# ISO 7870-9:2020

(Main)## Control charts — Part 9: Control charts for stationary processes

## Control charts — Part 9: Control charts for stationary processes

This document describes the construction and applications of control charts for stationary processes.

## Cartes de contrôle — Partie 9: Cartes de contrôle de processus stationnaires

### General Information

### Standards Content (Sample)

INTERNATIONAL ISO

STANDARD 7870-9

First edition

2020-06

Control charts —

Part 9:

Control charts for stationary

processes

Cartes de contrôle —

Partie 9: Cartes de contrôle de processus stationnaires

Reference number

ISO 7870-9:2020(E)

©

ISO 2020

---------------------- Page: 1 ----------------------

ISO 7870-9:2020(E)

COPYRIGHT PROTECTED DOCUMENT

© ISO 2020

All rights reserved. Unless otherwise specified, or required in the context of its implementation, no part of this publication may

be reproduced or utilized otherwise in any form or by any means, electronic or mechanical, including photocopying, or posting

on the internet or an intranet, without prior written permission. Permission can be requested from either ISO at the address

below or ISO’s member body in the country of the requester.

ISO copyright office

CP 401 • Ch. de Blandonnet 8

CH-1214 Vernier, Geneva

Phone: +41 22 749 01 11

Email: copyright@iso.org

Website: www.iso.org

Published in Switzerland

ii © ISO 2020 – All rights reserved

---------------------- Page: 2 ----------------------

ISO 7870-9:2020(E)

Contents Page

Foreword .iv

Introduction .v

1 Scope . 1

2 Normative references . 1

3 Terms and definitions, and abbreviated terms and symbols . 1

3.1 Terms and definitions . 1

3.2 Abbreviated terms and symbols . 2

3.2.1 Abbreviated terms . 2

3.2.2 Symbols . 2

4 Control charts for autocorrelated processes for monitoring process mean .3

4.1 General . 3

4.2 Residual charts . 3

4.3 Traditional control charts with adjusted control limits . 6

4.3.1 Modified EWMA chart . 6

4.3.2 Modified CUSUM chart . 8

4.4 Comparisons among charts for autocorrelated data . 8

5 Monitoring process variability for stationary processes . 9

6 Other approaches to deal with process autocorrelation .11

Annex A (informative) Stochastic process and time series .12

Annex B (informative) Performance of traditional control charts for autocorrelated data .15

Bibliography .20

© ISO 2020 – All rights reserved iii

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ISO 7870-9:2020(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 product and 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.

iv © ISO 2020 – All rights reserved

---------------------- Page: 4 ----------------------

ISO 7870-9:2020(E)

Introduction

Statistical process control (SPC) techniques are widely used in industry for process monitoring and

quality improvement. Various statistical control charts have been developed to monitor the process

mean and variability. Traditional SPC methodology is based on a fundamental assumption that process

data are statistically independent. Process data, however, are not always statistically independent from

each other. In the industry for continuous productions such as the chemical industry, most process data

on quality characteristics are self-correlated over time or autocorrelated. In general, autocorrelation

can be caused by the measurement system, the dynamics of the process, or both. In many cases, the

data can exhibit a drifting behaviour. In biology, random biological variation, for example the random

burst in the secretion of some substance that influences the blood pressure, can have a sustained effect

so that several consecutive measurements are all influenced by the same random phenomenon. In data

collection, when the sampling interval is short, autocorrelation, especially the positive autocorrelation

of the data, is a concern. Under such conditions, traditional SPC procedures are not effective and

appropriate for monitoring, controlling and improving process quality.

Autocorrelated processes can be classified in two kinds of processes, based on whether they are

stationary or nonstationary.

1) Stationary process – a direct extension of an independent and identically distributed (i.i.d.)

sequence. An autocorrelated process is stationary if it is in a state of “statistical equilibrium”. This

implies that the basic behaviour of the process does not change in time. In particular, a stationary

process has identical means and variances.

2) Nonstationary process.

Detailed information about stochastic process and time series can be found in Annex A.

To accommodate autocorrelated data, some SPC methodologies have been developed. Mainly, there are

two approaches. The first approach is to use a process residual chart after fitting a time series model or

other mathematical model to the data. Another more direct approach is to modify the existing charts,

for example by adjusting the control limits based on process autocorrelation.

The aim of this document is to outline the major process control charts for monitoring both of the

process mean and the process variance when the process is autocorrelated.

