signal does not

inform us about the variable or variables that have produced the shift. Several methods

for the interpretation of the out-of-control signal of the T2 control chart have been

developed. These methods can also be applied to the synthetic-T2 chart [Mason, Tracy

and Young (1997), Atienza, Tang and Ang (1998), Mason and Young (1999), Kourti

and MacGregor (2004), Aparisi, Avendaño and Sanz (2006)]. Applying these

techniques the global performance of the synthetic-T2 can be improved in comparison

with other multivariate charts that, as it will be show in the comparison made in this

paper, have better performance to detect some type of shifts, but the interpretation of its

out-of-control signal is not sufficiently studied.

4.- Optimization of the parameters of the synthetic-T2 chart

Two values of ARL are important for the design and performance of the synthetic-T2

chart, the in-control ARL [ 2 ( 0)S T

ARL d−

= ] and the out-of-control ARL

[ 2 ( 0)S T

ARL d−

≠ ]. The value of the in-control ARL is selected taking into account the

frequency of false alarms. The value of 2 ( 0)S T

ARL d−

≠ is important in order to rapidly

detect a shift in the mean vector of magnitude d.

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4.1. Obtaining the ARL values

The value of ARL for a given shift of magnitudeδ , for whatever synthetic chart, is

[Chen and Huang (2005a)]:

[ ] [ ] 1 1( ) * *1 (1 )s CRL LARL E ARL E CRL

q qδ = =

− − (4)

where q is the probability of a sample being non-conforming. Following (4), the ARL of

the synthetic-T2 chart is given by the next formula:

2 2

0 0

1 1( 0) ( , )* ( )* *1 (1 )CRL LS T T

ARL d ARL d LCsynt L ARL d LCsyntq q−

= = =− −

(5)

where 20 P( )iq T LCsynt= > .

Considering (4), when 0d ≠ , the value of ( 0)S TARL d− ≠ is obtained as:

2 2

1 1( 0) ( , )* ( ) *1 (1 )CRL LS T T

d d

ARL d ARL d LCsynt L ARL d LCsyntq q−

≠ = =− −

(6)

where 21 1 P( )d iq T LCsyntβ= − = − ≤ .

4.2. Optimization of the synthetic-T2 control chart

Equations (5) and (6) are employed to find the parameters of the optimum synthetic-T2

control chart to detect a shift d. The optimization process must find the values of L and

LCsynt that minimizes the value of 2 ( 0)S T

ARL d−

≠ for a given shift of magnitude d,

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taking into account the restriction of the specified in-control ARL, 2 ( 0)S T

ARL d−

= . The

cart that fulfils this design is considered to be optimum to detect a shift of size d in the

mean vector.

The search to find the solution consists of fixing a value of L and to seek a value of

LCsynt that fulfils equation (5), LCsyntL, given 2 ( 0)S T

ARL d−

= . With these values,

2 ( 0)S T

ARL d−

≠ is calculated following equation (6). The process starts with L = 1,

adding one unit, stopping when the value of 2 ( 0)S T

ARL d−

≠ cannot be reduced in

comparison with the previous value of L. The search procedure is halted when

2 21[ ( 0) 1, ] [ ( 0) , ]L LS T S TARL d L LCsynt ARL d L LCsynt+− −

≠ + ≥ ≠ .

Software for Windows ® has been developed. The program finds the parameters of the

optimum synthetic-T2 chart for a given shift magnitude d and a given in-control ARL.

The software is available upon request from the authors or can be downloaded from

http://xxxxx.xx. The output of this program is shown in Figure 3. This software allows

us to obtain the improvement of ARL against Hotelling’s T2 control chart, it draws the

ARL curves of T2 and synthetic-T2 charts, and the user can evaluate the ARL of the

synthetic-T2 chart for a given set of values of p, n, LCsynt, L and d.


As shown in Figure 3, to find the optimum design of the synthetic-T2 chart the user hast

to specify: the number of variables to be monitored, (p) the simple size (n), the desired

in-control ARL, 2 ( 0)S T

ARL d−

= , and the shift magnitude to be detected as soon as

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possible, d. Figure 3 shows the solution for the problem 2 ( 0) 200S T

ARL d−

= = , p = 3, n

= 4 and d = 0.75. The parameters of the optimum synthetic-T2 chart are: LCsynt =

9.5755 and L = 11, with 2 ( 0.75) 9.81S T

ARL d−

= = . The out-of-control ARL value for

the standard T2 chart is 20.406. Therefore, the reduction in the value of ARL for a shift

d = 0.75 employing the synthetic-T2 chart is 51.9%.

