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.
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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
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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
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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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