© ISO 2020 – All rights reserved v

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INTERNATIONAL STANDARD ISO 7870-9:2020(E)

Control charts —

Part 9:

Control charts for stationary processes

1 Scope

This document describes the construction and applications of control charts for stationary processes.

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, and abbreviated terms and symbols

3.1 Terms and definitions

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

3.1.1

autocovariance

internal covariance between members of series of observations ordered in time

3.1.2

control charts for autocorrelated processes

statistical process control charts applied to autocorrelated processes

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ISO 7870-9:2020(E)

3.2 Abbreviated terms and symbols

3.2.1 Abbreviated terms

ARL average run length

i.i.d. independent and identically distributed

SPC statistical process control

ACF autocorrelation function

AR(1) first order autoregressive process

EWMA exponentially weighted moving average

EWMAST exponentially weighted moving average for a stationary process

EWMS exponentially weighted mean squared deviation

CUSUM cumulative sum

3.2.2 Symbols

T index set for a stochastic process

μ true process mean

σ true process standard deviation

2 2

normal distribution with a mean of μ and variance of σ

N μσ,

()

γ autocovariance

ˆ estimator of autocovariance

γ

ρ autocorrelation

estimator of autocorrelation

ρˆ

ϕ dependent parameter of an AR(1) process

λ smoothing parameter for EWMA

r smoothing parameter for EWMS

τ time lag between two time points

2

S EWMS at t

t

2 2

S initial value of S

0 t

X random variable X at t

t

a random variable a at t in an AR(1) process

t

Δ step mean change as a multiple of the process standard deviation

arithmetic mean value of a sequence of x

x

s standard deviation of a sequence of x

ˆ prediction of X

X t

t

R residual at t

t

arithmetic mean value of R

R

t

S standard deviation of {R }

R t

Z EWMA statistic at t

t

Z initial value of Z

0 t

L value of the control limit for Z (expresses in number of standard deviation of Z )

Z t t

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ISO 7870-9:2020(E)

σ standard deviation of EWMA statistic

Z

σ standard deviation of the random variables a from white noise in an AR(1) process

a t

4 Control charts for autocorrelated processes for monitoring process mean

4.1 General

Many statisticians and statistical process control practitioners have found that autocorrelation in

process data has an impact on the performance of the traditional SPC charts. Similar to autocovariance

(see 3.1.1), autocorrelation is internal correlation between members of a series of observations ordered

in time. Autocorrelation can be caused by the measurement system, the dynamics of the process, or

both. In Annex B, the impact of positive autocorrelation on the performance of various traditional

control charts is demonstrated.

4.2 Residual charts

The residual charts have been used to monitor possible changes of the process mean. To construct a

residual chart, time series or other mathematical modelling has to be applied to the process data.

[1]

The residual chart requires modelling the process data and to obtain the process residuals . For a set

of time series data, xt;,=12,.,N , a time series or other mathematical model is established to fit the

{}

t

data. A residual at t is defined as:

ˆ

Rx=−x

tt t

where xˆ is the prediction of the time series at t based on a time series or other mathematical model.

t

Assuming that the model is true, the residuals are statistically uncorrelated to each other. Then,

traditional SPC charts such as X charts, CUSUM charts and EWMA charts can be applied to the residuals.

When an X chart is applied to the residuals, it is usually called an X residual chart. Once a change of the

mean in the residual process is detected, it is concluded that the mean of the process itself has been out-

of-control.

[2][3]

Similarly, the CUSUM residual chart and EWMA residual chart are proposed . See Reference [4] for

comparisons between residual charts and other control charts.

Advantage of the residual charts:

— a residual chart can be applied to any autocorrelated data, even if it is nonstationary. Usually, a

model is established with time series or other model fitting software.

Disadvantages of the residual charts:

— the residual charts do not have the same properties as the traditional charts. The X residual chart

for an AR(1) process (for an AR(1) process, see A.3.3) can have poor capability to detect a mean shift.

Reference [5] shows that when the process is positively autocorrelated, the X residual chart does not

perform well. Reference [6] shows that the detection capability of an X residual chart sometimes is

small comparing to that of an X chart;

— the residual charts require time series or other modelling. The user of a residual chart shall check

the validity of the model over time to reduce the mixed effect of modelling error and process change.