5.- Comparison versus other multivariate charts

5.1. Comparison versus Hotelling’s T2 chart

A comparison of performance between the synthetic-T2 and the standard T2 control

chart [ 2 ( )T

ARL d ] has been carried out. The optimum design of the synthetic-T2 chart

has been utilized in the comparison for different cases of

2 2( 0) ( 0)S T T

ARL d ARL d−

= = = , n, p and d. The ARL values are shown in the two first

columns of Tables 1 to &. The percentage of improvement for the ARL is calculated as:

2 2

2

( 0) ( 0)% of improvement *100

( 0)T S T

T

ARL d ARL dARL d

−≠ − ≠

=≠

(7)

The main important conclusions are:

1.- The synthetic-T2 control chart optimum for a shift d is always faster to detect this

shift than the Hotelling’s T2 chart. For large shifts, d > 2, the percentage of

improvement of the synthetic-T2 chart is better when the sample size is small. For large

sample sizes the performance of both charts is similar for large shifts. In some cases the

improvement is quite significant. For example,

when 2 2( 0) ( 0) 200S T T

ARL d ARL d−

= = = = , p = 2, n = 2 and d = 1.25 the improvement

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in the ARL is 53.66 %, with ARL values of 2 ( 1.25)T

ARL d = = 9.91

and 2 ( 1.25) 4.59S T

ARL d−

= = . For the same case the percentage of improvement is

larger than 40% for the range 0.75 2d≤ ≤ .

2.- If the synthetic-T2 chart is designed to minimise the ARL for a given shift d, the

optimized chart is also faster than the standard T2 chart for the rest of values of d,

0 3d< ≤ .

3.- The percentage of improvement of the synthetic-T2, for the same set of values of p, n

and d, increases as the value of ( 0)ARL d = is larger.

4.- For the same values of p and ( 0)ARL d = , the optimum parameters of the synthetic-

T2 charts for d are constants, independently of the sample size (if 2≥n ) and magnitude

of shift, d , if the value of d is large. The parameter L takes small values if the chart is

designed for large values of d, but L tends to be small if the chart is optimized for large

values of d.

5.- Influence of the sample size. For n = 1, 2 or 3, the optimum synthetic-T2 for a shift d

is always faster than the standard T2 for detecting shifts of magnitude 0 3d< ≤ . For

sample sizes 4n ≥ the performance of the optimum synthetic-T2 chart is better for small

and moderate shifts, but the improvement is marginal for large shifts. For example, if n

= 7 the improvement is negligible if 2.25d ≥ . This behaviour is logical because the

standard T2 shows ARL values close to one. As the sample size increases, the range of

values of d where the synthetic-T2 chart has better performance is narrower. Figure 4

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shows the ARL values of the optimum synthetic-T2 charts for a shift d. The curves

shown are for the cases ( 0) 200ARL d = = , p = 2, 7≤n and d = 0.25, 0.5,...,2.


For small and moderate shifts ( 0.75d ≤ ), and for the same set of values of ( 0)ARL d = ,

p and d, a larger sample size increases the percentage of improvement. However, when

1.75d ≥ the percentage of improvement of the synthetic-T2 chart is smaller when the

sample size increases.

6.- Influence of the number of variables. The percentage of improvement of the

synthetic-T2 chart depends of the number of variables, p. For the same values of n and d

the percentage of improvement is smaller when more variables are to be controlled if

0.75d ≤ . However, when 1.5d ≥ and 3≥n the percentage of improvement is larger

when the number of variables increases.

5.2. Comparison versus MEWMA, VSS-T2 and DS-T2 charts.

In this Section a comparison of the performance (ARL) of the optimum synthetic-T2

chart for a shift d, versus the optimum MEWMA, VSS-T2 y DS-T2 charts for the same d

is carried out. For the VSS-T2 and DS-T2 charts the restrictions 0( )E n n= and

1 2 20n n+ ≤ must be fulfilled. With these restrictions, first it is guaranteed that, in

average, the same sample size is employed. Secondly, it is very frequent that in order to

find the optimum charts to minimize ( 0)ARL d ≠ for the VSS-T2 and DS-T2 charts, large

values of n2 are obtained, producing solutions that may be not feasible in the real

application.

