Efficient Monitoring of Autoregressive and Moving Average Process
using HWMA Control Chart
YUPAPORN AREEPONG1,*, SAOWANIT SUKPARUNGSEE1,
TANAPAT ANUSAS-AMORNKUL2
1Department of Applied Statistics, Faculty of Applied Science,
King Mongkut’s University of Technology North Bangkok,
Bangkok 10800,
THAILAND
2Department of Computer and Information Science, Faculty of Applied Science,
King Mongkut’s University of Technology North Bangkok,
Bangkok 10800,
THAILAND
*Corresponding Author
Abstract: - Quality control is an essential process for manufacturing and industry because it enhances product
quality, consumer satisfaction, and overall profitability. Among many other statistical process control tools,
quality practitioners typically employ control charts to monitor the industrial process and detect production
changes. Control charts are widely used to detect flaws in many applications, such as distributed circuits and
systems, electronic devices, and systems and signals. In this study, we derived an explicit formula for Average
Run Length (ARL) of the Homogenously Weighted Moving Average control chart (HWMA) under the ARMA
(p,q) process. The accuracy was checked using the numerical integral equation (NIE) technique. The finding
showed that the explicit formulas and numerical solutions presented an outstanding level of agreement.
However, the computational time for the explicit formulas was approximately one second, which was less than
that required for the NIE. Moreover, the performance efficiency of the HWMA control chart is compared with
the cumulative sum control chart for ARMA (p, q) processes including ARMA (2,1), ARMA (2,3), and ARMA
(1,1) processes. The results found that the HWMA control chart performance is found to be preferable to the
CUSUM control chart performance. Additionally, the explicit formula of the HWMA control chart was
implemented in a practical application of the count of nonconformities in printed circuit boards (PCBs).
Key-Words: - Average Run Length, ARMA process, Homogenously Exponentially Weighted Moving Average,
numerical integral equation, explicit formula, exponential white noise.
Received: June 25, 2023. Revised: December 13, 2023. Accepted: January 15, 2024. Published: March 14, 2024.
1 Introduction
Control charts are frequently employed in the field
of statistical process control (SPC) to identify and
rectify process variations. Control charts have
extensive use in various applications, such as
electronics, distributed circuits and systems,
electronic devices, and systems and signals, to
detect changes in production, [1], [2]. In addition,
control charts play a crucial role in the
manufacturing of electronic components as they
enable early detection of deviations from critical
parameters. This enables the implementation of
corrective measures that ensure the quality of the
product. The study [3], is indeed one of the most
widely used tools in SPC. This control chart is
designed to monitor and control the variability of a
process over time. It is a fundamental tool in
quality management and process improvement. The
study [4], proposed a Cumulative Sum (CUSUM)
control chart, which can detect minute shifts or
changes in a process. Traditional Shewhart control
chart may not be as sensitive to small changes as it
was intended to detect significant shifts in the
process mean or dispersion. When identifying and
signaling the presence of smaller process changes,
the CUSUM control chart is especially useful.
Later, the study [5], introduced the Exponentially
Weighted Moving Average (EWMA) control chart.
It was designed to detect shifts in the process mean,
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but it focused on recent observations. This makes it
particularly sensitive to small, persistent shifts in
the process. Later, Homogeneous Weighted
Moving Average (HWMA) control charts were
recently developed by [6] through the placement of
weights on observations, which differ from the
ones used in EWMA. In addition, as a competitive
alternative to the EWMA control chart, the HWMA
control chart was designed to detect minor to
moderate mean changes for zero-state performance.
Additionally, the efficacy of HWMA control charts
is examined in terms of the effects of non-normal
data. In addition [6], demonstrated that the HWMA
control chart exhibited superior performance to
both the CUSUM and EWMA control charts. The
authors therefore intended to assess the control
charts' ability to detect process changes by
providing a precise formula for the average run
length of a HWMA control chart.
In industrial processes, statistical process control is
typically implemented to identify process variation
and measure process performance. Autocorrelated
observations have been observed in numerous
processes, including chemical manufacturing,
production process, wood product manufacturing,
and waste-water processing, [7]. Consequently, the
autocorrelation data will be associated with a
statistical model. The commonly used model in the
real world is the Autoregressive Moving Average
(ARMA) model, derived from [8] and is
particularly applicable to stationary data. It is
critical that the appropriate control charts can be
applied to these data. The research presents a
special case of exponential white noise, which had
previously been investigated by [9], [10], [11].
The performance of control charts can be
compared with the average run length (ARL).
