# ISO 7870-7:2020

(Main)## Control charts — Part 7: Multivariate control charts

## Control charts — Part 7: Multivariate control charts

This document describes the construction and use of multivariate control charts in statistical process control (SPC) and establishes methods for using and understanding this generalized approach to control charts where the characteristics being measured are from variables data. The use of principal component analysis (PCA) and partial least squares (PLS) in the field of multivariate statistical process control is not presented in this document NOTE The document describes the current state of the art of multivariate control charts that are being applied in practice nowadays. It does not describe the current state of scientific research on the topic.

## Cartes de contrôle — Partie 7: Cartes de contrôle multivariées

### General Information

### Standards Content (Sample)

INTERNATIONAL ISO

STANDARD 7870-7

First edition

2020-02

Control charts —

Part 7:

Multivariate control charts

Cartes de contrôle —

Partie 7: Cartes de contrôle multivariées

Reference number

ISO 7870-7:2020(E)

©

ISO 2020

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ISO 7870-7:2020(E)

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© ISO 2020

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ISO 7870-7:2020(E)

Contents Page

Foreword .iv

Introduction .v

1 Scope . 1

2 Normative references . 1

3 Terms and definitions . 1

4 Abbreviated terms and symbols . 2

4.1 Abbreviated terms . 2

4.2 Symbols . 2

5 Purpose and classification of multivariate control charts . 4

5.1 Purpose and applying conditions for multivariate control charts . 4

5.2 Classification of multivariate control charts . 5

6 Multivariate control charts with unweighted averages for mean shift .6

6.1 General . 6

6.2 Control charts for the process mean (n>1) . 7

2

6.2.1 χ control chart when pre-specified parameter values are known . 7

2

6.2.2 T control chart when pre-specified parameter values are unknown . 8

6.3 Control charts for the process mean (n=1) . 8

2

6.3.1 χ control chart when pre-specified parameter values are known . 8

2

6.3.2 T control chart when pre-specified parameter values are unknown . 9

6.4 Summary and selection of multivariate control charts with unweighted averages

for mean shifts . 9

6.5 Test for assignable causes .10

7 Multivariate control charts with weighted averages for mean shifts .11

8 Control charts for the process dispersion .12

9 Interpretation of an out-of-control signal .13

Annex A (informative) Example of multivariate statistical process control .14

Annex B (informative) Example of MEWMA control chart .17

Annex C (informative) Estimation of μ and Σ .23

Bibliography .25

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ISO 7870-7:2020(E)

Foreword

ISO (the International Organization for Standardization) is a worldwide federation of national standards

bodies (ISO member bodies). The work of preparing International Standards is normally carried out

through ISO technical committees. Each member body interested in a subject for which a technical

committee has been established has the right to be represented on that committee. International

organizations, governmental and non-governmental, in liaison with ISO, also take part in the work.

ISO collaborates closely with the International Electrotechnical Commission (IEC) on all matters of

electrotechnical standardization.

The procedures used to develop this document and those intended for its further maintenance are

described in the ISO/IEC Directives, Part 1. In particular the different approval criteria needed for the

different types of ISO documents should be noted. This document was drafted in accordance with the

editorial rules of the ISO/IEC Directives, Part 2 (see www .iso .org/ directives).

Attention is drawn to the possibility that some of the elements of this document may be the subject of

patent rights. ISO shall not be held responsible for identifying any or all such patent rights. Details of

any patent rights identified during the development of the document will be in the Introduction and/or

on the ISO list of patent declarations received (see www .iso .org/ patents).

Any trade name used in this document is information given for the convenience of users and does not

constitute an endorsement.

For an explanation of the voluntary nature of standards, the meaning of ISO specific terms and

expressions related to conformity assessment, as well as information about ISO's adherence to the

World Trade Organization (WTO) principles in the Technical Barriers to Trade (TBT), see www .iso .org/

iso/ foreword .html.

This document was prepared by Technical Committee ISO/TC 69, Applications of statistical methods,

Subcommittee SC 4, Applications of statistical methods in process management.

A list of all parts in the ISO 7870 series can be found on the ISO website.

Any feedback or questions on this document should be directed to the user’s national standards body. A

complete listing of these bodies can be found at www .iso .org/ members .html.