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Tables 1 to 6 show the optimum parameters of the control charts that are used in the

comparison and the value of ( 0)ARL d ≠ for d = 0.25, 0.5, … 2.5 and for some

combinations of p and n when ARL(d = 0) = 200. The authors have obtained more

tables for the following cases: p = 2, 3, 4, 5 and 10, n = 1, 2, 3, 5 and 7, ARL(d = 0) =

200 and 400 and d = 0.25, 0.5, … 2.5.

[INSERT TABLES 1 to 6 OVER HERE]

The optimum synthetic-T2 chart for a shift d is faster than the MEWMA and VSS-T2

charts when p = 2 and n is large. For example, when p = 2 and n = 7 the synthetic-T2

chart is superior or equal than the MEWMA chart if 0.75d ≥ . On the other hand,

considering the same case, the synthetic-T2 chart is superior or similar than the VSS-T2

chart when 1d ≥ . However, the range where the optimum synthetic-T2 chart is superior

to these charts worsens as the sample size decreases and the number of variables

increases. For example, if p = 10 and n = 1, the 2 ( 0)S T

ARL d−

≠ is lower than the ARL of

MEWMA if 2.75d ≥

There are some cases where the MEWMA and VSS-T2 charts optimum for a shift d

have an ARL value larger or equal than the standard T2 chart. However, the synthetic-T2

and DS-T2 charts optimum for a shift d always have lower or equal ARL values in

comparison with the standard T2 chart. If the synthetic-T2 chart is designed to minimise

the ARL for a shift d, for the rest of size of shifts, 0 3d< ≤ , it will be always faster to

detect these shifts than the T2 chart. This behaviour does not happen sometimes with

the MEWMA and VSS-T2 charts.

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In general, the performance of the MEWMA y VSS-T2 charts is better for detecting

small shifts than the synthetic-T2 chart, but this is may not be true for moderate and

large shifts. For a given value of ( 0)ARL d = and number of quality variables, the range

where it is better to use the synthetic-T2 chart instead of the MEWMA control chart

increases as the sample size increases. The synthetic-T2 chart is master thatn the VSS-T2

chart when d = 0.25. However, for this same value, the lowest ARL value is obtained by

the DS-T2chart. For very small shifts the DS-T2 chart has better performance than the

MEWMA.

When 3n ≥ and 10p ≤ the synthetic-T2 chart is faster than the MEWMA and VSS-T2

charts for shifts of magnitude 1.75d ≥ , although the range of d where the performance

of the synthetic-T2 is superior is incremented when the sample size increases and the

number of variables is smaller. Figure 5 shows the ARL values for the optimum designs

for a shift d, considering all the charts involved in comparison. The case shown in the

figure is ARL(d = 0) = 200 and d = 0.75, 1,…,2.5.


The DS-T2 chart has better performance than the synthetic-T2 for 1d ≤ , independently

of the sample size and the number of variables. The performance of these two charts, for

moderate and large shifts, is again very similar as the sample size increases and the

number of variables decreases.

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Figure 6 shows the ARL curves of the optimum charts designed to have the best

performance for 0.75d = , for the case ARL(d = 0) = 200, p = 2 and n = 7 (Table 2). It

is possible to see that the ARL curve of the synthetic-T2 chart is always below the curve

of the Hotelling’s T2 chart. However, this does not happen with the curves of the

MEWMA and VSS-T2 charts. The ARL of the synthetic-T2 is lower than the ARL of

the MEWMA and VSS-T2 charts when d ≥ 0.8. The DS-T2 control chart is the chart that

has the better performance to detect simultaneously very small and large shifts. For

moderate and large shifts, the performance of the synthetic-T2 chart is similar to the DS-

T2 control chart.


6.- Conclusions.

In this article the synthetic-T2 control chart has been developed, finding a procedure to

obtain the optimum parameters of this chart to minimize the out-of-control ARL, given

a value of in-control ARL value and a magnitude of shift in the mean vector.

It may be advisable to employ the synthetic-T2 control chart as an alternative to the

standard T2 chart, because the first one always detects faster the shifts in the mean

vector for all the values of magnitude of shift. One important property of the synthetic-

T2 chart is that a optimized chart for a given magnitude of shift always gives better

performance, for all the values of shift, than the equivalent Hotelling’s control chart.

Although its real application can be more complicated than the standard T2 chart, the

improved performance, and the possibility of continuing employing the tools for the

20

interpretation of the out-of-control signal of the Hotelling’s T2 chart, make the

utilization of the synthetic-T2 control chart an interesting one. For example, the user

interested in finding out the variables that have produced the out-of-control signal can

apply the MTY decomposition method [Mason, Tracy and Young (1997) and Mason

and Young (1999)] or the use of neural networks [Aparisi, Avendaño and Sanz (2006)].