An example is illustrated in which the data, with a size of 50, are the daily measurements of the viscosity

[7]

of a coolant in an aluminium cold rolling process . Figure 1 shows the data with a decreasing trend. It

is suspected that the measurements are not independent. Figure 2 shows the sample autocorrelation

function (ACF) for lags from 0 to 12. For sample autocorrelation and ACF, see A.4.2 and A.5 in Annex A,

and Reference [8]. As indicated in A.5, under the assumption for an i.i.d. normal sequence, approximately

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ISO 7870-9:2020(E)

95 % of the sample autocorrelations with a lag larger than one should fall between the bounds of

±19, 650 . Based on that, the data are not independent. Reference [7] provides a model with the

predicted viscosity at a period t given by:

ˆ

xa=+bx ++cx dx +ex , t=15,., 0

tt−−12tt−−34t

Key

X observation

Y viscosity

Figure 1 — Example

Key

X lag

Y autocorrelation

Figure 2 — Sample autocorrelations for the series of daily measurements of viscosity and

an approximate 95 % confidence band

4 © ISO 2020 – All rights reserved

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ISO 7870-9:2020(E)

ˆ

For the estimates of a, b, c, and d given in Reference [7], the residuals are calculated by Rx=−x ,

tt t

t = 1,., 46 which are shown in Figure 3. To test whether the residuals are independent from each other,

the ACF with a confidence band is again applied and shown in Figure 4. Since the residuals are

determined to be not autocorrelated, a X chart with 3σ control limits (RS±3 , where R is the average

R

of {R } and S is the standard deviation of {R }) applies to the residuals, as shown in Figure 3. It is

t R t

concluded that the mean of the residuals, as well as the process, is in control.

Key

X time

Y residual

Figure 3 — Residuals of the viscosity series and the X chart with 3σ control limits

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ISO 7870-9:2020(E)

Key

X lag

Y autocorrelation

Figure 4 — Sample autocorrelation of the residuals of viscosity series and

an approximate 95 % confidence band

4.3 Traditional control charts with adjusted control limits

4.3.1 Modified EWMA chart

Comparing to the residual charts, a more direct approach is to modify the existing charts by adjusting

the control limits without time series modelling. Some methods based on this approach, however, are

[9]

restricted to specific processes, for example AR(1) processes . Reference [10] proposes monitoring

EWMA for a stationary process, an EWMAST chart, which can be applied to a stationary process in

[10]

general. The chart is constructed by charting the EWMA statistic :

ZZ=−()1 λλ+ X (1)

tt−1 t

where

Z = μ is the process mean;

0

λ is the smoothing constant (0 < λ ≤ 1).

2

Assume that the process Xt;,=12,.,N is stationary with mean μ and variance σ . When t is large,

{}

t

the variance of Z is approximated by:

t

M

λ kM2 −k

()

22

σ ≈ σρ12+ k 11−λλ−−1 (2)

()() ()

z ∑

2−λ

k=1

where M is an integer and ρ()k is the process autocorrelation at lag of k. Note that when the process is

2

not autocorrelated, σ is of the same form as that for the traditional EWMA chart. Assuming that X is

z t

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ISO 7870-9:2020(E)

normally distributed, Z is also normally distributed with a mean of μ. The EWMAST chart is constructed

t

by charting Z . The centre line is at μ and the L σ control limits are given by:

t Z

μσ±L .

zz

[10]

In general, λ = 0,2 is recommended , and L usually equals two or three. When μ, σ and the

Z

autocorrelations are unknown, they are usually estimated by the arithmetic mean, x , sample standard

ˆ

deviation, s, and sample autocorrelations, ρ k , respectively based on some historical data of {X }

()

t

when the process is under control. When a set of historical data are used to estimate the autocorrelations,

some rules of thumb can be followed. Reference [11] (p. 32) suggests that useful estimates of ρ(k) can

only be made if the data size N is roughly 50 or more and k ≤ N/4. Thus, M in Formula (2) should be large

enough to make the approximation in Formula (2) usable and at the same time less than N/4 to avoid

large estimation errors of autocorrelations. Based on simulation, when N ≥ 100, M = 25 is

[10]

recommended .

2

An example is illustrated, in which data from an AR(1) process with φ = 0,5, process variance σ = 1,

and length of 200 are simulated. The white noise (see A.3.2) is normally distributed. The process mean

is zero for the first 100 observations. Beginning at the observation number 101, the process mean has a

step mean change from 0 to 1 or 1σ. The plot of the simulated data is shown in Figure 5.

Key

X time

Figure 5 — Realization of the AR(1) process used to illustrate the EWMAST chart

Treating the period of the first 100 data points as stationary, the mean, the process standard deviation,

and the sample autocorrelations are estimated. x=−01, 0 , s = 0,91, and ρˆ()k ,(,k=12., 5) are obtained.