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ISO 7870-7:2020(E)

Introduction

When a number of quality characteristics are to be controlled simultaneously, the usual practice has

been to maintain a separate (univariate) chart for each characteristic. Unfortunately, this can give

misleading results when the characteristics are highly correlated. Process monitoring of problems in

which several related variables are of interest are collectively known as multivariate statistical process

control (MSPC). The most useful tools of multivariate statistical process control are multivariate

control charts. Multivariate control charts are applied for statistical process evaluation and control

under the consideration of dependability between quality characteristics.

The function of a multivariate statistical process control system is to provide a statistical signal when

assignable causes of variation are present. The systematic elimination of assignable causes of excessive

variation, through continuous determined efforts, brings the process into a state of statistical control.

Once the process is operating in statistical control, its performance is predictable and its capability to

meet the specifications can then be assessed.

The main purpose of this document is to show how multivariate control charts can be used for process

control in terms of SPC and how the state of process stability can be assessed in a multivariate way.

ISO 22514-6 provides a calculation method for capability statistics for process parameters or product

characteristics following a multivariate normal distribution or approximately multivariate normal.

Multivariate charts are based on multivariate characteristics where more than one characteristic is to

be monitored in connection with others. In practice, a multivariate control chart is always applied with

1)

the support of software, such as Minitab, JMP, and Q-DAS .

1) MINITAB is the trade name of a product supplied by Minitab Inc. JMP is the trade name of a product supplied by

SAS Institute Inc. Q-DAS is the trade name of a product supplied by Q-DAS GmbH. This information is given for the

convenience of users of this document and does not constitute an endorsement by ISO of these products.

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INTERNATIONAL STANDARD ISO 7870-7:2020(E)

Control charts —

Part 7:

Multivariate control charts

1 Scope

This document describes the construction and use of multivariate control charts in statistical process

control (SPC) and establishes methods for using and understanding this generalized approach to

control charts where the characteristics being measured are from variables data.

The use of principal component analysis (PCA) and partial least squares (PLS) in the field of multivariate

statistical process control is not presented in this document

NOTE The document describes the current state of the art of multivariate control charts that are being

applied in practice nowadays. It does not describe the current state of scientific research on the topic.

2 Normative references

The following documents are referred to in the text in such a way that some or all of their content

constitutes requirements of this document. For dated references, only the edition cited applies. For

undated references, the latest edition of the referenced document (including any amendments) applies.

ISO 3534-2, Statistics — Vocabulary and symbols — Part 2: Applied statistics

3 Terms and definitions

For the purposes of this document, the terms and definitions given in ISO 3534-2 apply.

ISO and IEC maintain terminological databases for use in standardization at the following addresses:

— ISO Online browsing platform: available at https:// www .iso .org/ obp

— IEC Electropedia: available at http:// www .electropedia .org/

3.1

multivariate characteristics

multivariate quantity where the set of features consists of d quantities that are alone or combined with

the quality of a product

Note 1 to entry: Following ISO 7870-2, these quantities are denoted as quality characteristics X where i = 1, 2, …,d.

i

Note 2 to entry: The observation of multivariate characteristics can be expressed as the vector

T

x=(x , x , …, x ) . Thus, a multivariate quantity can be considered as a feature vector of a product. The value of the

1 2 d

multivariate quantity is represented by a point in the d-dimensional feature space.

Note 3 to entry: All single quantities combined in the multivariate vector can be measured in the same product

or object.

Note 4 to entry: If the multivariate quantity is described by means of statistics, the vector is considered as a

random vector following a d-dimensional multivariate distribution.

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ISO 7870-7:2020(E)

3.2

confidence region

d-dimensional region for a multivariate characteristics of d-dimension and defined for a specified

confidence level

Note 1 to entry: The region is limited by lines, surfaces or hyper-surfaces in the d-dimensional space.

Note 2 to entry: Form and size of the region are defined by one or more parameters.

4 Abbreviated terms and symbols

4.1 Abbreviated terms

SPC statistical process control

MSPC multivariate statistical process control

PCA principal component analysis

PLS partial least squares

UCL upper control limit

LCL lower control limit

ARL average run length

EWMA exponential weighted moving average

MEWMA multivariate exponential weighted moving average

4.2 Symbols

B the 1 − α quantile of beta distribution with degree of freedom ν and ν

1 2

1−αν,,ν

12

d number of dimensions for multivariate characteristics

2 2

the statistic plotted of a phase IIχ control chart

D

j

E(|S|) mean of |S|

F the 1 − α quantile of F distribution with degree of freedom ν and ν

1 2

1−αν,,ν

12

h upper control limit of MEWMA control chart

L lower control limit

CL

m number of subgroups

n size of each subgroup

N (μ,Σ) d-dimensional normal distribution with μ and Σ

d

s covariance between the a-th and b-th quality characteristics with n=1

ab

s covariance between the a-th and b-th quality characteristics in the j-th subgroup

abj

with n>1

2

variance of the i-th quality characteristic with n=1

s

i

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ISO 7870-7:2020(E)

