ISO 7870-9:2020

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 2020
© ISO 2020
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ii © ISO 2020 – All rights reserved

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
Foreword
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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.
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iv © ISO 2020 – All rights reserved

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.
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
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
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
2 © ISO 2020 – All rights reserved

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
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

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
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;
λ is the smoothing constant (0 < λ ≤ 1).
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 ≈ σρ12+ 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
not autocorrelated, σ is of the same form as that for the traditional EWMA chart. Assuming that X is
z t
6 © ISO 2020 – All rights reserved

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 .
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.
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
...