ˆ

With M = 25 and λ = 0,2 in Formula (2), the standard deviation of Z is estimated by σ = 02, 4 . Figure 6

t

z

shows the EWMAST chart with the centre line at x =−01, 0 and the 3σ control limits given by

ˆ

x ±=3σ −0, 81; 0,60 . The chart gives a signal indicating a mean increase starting at observation

()

z

number 110.

© ISO 2020 – All rights reserved 7

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ISO 7870-9:2020(E)

Key

X time

Y EWMA

Figure 6 — EWMAST chart applies to the simulated data

with a mean increase displayed in Figure 5

4.3.2 Modified CUSUM chart

Reference [12] considers charting the raw data directly by a CUSUM chart when the process

autocorrelation is low. When the autocorrelation is high, the use of transformed observations is

considered. Other approaches are proposed to apply modified CUSUM charts to AR(1) processes or

[9][13]

some other time series .

4.4 Comparisons among charts for autocorrelated data

There are comparisons among some control charts for autocorrelated data. References [9] and [4]

compare the X chart, X residual chart, CUSUM residual chart, EWMA residual chart, and EWMAST chart

for stationary AR(1) processes by simulations. The EWMAST chart performs better than the

CUSUM residual and EWMA residual charts. Overall, it also performs better than the X chart and

X residual chart. The comparisons also show that the CUSUM residual and EWMA residual charts

perform almost the same. The CUSUM residual an

**...**

FINAL

INTERNATIONAL ISO/FDIS

DRAFT

STANDARD 7870-9

ISO/TC 69/SC 4

Control charts —

Secretariat: DIN

Voting begins on:

Part 9:

2020-04-13

Control charts for stationary

Voting terminates on:

processes

2020-06-08

Cartes de contrôle —

Partie 9: Cartes de contrôle des processus stationnaires

RECIPIENTS OF THIS DRAFT ARE INVITED TO

SUBMIT, WITH THEIR COMMENTS, NOTIFICATION

OF ANY RELEVANT PATENT RIGHTS OF WHICH

THEY ARE AWARE AND TO PROVIDE SUPPOR TING

DOCUMENTATION.

IN ADDITION TO THEIR EVALUATION AS

Reference number

BEING ACCEPTABLE FOR INDUSTRIAL, TECHNO-

ISO/FDIS 7870-9:2020(E)

LOGICAL, COMMERCIAL AND USER PURPOSES,

DRAFT INTERNATIONAL STANDARDS MAY ON

OCCASION HAVE TO BE CONSIDERED IN THE

LIGHT OF THEIR POTENTIAL TO BECOME STAN-

DARDS TO WHICH REFERENCE MAY BE MADE IN

©

NATIONAL REGULATIONS. ISO 2020

---------------------- Page: 1 ----------------------

ISO/FDIS 7870-9:2020(E)

COPYRIGHT PROTECTED DOCUMENT

© ISO 2020

All rights reserved. Unless otherwise specified, or required in the context of its implementation, no part of this publication may

be reproduced or utilized otherwise in any form or by any means, electronic or mechanical, including photocopying, or posting

on the internet or an intranet, without prior written permission. Permission can be requested from either ISO at the address

below or ISO’s member body in the country of the requester.

ISO copyright office

CP 401 • Ch. de Blandonnet 8

CH-1214 Vernier, Geneva

Phone: +41 22 749 01 11

Fax: +41 22 749 09 47

Email: copyright@iso.org

Website: www.iso.org

Published in Switzerland

ii © ISO 2020 – All rights reserved

---------------------- Page: 2 ----------------------

ISO/FDIS 7870-9:2020(E)

Contents Page

Foreword .iv

Introduction .v

1 Scope . 1

2 Normative references . 1

3 Terms and definitions, and abbreviated terms and symbols . 1

3.1 Terms and definitions . 1

3.2 Abbreviated terms and symbols . 2

3.2.1 Abbreviated terms . 2

3.2.2 Symbols . 2

4 Control charts for autocorrelated processes for monitoring process mean .3

4.1 General . 3

4.2 Residual charts . 3

4.3 Traditional control charts with adjusted control limits . 6

4.3.1 Modified EWMA chart . 6

4.3.2 Modified CUSUM chart . 8

4.4 Comparisons among charts for autocorrelated data . 8

5 Monitoring process variability for stationary processes . 9

6 Other approaches to deal with process autocorrelation .11

Annex A (informative) Stochastic process and time series .12

Annex B (informative) Performance of traditional control charts for autocorrelated data .15