2

variance of the i-th quality characteristic in the j-th subgroup with n>1

s

ij

2 2

s average of s over all m subgroups for the i-th quality characteristic with n>1

i ij

s average of s over all m subgroups for the covariance between the a-th and b-th

abj

ab

quality characteristics with n>1

S sample variance-covariance matrix with n=1

S sample variance-covariance matrix with n>1

|S| determinant of the sample variance-covariance matrix S

2 2

the statistic plotted of a phase I T -chart.

T

j

2

2

the statistic plotted of a phase II T -chart

T

f

tr trace operator

U upper control limit

CL

V(|S|) variance of |S|

x the j-th observation on the i-th quality characteristic with n=1

ij

x the k-th observation in the j-th subgroup on the i-th quality characteristic with n>1

ijk

x mean of the i-th quality characteristic in the j-th subgroup with n>1

ij

x average of x over all m subgroups for the i-th quality characteristic with n>1

i ij

x an observation vector

x vector of j-th observation with n=1

j

x vector of a future individual observation with n=1

f

x

sample mean vector with n=1

x mean of the j-th rational subgroup with n>1

j

x mean of a future rational subgroup with n>1

f

x sample mean vector with n>1

{}x i-th element of the vector x

i

2

the statistic plotted of MEWMA control chart.

Y

j

Z MEWMA statistic

j

2

2

the 1 − α quantile of χ distribution with degree of freedom ν

χ

1−αν,

δ shift size of the mean vector

λ MEWMA moving parameter vector

λ EWMA moving parameter, 0 < λ ≤ 1

μ mean vector of multivariate characteristics

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ISO 7870-7:2020(E)

μ pre-specified mean vector of multivariate characteristics

0

ρ

correlation coefficient between y and y

yy, 1 2

12

Σ variance-covariance matrix of multivariate characteristics

Σ pre-specified variance-covariance matrix of multivariate characteristics

0

Σ variance-covariance matrix of MEWMA statistic Z

Z j

j

−1

(·) inverse operator

T

(·) transpose operator

5 Purpose and classification of multivariate control charts

5.1 Purpose and applying conditions for multivariate control charts

There are many situations in which the simultaneous monitoring or control of two or more related

quality characteristics is necessary. The difficulty with using independent univariate control charts is

illustrated in Figure 1. Only two quality characteristics (y , y ) are considered for ease of illustration.

1 2

Suppose that, when the process is in a state of statistical control where only common cause variation is

present, both y and y follow a normal distribution but are correlated (ρ = −0,94) as illustrated in

1 2

yy,

12

the joint plot of y vs. y in Figure 1. The ellipse represents a contour for the in-control process, with

1 2

0,997 3-quantile, corresponding to risk of a false alarm of 0,002 7 in the Shewhart chart, and the points

represent a set of individual observations from this distribution. The same observations are also

plotted in Figure 1 as individual Shewhart control charts on y and y vs. the observation number (time)

1 2

with their corresponding upper and lower control limits (the 0,998 65-quantiles).

By looking at each of the individual Shewhartcontrol charts, the process appears to be clearly in a state

of statistical control, and none of the points give any indication of a problem. The true situation is only

revealed in the bivariate y vs. y plot where it is seen that the lot of product indicated by the ⊗ is

1 2

clearly outside the confidence region and is clearly different from the normal “in-control” population of

the product.

If the quality characteristics are not independent, which usually would be the case if they relate

to the same product, there is no easy way to measure the distortion in the joint control procedure.

Process-monitoring problems in which several related variables are of interest are sometimes called

multivariate quality control problems. This subject is particularly important, as automatic inspection

procedures make it relatively easy to measure many parameters on each unit of product manufactured.

For example, many chemical and process plants and semi-conductor manufacturers routinely maintain

manufacturing databases with the process and quality data on hundreds of variables. Monitoring or

analysing these data with univariate SPC procedures is often ineffective. Multivariate control charts

are applied for statistical process evaluation and control under consideration of dependability between

the product or process characteristics.