The synthetic-T2 chart may provide good alternative to the DS-T2 chart in order to

detect moderate and large shifts in the mean vector, having the advantage of being a

chart easier to employ in the real application. For the detection of very small shifts (d =

0.25) we recommend the DS-T2. In general, for very small shifts the synthetic-T2 chart

cannot compete with the MEWMA, VSS-T2, and DSS-T2 charts, but it has a much

better performance in comparison with the Hotelling’s T2 chart. For the case of

moderate and large shifts there cases where the synthetic-T2 chart is superior that the

MEWMA or VSS-T2 charts.

Acknowledgements

This work has been supported by the Ministry of Education and Science of Spain,

research project number DPI2006-06124 including European FEDER funding, and the

support of the ITESM-Foundation Carolina agreement.

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26

Figure 1. Sub-charts of the synthetic-T2 Control chart: T2 sub-chart and CRL sub-chart.

Figure 2. Values of CRL between two non-conforming samples.

conforming sample

nonconforming sample

In-control state

T2 sub-chart

L

CR

LSignal out of control

CRL sub-chart

LCsynt

0t

t1

t = 0

Conforming sample Nonconforming sample

CRL1 = 5CRL2 = 4

CRL3 = 7CRL4 = 6

27

Figure 3. Main window of the developed software to find the optimum synthetic-T2 and synthetic- X control

charts.

Figure 4. ARL of the optimum synthetic-T2 chart for a shift d. ( 0) 200S TARL d− = = , p = 2.

n = 1

n = 2

n = 3

n = 5n = 7

28

Figure 5. ARL of the optimum multivariate control charts for a given magnitude of shift d. ARL(d = 0) = 200.

Figure 6. ARL curves of the optimum control charts for a shift 0.75d = . ARL(d = 0) = 200, p = 2, n = 7.

29

T2 Synthetic-T2 MEWMA VSS-T2 DS-T2

(CL) ( LCsynt, L ) ( h, r ) ( CL, w, n1, n2 ) ( h, h1, w1, n1, n2 )d n1+n2 <= 20 n1+n2 <= 20

200 200 200 200 20010.5966148.34 129.27 41.75 138.28 16.11

(8.942,50) (7.372,0.05) (10.597, 5.75,1,19) (11.25,19.16,1.39,1,2)76.86 51.37 17.03 39.95 21.36

(8.499,30) ( 9.067,0.13) (10.597,5.78,1,19) (6.00,13.96,5,73,1,18)37.01 19.88 9.24 12.55 9.48

(7.987,17) ( 9.366,0.16) (10.597,5.41,1,16) (6.69,14.77,5.11,1,3)18.48 8.80 6.03 6.22 5.79

(7.496,10) (9.95,0.26) (10.597,4.39,1,10) (9.59,11.27,4.71,1,11)9.91 4.59 4.34 3.93 3.62

(7.161,7) ( 10.28,0.38) (10.597, 3.89,1,8) (9.87,12.62,3.20,1,5)5.76 2.79 3.27 2.86 2.64

(6.842,5) (10.36, 0.43) (10.597,3.22,1,6) (9.45,11.58,4.34,1,9)3.65 1.93 2.61 2.28 1.96

(6.630,4) (10.51,0.59) (10.597,2.77,1,5) (11.27,11.12,3.83,1,7)2.51 1.49 2.10 1.92 1.50

(6.355,3) (10.5485,0.67) (10.597,2.20,1,4) (13.25,11.01,2,74,1,4)1.87 1.26 1.73 1.68 1.26

(5.966,2) (10.57,0.74) (10.597,1.39,1,3) (13.59,10.95,2.74,1,4)1.50 1.13 1.47 1.49 1.20

(5.966,2) (10.58, 0.79) (10.597,1.39,1,3) (14.12,10.82,3.18,1,5)

2

2.25

2.5

1

1.25

1.5

1.75

0

0.25

0.5

0.75

Table 1. ARL of the optimum multivariate control charts for a shift d. ARL(d = 0) = 200, p = 2, n = 2.