Bibliography .20

© ISO 2020 – All rights reserved iii

---------------------- Page: 3 ----------------------

ISO/FDIS 7870-9:2020(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 product and 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.

iv © ISO 2020 – All rights reserved

---------------------- Page: 4 ----------------------

ISO/FDIS 7870-9:2020(E)

Introduction

Statistical process control (SPC) techniques are widely used in industry for process monitoring and

quality improvement. Various statistical control charts have been developed to monitor the process

mean and variability. Traditional SPC methodology is based on a fundamental assumption that process

data are statistically independent. Process data, however, are not always statistically independent from

each other. In the industry for continuous productions such as the chemical industry, most process data

on quality characteristics are self-correlated over time or autocorrelated. In general, autocorrelation

can be caused by the measurement system, the dynamics of the process, or both. In many cases, the

data can exhibit a drifting behaviour. In biology, random biological variation, for example the random

burst in the secretion of some substance that influences the blood pressure, can have a sustained effect

so that several consecutive measurements are all influenced by the same random phenomenon. In data

collection, when the sampling interval is short, autocorrelation, especially the positive autocorrelation

of the data, is a concern. Under such conditions, traditional SPC procedures are not effective and

appropriate for monitoring, controlling and improving process quality.

Autocorrelated processes can be classified in two kinds of processes, based on whether they are

stationary or nonstationary.

1) Stationary process – a direct extension of an independent and identically distributed (i.i.d.)

sequence. An autocorrelated process is stationary if it is in a state of “statistical equilibrium”. This

implies that the basic behaviour of the process does not change in time. In particular, a stationary

process has identical means and variances.

2) Nonstationary process.

Detailed information about stochastic process and time series can be found in Annex A.

To accommodate autocorrelated data, some SPC methodologies have been developed. Mainly, there are

two approaches. The first approach is to use a process residual chart after fitting a time series model or

other mathematical model to the data. Another more direct approach is to modify the existing charts,

for example by adjusting the control limits based on process autocorrelation.

The aim of this document is to outline the major process control charts for monitoring both of the

process mean and the process variance when the process is autocorrelated.

© ISO 2020 – All rights reserved v

---------------------- Page: 5 ----------------------

FINAL DRAFT INTERNATIONAL STANDARD ISO/FDIS 7870-9:2020(E)

Control charts —

Part 9:

Control charts for stationary processes

1 Scope

This document describes the construction and applications of control charts for stationary processes.

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, and abbreviated terms and symbols

3.1 Terms and definitions

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

3.1.1

covariance stationary time series

weakly stationary time series

time series characterized by a constant process mean, a constant process variance and a covariance

function which only depends on the difference of the time indices

3.1.2

autocovariance

internal covariance between members of series of observations ordered in time

3.1.3

control charts for autocorrelated processes

statistical process control charts applied to autocorrelated processes

© ISO 2020 – All rights reserved 1

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ISO/FDIS 7870-9:2020(E)

3.2 Abbreviated terms and symbols

3.2.1 Abbreviated terms

ARL Average run length

i.i.d. independent and identically distributed

SPC Statistical process control

ACF Autocorrelation function

AR(1) First order autoregressive process

EWMAST chart exponentially weighted moving average chart for a stationary process

EWMS chart exponentially weighted mean square chart

3.2.2 Symbols

T index set for a stochastic process

μ true process mean

σ true process standard deviation

2

2

N μσ, normal distribution with a mean of μ and variance of σ

()

γ autocovariance

estimator of autocovariance

γˆ

ρ autocorrelation

ˆ estimator of autocorrelation

ρ

ϕ dependent parameter of an AR(1) process

λ smoothing parameter for EWMA

r smoothing parameter for EWMS

τ time lag between two time points

2

S EWMS at t

t

2 2

S initial value of S

0 t

X random variable X at t

t

a random variable a at t in an AR(1) process

t

Δ step mean change as a multiple of the process standard deviation

arithmetic mean value of a sequence of x

x

s standard deviation of a sequence of x

ˆ prediction of X

X

t

t

R residual at t

t

arithmetic mean value of R

R

t

S standard deviation of {R }

R i

Z EWMA statistic at t

t

Z initial value of Z

0 t

L value of the control limit for Z (expresses in number of standard deviation of Z )

Z t t

σ standard deviation of EWMA statistic

Z

σ standard deviation of the random variables a from white noise in an AR(1) process

a t

2 © ISO 2020 – All rights reserved

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ISO/FDIS 7870-9:2020(E)

4 Control charts for autocorrelated processes for monitoring process mean

4.1 General

Many statisticians and statistical process control practitioners have found that autocorrelation in

process data has an impact on the performance of the traditional SPC charts. Similar to autocovariance

(see 3.1.2), autocorrelation is internal correlation between members of a series of observations ordered

in time. Autocorrelation can be caused by the measurement system, the dynamics of the process, or

both. In Annex B, the impact of positive autocorrelation on the performance of various traditional

control charts is demonstrated.