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ISO 7870-7:2020(E)

Key

y mean of y U upper control limit of y

1 2

1 CL

2

mean of y lower control limit of y

y L

2 2

2 CL

2

j

observation number point corresponding to y

1

upper control limit of y point corresponding to y

U

1 2

CL

1

L

lower control limit of y point corresponding to (y ,y )

CL 1 1 2

1

Figure 1 — Quality control of two variables

Multivariate control charts work well when the number of process variables is not too large – ten or

fewer. As the number of variables grows, however, traditional multivariate control charts lose efficiency

with regard to shift detection. A popular approach in these situations is to reduce the dimensionality of

the problem. This can be done with the use of projection methods such as principal component analysis

(PCA) or partial least squares (PLS). These two methods are based on building a model from a historical

data set, that is assumed to be in control. After the model has been built, a future observation is checked

as to whether it fits well or not in the model.

In the SPC univariate case, the normal distribution is generally used to describe the behaviour of

a continuous quality characteristic. The same approach can be used in the multivariate case. A

multivariate normal distribution is applied as the basic assumption for a multivariate characteristics.

5.2 Classification of multivariate control charts

If the multivariate characteristics is considered to be a random vector with a multivariate normal

distribution, this distribution is characterized by a mean vector μ and a variance-covariance matrix

Σ (see Annex C). Obviously from the viewpoint of the application of multivariate process control,

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ISO 7870-7:2020(E)

multivariate control charts can be applied to monitor the mean shift and process dispersion separately.

Thus, for the application, multivariate control charts can be classified as follows:

a) multivariate control charts for mean shift;

b) multivariate control charts for process dispersion.

For the mean shift, multivariate control charts with unweighted averages are analogous to the Shewhart

X chart or chart for individuals. They use information only from the current sample and are relatively

insensitive to small and moderate shifts in the mean vector. Multivariate control charts with weighted

averages such as multivariate EWMA control chart can be used to overcome this problem. Just like

EWMA charts are generally used for detecting small shifts in the process mean and they usually detect

shifts of 0,5 sigma to 2 sigma much faster. Thus, multivariate control charts for mean shift can be

classified as follows:

2 2

i) multivariate control charts with unweighted averages (see Clause 6), such as χ and T chart;

ii) multivariate control charts with weighted averages (see Clause 7), such as multivariate EWMA

control chart.

Figure 2 is given to show how to select multivariate control charts.

Figure 2 — Multivariate control chart selection flow chart

6 Multivariate control charts with unweighted averages for mean shift

6.1 General

For each of the multivariate control charts, there are two distinct situations:

a) when no pre-specified process parameter values are given, and

b) when pre-specified process parameter values are given.

The pre-specified or known process parameter values can be defined by target values or by

requirements or by estimated values that have been determined by the data under the condition of a

process in control.

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ISO 7870-7:2020(E)

There are two distinct phases of control charting practice.

i) Phase I: control charts are used for retrospective testing of whether the process was in-control

when the first subgroups were being drawn. Once this is accomplished, the control chart is used

to define what is meant by a process being statistically in-control. This is referred to as the

retrospective use of control charts;

ii) Phase II: control charts are used for testing whether the process remains in-control when future

subgroups are drawn. In this phase, the charts are used as aids to the practitioners in monitoring

the process for any changes from an in-control state.

Another crucial matter is the subgroup size n of each rational subgroup. If n=1, then special care must

be taken. Thus, four possibilities are considered:

— phase I and n=1, working with individual observations;

— phase I and n>1, working with rational subgroups;

— phase II and n=1, working with individual observations;

— phase II and n>1, working with rational subgroups.

6.2 Control charts for the process mean (n>1)

2

6.2.1 χ control chart when pre-specified parameter values are known

Assume that the vector x follows a d-dimensional normal distribution, denoted as N (μ ,Σ ), and there

d 0 0

are m subgroups each of size n>1 available from the process. Furthermore, assume that the observation

vectors x are not time dependent. A control chart can be based on the sequence of the following statistic:

T

2 −1

Dn=−xxμμΣΣμ−μ j = 1,2, ., m (1)

() ()

jj 00 j 0

Here x is the vector of the mean of the j-th rational subgroup, where μ and Σ are the known vector of

0 0

j

means and the known variance-covariance matrix, respectively.

2

The D statistic represents the weighted distance (Mahalanobis distance) of any point from the target

j

2

μ . If the value of the test statistic D plots above upper control limit, the chart signals a potential out-

0

j

of-control process. In general, control charts have both upper and lower control limits. However, in this

case only an upper control limit is used, because extreme values of the statistic correspond to points far

remote from the target μ , whereas small or zero values of the statistic correspond to points close to

0

the target μ .