200 200 200 200 20010.596684.35 58.32 18.27 66.17 10.36

(8.556,32) (8.471, 0.09) (10.597, 2.20,1,19) (12.02,11.90,0.56,1,8)21.98 10.70 6.65 9.06 7.81

(7.666,12) (9.769,0.22) (10.597,2.19,1,19) (9.73,15.27,2.92,4,13)7.04 3.33 3.62 3.30 2.85

(6.842,5) (10.195,0.34) (10.597,2.33,2,18) (10.19,11.88,5.07,6,13)3.02 1.68 2.36 2.10 1.52

(6.355,3) (10.41,0.47) ( 10.597,3.01,5,14) (9.63,14.34,4.59,6,10)1.72 1.20 1.63 1.54 1.18

(5.966,2) (10.57,0.74) (10.597,3.89,6,13) (10.89,12.36,2.58,4,11)1.24 1.05 1.25 1.25 1.04

( 5.966,2) (10.59,0.87) (10.597,0.00,6,7) (10.25,15.36,3.88,6,7)1.07 1.01 1.08 1.08 1.02

(5.966,2) (10.594,0.91) (10.597,1.39,6,8) (17.94,10.63,5.15,6,14)1.02 1.00 1.02 1.03 1.00

(5.966,2) (10.59,0.86) (10.597,3.22,6,11) (15.23,10.75,3.53,5,12)1.00 1.00 1.00 1.01 1.00

(5.966,2) (10.59,0.90)) (10.597,2.20,6,9) (12.56,11.33,2.74,5,8)1.00 1.00 1.00 1.00 1.00

(5.966,2) (10.59,1.00) (10.597,3.89,6,13) (13.54,10.94,3.04,4,14)

2

2.25

2.5

1

1.25

1.5

1.75

0

0.25

0.5

0.75


30



200 200 200 200 20016.7496168.96 155.51 52.82 164.11 7.28

(14.719,48) (13.44,0.06) (16.750,10.79,1,19) (18.85,17.33,0.04,1,1,)109.20 82.58 20.96 66.17 31.07

(14.386,35) (14.1, 0.08) (16.750,10.62,1,19) (10.17,27.13,10.94,1,19)61.43 37.53 11.69 20.27 15.13

(13.928,23) (15.31,0.15) (16.750,10.79,1,19) (12.43,20.15,9.68,1,12)33.11 17.15 7.58 9.46 7.61

(13.448,15) (15.87,0.22) (16.750,9.73,1,13) (12.80,24.26,8.62,1,8)18.07 8.53 5.38 5.52 5.07

(12.980,10) (16.19,0.29) (16.750, 8.62,1,9) (13.73,18.84,9.18,1,10)10.28 4.75 4.07 3.78 3.22

(12.561,7) (16.366,0.35) ( 16.750, 7.82,1,7) (14.26,22.20,7.80,1,6)6.20 2.97 3.22 2.87 2.61

(12.159,5) (16.51,0.42) (16.750, 7.28,1,6) (16.99,17.66,7.76,1,6)3.99 2.07 2.63 2.37 2.06

(11.890,4) (16.609,0.51) (16.750,5.73,1,4) (14.60,19.25,8.22,1,7)2.76 1.59 2.18 2.08 1.47

( 11.538,3) (16.661, 0.58) ( 16.750,4.35,1,3) (18.81,17.69,5.69,1,3)2.04 1.32 1.82 1.74 1.22

(11.538,3) (16.734,0.79) (16.750,4.35,1,3) (22.96,16.92,5.69,1,3)2.5

2

2.25

1.5

1.75

1

1.25

0.5

0.75

0.25

0




200 200 200 200 20016.7496116.74 90.81 23.00 101.11 6.36

(14.445,37) (13.8,0.07) (16.750,5.73,1,19) (15.95,28.51,0.02,1,6)38.86 20.87 8.38 16.81 14.47

(13.590,17) (15.745,0.20) (16.750,5.73,1,19) (15.63,20.23,8.02,5,13)12.71 5.88 4.51 4.51 3.95

(12.719,8) (16.34,0.34) ( 16.750, 5.73,1,19) (15.24,23.07,8.24,5,14)4.97 2.47 2.92 2.62 1.95

(11.890,4) (16.618,0.52) (16.750,5.73,2,17) (15.23,20.11,8.43,5,15)2.46 1.48 2.04 1.86 1.39

(11.538,3) (16.7,0.66) (16.750,6.98,5,14) (16.08,28.52,7.08,4,14)1.55 1.14 1.52 1.44 1.10

(11.037,2) (16.71,0.69) (16.750,7.29,6,11) (16.77,20.13,6.61,5,8)1.19 1.04 1.22 1.19 1.02

(11.037,2) (16.72,0.73) (16.750,0.00,6,7) (23.39,16.90,5.69,5,6)1.05 1.01 1.06 1.05 1.01