4.2 Residual charts

The residual charts have been used to monitor possible changes of the process mean. To construct a

residual chart, time series or other mathematical modelling has to be applied to the process data.

[1]

The residual chart requires modelling the process data and to obtain the process residuals . For a set

of time series data, xt;,=12,.,N , a time series or other mathematical model is established to fit the

{}

t

data. A residual at t is defined as:

Rx=−xˆ

tt t

ˆ

where x is the prediction of the time series at t based on a time series or other mathematical model.

t

Assuming that the model is true, the residuals are statistically uncorrelated to each other. Then,

traditional SPC charts such as X charts, CUSUM, and EWMA charts can be applied to the residuals.

When an X chart is applied to the residuals, it is usually called an X residual chart. Once a change of the

mean in the residual process is detected, it is concluded that the mean of the process itself has been out-

of-control.

[2][3]

Similarly, the CUSUM residual chart and EWMA residual chart are proposed . See Reference [4] for

comparisons between residual charts and other control charts.

Advantage of the residual charts:

— a residual chart can be applied to any autocorrelated data, even if it is nonstationary. Usually, a

model is established with time series or other model fitting software.

Disadvantages of the residual charts:

— the residual charts do not have the same properties as the traditional charts. The X residual chart

for an AR(1) process (for an AR(1) process, see A.3.3) can have poor capability to detect a mean shift.

Reference [5] shows that when the process is positively autocorrelated, the X residual chart does not

perform well. Reference [6] shows that the detection capability of an X residual chart sometimes is

small comparing to that of an X chart;

— the residual charts require time series or other modelling. The user of a residual chart shall check

the validity of the model over time to reduce the mixed effect of modelling error and process change.

An example is illustrated in which the data, with a size of 50, are the daily measurements of the viscosity

[7]

of a coolant in an aluminium cold rolling process . Figure 1 shows the data with a decreasing trend. It

is suspected that the measurements are not independent. Figure 2 shows the sample autocorrelation

function (ACF) for lags from 0 to 12. For sample autocorrelation and ACF, see A.4.2 and A.5 in Annex A,

and Reference [8]. As indicated in A.5, under the assumption for an i.i.d. normal sequence, approximately

95 % of the sample autocorrelations with a lag larger than one should fall between the bounds of

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±19, 650 . Based on that, the data are not independent. Reference [7] provides a model with the

predicted viscosity at a period t given by:

xaˆ =+bx ++cx dx +ex , t=15,., 0

tt−−12tt−−34t

Key

X observation

Y viscosity

Figure 1 — Example

Key

X lag

Y autocorrelation

Figure 2 — Sample autocorrelations for the series of daily measurements of viscosity and

an approximate 95 % confidence band

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ˆ

For the estimates of a, b, c, and d given in Reference [7], the residuals are calculated by Rx=−x ,

tt t

t = 1,., 46 which are shown in Figure 3. To test whether the residuals are independent from each other,

the ACF with a confidence band is again applied and shown in Figure 4. Since the residuals are

determined to be not autocorrelated, a X chart with 3σ control limits (RS±3 , where R is the average

R

of {R } and S is the standard deviation of {R }) applies to the residuals, as shown in Figure 3. It is

t R t

concluded that the mean of the residuals, as well as the process, is in control.

Key

X time

Y residual

Figure 3 — Residuals of the viscosity series and the X chart with 3σ control limits

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Key

X lag

Y autocorrelation

Figure 4 — Sample autocorrelation of the residuals of viscosity series and

an approximate 95 % confidence band

4.3 Traditional control charts with adjusted control limits

4.3.1 Modified EWMA chart

Comparing to the residual charts, a more direct approach is to modify the existing charts by adjusting

the control limits without time series modelling. Some methods based on this approach, however, are

[9]

restricted to specific processes, for example AR(1) processes . Reference [10] proposes monitoring

EWMA for a stationary process (EWMAST) chart, which can be applied to a stationary process in

[10]

general. The chart is constructed by charting the EWMA statistic :

ZZ=−()1 λλ+ X (1)

tt−1 t

where

Z = μ is the process mean;

0

λ is the smoothing constant (0 < λ ≤ 1).