0

2

2

The D statistic follows a χ -distribution with d degrees of freedom. Thus, a multivariate Shewhart

j

control chart for process mean, with known mean vector μ and variance-covariance matrix Σ , has the

0 0

following upper control limit:

2

U =χ (2)

CL

1−α,d

For the determination of the upper control limit α can be chosen as 0,1 %, 0,2 %, 0,5 %, or 1 % under the

consideration of a practical application. For example, the selection of 0,2 % means that there is

approximately a 0,2 % risk of a false alarm, or an average of twice in a thousand, of a plotted point

2

corresponding to the D statistic being outside of the upper control limit when the process is in-control.

j

2

This control chart is called a phase II χ control chart.

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ISO 7870-7:2020(E)

2

6.2.2 T control chart when pre-specified parameter values are unknown

When more than 20 subgroups are already obtained, and multivariate control charts are used to

monitor the process, the sample mean vector x is estimated by the average of all subgroup means. The

sample variance-covariance matrix S is estimated by the d × d average of subgroup variance-covariance

matrices. See Annex C.1.

If μ is replaced x , and Σ is replaced by S , with n>1 and x is the mean of the j-th rational subgroup, a

0 0

j

[4]

control chart can be based on the sequence of the following statistic :

21T −

Tn=−()xx Sx()−x j = 1, 2, ., m (3)

jj j

2

for the j-th subgroup. Then the Tc/(dm,,n) statistic follows an F distribution with d and (mn–m–

j 0

−1

d+1) degrees of freedom. Here, cd(,mn,)=−[(dm 11)(nm−−)]()nm−+d 1 .

0

Thus, a multivariate Shewhart control chart for process mean with unknown parameters has the

following upper control limit:

dm()−−11()n

U = F (4)

CL 11−−α ,,dmnm−+d

mn−−md+1

2

This control chart is called a phase I T -chart.

If μ is replaced by x , and Σ is replaced by S , with n>1 and x is the mean of a future rational

0 0

f

subgroup, a control chart can be based on the sequence of the following statistic:

2 T1−

Tn=−()xx Sx()−x (5)

f ff

2

Then the Tc/(dm,,n) statistic follows an F-distribution with d and (mn–m–d+1) degrees of freedom,

1

f

−1

where cd(,mn,)=+[(dm 11)(nm−−)]()nm−+d 1 and m is used to show the number of subgroups that

1

belong to phase I.

Thus, a multivariate Shewhart control chart for process mean with unknown parameters, has the

following upper control limit:

dm()+−11()n

U = F (6)

CL 11−−α ,,dmnm−+d

mn−−md+1

2

This control chart is called a phase II T -chart.

6.3 Control charts for the process mean (n=1)

2

6.3.1 χ control chart when pre-specified parameter values are known

For charts constructed using individual observations (n=1), a control chart can be based on the

sequence of the following statistic:

T

2 −1

D =−xxμμΣΣμ−μ j = 1, 2, ., m (7)

() ()

jj 00 j 0

where x is the j-th, j =1, 2, …, m, observation following N (μ ,Σ ), where μ and Σ are the known vector

j d 0 0 0 0

of means and the known variance-covariance matrix, respectively. Moreover, assume that the

2 2

observations x are not time dependent. The D statistic follows a χ -distribution with d degrees of

j j

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ISO 7870-7:2020(E)

freedom. Thus, a multivariate Shewhart control chart for process mean with known mean vector μ

0

and variance-covariance matrix Σ has the following upper control limit:

0

2

U =χ (8)

CL 1−α,d

2

This control chart is called a phase II χ control chart.

2

6.3.2 T control chart when pre-specified parameter values are unknown

When more than 20 observed vectors on multivariate characteristics are already obtained and

multivariate control charts are used to monitor the process, the sample mean vector x and the sample

variance-covariance vector S are estimated. See Annex C.2.

If μ is replaced by x , Σ is replaced by S, and x is the j-th individual observation which is not

0 0

j

independent of the estimators x and S, a control chart can be based on the sequence of the following

statistic:

21T −

T =−()xx Sx()−x j = 1, 2, ., m (9)

jj j

2

1 21()m−

2

then the Td/(m) statistic follows beta distribution with d/2 and −−d 1 degrees of

i 0

2 34m−

21−

freedom, where dm()=−()mm1 .

0

Thus, a multivariate Shewhart control chart for process mean with unknown parameters has the

following u

**...**

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