(11.037,2) (16.75, 0.91) (16.750,0.00,6,7) (18.46,17.57,7.51,5,11)1.01 1.00 1.01 1.01 1.00

(11.037,2) (16.75,0.97) (16.750,0.00,6,7) (17.44,18.17,0.49,3,4)1.00 1.00 1.00 1.00 1.00

(11.037,2) (16.75,0.92) (16.750,0.00,6,7) (23.53,16.93,5.10,5,5)2.5

2

2.25

1.5

1.75

1

1.25

0.25

0.5

0.75

0


31



200 200 200 200 20025.1882179.24 169.43 64.15 176.60

(22.688,45) (17.214,0.02) ( 25.188,17.96,1,19)132.28 108.01 26.58 92.58 42.16

(22.439,37) (21.8,0.07) (25.188,17.50,1,19) (17.21,35.76,18.05,1,19)85.15 57.75 14.73 31.84 19.76

( 22.029,27) (22.71, 0.1) (25.188,17.95,1,19) (17.71,29.98,18.08,1,19)50.78 28.94 9.43 13.54 10.84

(21.557,19) (23.81,0.17) ( 25.188,17.37,1,16) (19.77,48.54,15.98,1,10)29.43 14.72 6.68 7.63 7.11

(21.034,13) ( 24.09, 0.2) (25.188,15.99,1,11) (20.80,27.54,16.56,1,12)17.14 7.98 5.01 4.97 4.39

(20.513,9) (24.754, 0.34) (25.188,14.72,1,8) (20.87,32.66,15.22,1,8)10.27 4.71 3.91 3.62 3.53

(20.150,7) ( 24.8725,0.39) (25.188,13.44,1,6) (20.19,42.64,15.61,1,9)6.42 3.05 3.19 2.84 2.40

(19.657,5) (25.022,0.49) (25.188,12.55,1,5) (22.86,30.76,13.43,1,5)4.23 2.16 2.65 2.35 1.93

(19.324,4) (25.052,0.52) (25.188,12.55,1,5) (23.88,27.88,13.40,1,5)2.96 1.66 2.30 2.01 1.45

(18.890,3) (25.17,0.77) (25.188,11.32,1,4) (25.64,27.62,11.29,1,3)

2

2.25

2.5

1

1.25

1.5

1.75

0

0.25

0.5

0.75

Table 5. ARL of the optimum multivariate control charts for a shift d. ARL(0) = 200, p = 10, n = 2.



200 200 200 200 20025.1882138.83 115.91 29.36 127.60 6.91

(22.474,38) (20.80,0.05) (25.188,11.32,1,19) (25.96,29.05,0.04,2,5)58.29 34.63 10.54 31.97 24.52

(21.693,21) (23.296,0.13) (25.188,11.60,2,18) (23.55,33.55,15.61,6,9)21.09 10.03 5.58 7.05 7.59

(20.799,11) (24.294,0.23) (25.188,11.60,2,18) (26.27,26.16,16.18,6,11)8.13 3.77 3.70 3.34 2.94

(19.925,6) ( 24.48, 0.26) ( 25.188,11.67,3,16) (25.93,26.88,12.50,4,12)3.70 1.95 2.49 2.21 1.77

(19.324,4) (25.1, 0.58) (25.188,12.19,4,15) (25.86,26.56,12.33,3,15)2.07 1.33 1.82 1.70 1.22

(18.890,3) (25.17,0.77) ( 25.188,14.14,6,12) (37.27,25.24,13.34,5,10)1.41 1.10 1.39 1.37 1.12

(18.266,2) (25.17,0.78) (25.188,14.14,6,12) (21.79,31.32,17.11,6,14)1.14 1.03 1.14 1.14 1.05

(18.266,2) (25.18,0.85) (25.188,0.00,6,7) (23.91,27.78,14.91,5,15)1.04 1.00 1.04 1.04 1.01

(18.266,2) (25.19,0.97) (25.188,0.00,6,7) (26.24,26.58,13.74,5,11)1.01 1.00 1.01 1.01 1.00

(18.266,2) (25.187,0.94) (25.188,0.00,6,7) (26.49,28.28,0.14,3,4)

2

2.25

2.5

1

1.25

1.5

1.75

0

0.25

0.5

0.75


Documents

The design of the multivariate synthetic-T 2 control chart. · 1 The design of the multivariate synthetic-T2 control chart. Francisco Aparisi1 and Marco A. de Luna2. 1Departamento