2

Assume that the process Xt;,=12,.,N is stationary with mean μ and variance σ . When t is large,

{}

t

the variance of Z is approximated by:

t

M

λ kM2 −k

()

22

σ ≈ σρ12+ k 11−λλ−−1 (2)

()() ()

z ∑

2−λ

k=1

where M is an integer and ρ()k is the process autocorrelation at lag of k. Note that when the process is

2

not autocorrelated, σ is of the same form as that for the traditional EWMA chart. Assuming that X is

z t

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normally distributed, Z is also normally distributed with a mean of μ. The EWMAST chart is constructed

t

by charting Z . The centre line is at μ and the L σ control limits are given by:

t Z

μσ±L .

zz

[10]

In general, λ = 0,2 is recommended , and L usually equals two or three. When μ, σ and the

Z

autocorrelations are unknown, they are usually estimated by the arithmetic mean, x , sample standard

ˆ

deviation, s, and sample autocorrelations, ρ k , respectively based on some historical data of {X }

()

t

when the process is under control. When a set of historical data are used to estimate the autocorrelations,

some rules of thumb can be followed. Reference [11] (p. 32) suggests that useful estimates of ρ(k) can

only be made if the data size N is roughly 50 or more and k ≤ N/4. Thus, M in Formula (2) should be large

enough to make the approximation in Formula (2) usable and at the same time less than N/4 to avoid

large estimation errors of autocorrelations. Based on simulation, when N ≥ 100, M = 25 is

[10]

recommended .

2

An example is illustrated, in which data from an AR(1) process with φ = 0,5, process variance σ = 1,

and length of 200 are simulated. The white noise (see A.3.2) is normally distributed. The process mean

is zero for the first 100 observations. Beginning at the observation number 101, the process mean has a

step mean change from 0 to 1 or 1σ. The plot of the simulated data is shown in Figure 5.

Key

X time

Figure 5 — Realization of the AR(1) process used to illustrate the EWMAST chart

Treating the period of the first 100 data points as stationary, the mean, the process standard deviation,

and the sample autocorrelations are estimated. x=−01, 0 , s = 0,91, and ρˆ()k ,(,k=12., 5) are obtained.

ˆ

With M = 25 and λ = 0,2 in Formula (2), the standard deviation of Z is estimated by σ = 02, 4 . Figure 6

t

z

shows the EWMAST chart with the centre line at x =−01, 0 and the 3σ control limits given by

ˆ

x ±=3σ −0, 81; 0,60 . The chart gives a signal indicating a mean increase starting at observation

()

z

number 110.

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Key

X time

Y EWMA

Figure 6 — EWMAST chart applies to the simulated data

with a mean increase displayed in Figure 5

4.3.2 Modified CUSUM chart

Reference [12] considers charting the raw data directly by a CUSUM chart when the process

autocorrelation is low. When the autocorrelation is high, the use of transformed observations is

considered. Other approaches are proposed to apply modified CUSUM charts to AR(1) processes or

[9][13]

some other time series .

4.4 Comparisons among charts for autocorrelated data

There are comparisons among some control charts for autocorrelated data. References [9] and [4]

compare the X chart, X residual chart, CUSUM residual chart, EWMA residual chart, and EWMAST chart

for stationary AR(1) processes by simulations. The EWMAST chart performs better than the

CUSUM residual and EWMA residual charts. Overall, it also performs better than the X chart and

X residual chart. The comparisons also show that the CUSUM residual and EWMA residual charts

perform almost the same. The CUSUM residual and EWMA residual charts perform better than

the X residual chart when the process autocorrelation is not strong. On the contrary, when the

autocorrelation is strong, the X residual chart performs better than the other residual charts. When the

process autocorrelation is very strong, i.e. the process is near nonstationary, the EWMAST chart still

performs relatively better than other charts.

An obvious advantage of using EWMAST chart is that there is no need to build a time series model for

stationary process data. The implementation of an EWMAST chart only requires the estimation of the

process mean, standard deviation, and autocorrelations obtained when the process is under control. In

summary, when the process is autocorrelated and stationary, it is recommended to use EWMAST chart

to monitor the process mean.

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5 Monitoring process variability for stationary processes

Reference [14] considers two control charts for monitoring the process variability: one is based on the

exponentially weighted mean squared deviation from the target, called the exponentially weighted

mean square (EWMS) chart, and the other is based on an exponentially weighted moving variance in

which the process mean is estimated using an EWMA chart of the observations, called the exponentially

weighted moving variance (EWMV) chart.

2

Assume that {,Xt=12, .} is a process with mean μ and process variance σ and jointly normally

t

distributed. The exponential weighted moving mean square error is defined as:

2 22

Sr=−()1 Sr+−()X μ

tt−1 t

where

t = 1,2,. ;

r is a smoothing parameter (0 < r ≤ 1).

22 2

Usually, let S =σ be the process variance. From the above, S is an estimator of the process mean

0 t

2

22

square error at time t. The EWMS chart is constructed by charting S with the centre line at S =σ ,

t

0

2

and the control limits are determined by σ and a Chi-squared distribution with its degrees of freedom

being a function of r for each t. Reference [14] proposes applying EWMS chart to an i.i.d. sequence and

the processes, which can be represented as an AR(1) process plus white noise. Reference [15] proposes

using residual chart to monitoring possible variance changes for a processes which is an AR(1) process

plus white noise.

Reference [16] extends the EWMS chart to the case of stationary processes. Combining with the

EWMAST chart, an EWMS chart can be used to detect possible variance change for a stationary process.

For illustration on the EWMS chart, a constructed example is presented. A realization from an AR(1)

process is generated with mean μ = 0 and the dependence parameter ϕ = 0,5. The process variance

2 2 2

is σ = 1 from t = 1 to t = 150, σ = 0,5 from t = 151 to t = 300, and σ = 2 from t = 301 to t = 450. The

observed process is displayed in Figure 7.

An EWMAST chart is applied to the simulated data with the chart parameter λ = 0,2. The standard

deviation of the EWMA statistic in the EWMAST chart based on Formula (2) is 0,51. The chart with 3σ

control limits, shown in Figure 8, shows that although there are nine points between t from 372 to

448 out of the control limits, the process mean seems stable. Thus, the process is treated to have a

constant mean.

For the EWMS chart, r = 0,05 and α = 0,05 are chosen, which give the asymptotic lower and upper control

limits to be 0,52 and 1,64, respectively. Decreases in the mean square error are detected from t = 158

and other points, and increases from t = 329 and other points, as shown in Figure 9. Since it is shown in

Figure 8 that the process mean seems stable, it is concluded that the process variance changed.

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Key

X time

Y X

Figure 7 — Realization of the AR(1) process used to illustrate the EWMS procedure where

the process mean is fixed at 0, but the process variance changes two times

Key

X time

Y EWMA

Figure 8 — EWMAST chart with control limits for the time series displayed in Figure 7

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Key

X time

Y EWMS

Figure 9 — EWMS chart with control limits for the time series displayed in Figure 7

6 Other approaches to deal with process autocorrelation

In Clauses 4 and Clause 5, various process control charts to accommodate the autocorrelation of the

process data are discussed. As an alternative to accommodating, the effect of the autocorrelation can

be reduced by some data treatment mechanism. Reference [17] discusses the effects of the choice of

the sampling interval on some process data. When the process is stationary and the samples are taken

less frequently in time, the autocorrelation of the sampled data decreases. Thus, when the sampling

interval is sufficiently large, the data appear to be uncorrelated. However, this approach discards the

intermediate data and therefore increases the possibility of missing important events in the process.

Instead of choosing a large sampling interval, moving averages of process with a fixed window size

can be formed. Reference [18] shows that, when a process is stationary and satisfies some regularity

conditions, the non-overlapping means or batch means are asymptotically independent and normally

distributed. Thus, when the batch size is large enough, the batch means can be treated as white

noise. For some specific stationary processes, numerous papers discuss the process behaviour of the

subsample means or batch means, and the related charts for batch means. In Reference [19], the effect

of using generalized moving averages of a stationary process to reduce its autocorrelation and its

applications to process control charts are discussed.

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Annex A

(informative)

Stochastic process and time series

A.1 General

[8]

A stochastic process {;Xt∈T} is a collection of random variables, where T is an index set . When T

t

represents time, the stochastic process is referred to as a time series. When T takes on a discrete set of

values, e.g. T =±{01,,, ±2 .} , the process is said to be a discrete time series. In this document, only

discrete time series with equal time space are considered. A discrete time series xx,,.,x can be

12 n

viewed as the values taken by a sequence of random variables XX,,.,X . The sequence of xx,,.,x

12 n 12 n

is called a realization of XX,,.,X .

12 n

A.2 Autocovariance and

**...**

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