Numerous researchers have demonstrated that the
ARL can be examined through a variety of
approaches. The study [12], evaluated the ARL
using the numerical integral equation for the
CUSUM control chart. This research studied the
autocorrelation data based on the ARFI process
with exogenous variables. Later, the run length
distribution is evaluated using the Markov chain
and integral equation techniques to determine the
cumulative sum (CUSUM) and the EWMA control
charts, [13]. The outcome demonstrated that both
methods gave the same approximations for the
ARL if the integral in the integral equation is
approximated using the product midpoint rule.
ARL was proposed in the study [14], for the
CUSUM control chart based on SAR(P)L with a
trend process. The explicit formula outperformed
the numerical integration regarding computational
time. To identify variations in the mean of a long-
memory model with exponential white noise
running on a CUSUM control chart, the study [15],
compared the abilities of the ARL derivations
produced through the NIE approach and explicit
formulations. When observations were seasonal
autoregressive models with an exogenous variable,
SARX(P,r)L with exponential white noise, the study
[16], produced the explicit formula for the CUSUM
control chart. In addition, the numerical integral
equation approximates the numerical integral
equation of ARL using the Midpoint, Trapezoidal,
Simpson, and Gaussian rules. The study [17],
introduced a novel explicit formula for ARL
utilizing the Fredholm integral equation method in
the setting of an EWMA control chart that focused
on the MAX(q,r) process. The study [18], proposed
an exact formula for ARL on the EWMA control
chart under the MAX(q,r) process. The study [19],
constructed the explicit formula for the average run
length (ARL) of the CUSUM control chart based
on a SAR(P)L model with an exogenous variable.
For long-memory models, such as fractionally
integrated MAX processes (FIMAX) on the
EWMA control chart, the study [20], provided an
approximation numerical evaluation of the ARL.
The study [21], focused on the Moving Average
with exogenous factors (MAX) process to provide
the closed-form formulas of ARL for a two-sided
Extended EWMA scheme. Furthermore, the ARL
using the explicit formula was compared using the
numerical integral equation (NIE) method for
verification. The study, [22], developed a
mathematical expression and NIE method to
calculate the ARL for the MA(q) process on a
Double EWMA scheme.
Finally, this research is to develop a precise
analytical expression for the ARL on the HWMA
design and focus on the Autoregressive Moving
Average (ARMA) model with order p and q.
Furthermore, a comprehensive evaluation of the
effectiveness of the CUSUM chart is also
presented. Moreover, the efficacy of the HWMA
chart has been studied using the dataset on the
count of nonconformities in printed circuit boards.
2 Materials and Methods
2.1 The Homogenously Weighted Moving
Average Control Chart
The statistical properties of the HWMA control
chart which proposed by [6], can be mathematically
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represented using the recursive formula, as shown
in Eq.(1).
1
(1 ) ,
t t t
H Y Y
for
1,2,3,...t
(1)
where
t
Y
is a sequence of the ARMA(p,q) process
with exponential white noise, and the starting value
0
Y
is an initial value.
The control limits of the HWMA control chart
comprise
Upper and Lower control limits are:
2
2
1
22
2
1
,1
(1 )
[ ], 1
( 1)
t
Bt
n
UCL
Bt
nt
2
2
1
22
2
1
,1
(1 )
[ ], 1
( 1)
t
Bt
n
LCL
Bt
nt
where
1
B
represents the width of the control limits.
The HWMA stopping time (
h
) is defined as
0; ,
bt
t H b
for
.b
where
b
is the stopping time and
b
is the UCL.
2.2 The Cumulative Sum Control Chart
The Cumulative Sum (CUSUM) control chart,
which, [4], developed, is a quality control tool
utilized to identify small differences in the process
mean. The statistics
()
t
C
of the CUSUM control
chart can be mathematically represented as follows,
utilizing the algorithm described in Eq. (2).
1, 1, 2,3,..
t t t
C C Y a t
(2)
where
a
is non-zero constant,
0
C
is the initial
value of CUSUM;
[0, ]l
and the stopping time
of the CUSUM control chart is defined as
{ 0; }
st
t C l
and
l
is the UCL.
3 The ARL of HWMA Control Chart
3.1 The Explicit formula of ARL for
ARMA(p,q) process
An ARMA(p,q) process can be derived as
0 1 1 2 2 1 1
22
...
...
t t t p t p t t
t q t q
Y Y Y Y
(3)
where
t
Y
is a sequence of the ARMA(p,q) with
exponential white noise,
i
is an autoregressive
parameter,
i
is moving average parameter, the
starting value
0
Y
is an initial value;
[0, ]b
where
b
is a control limit of the HWMA control
chart.
From the recursion of HWMA statistics in Eq. (1),
1
(1 )
t t t
H Y Y
and
0 1 1 2 2 1 1
...
t t t p t p t t
Y Y Y Y
22
...
t q t q
Therefore, the HEWMA control chart for the
ARMA(p,q) process can be written as,
0 1 1 2 2 1 1
( ...
t t t p t p t t
H Y Y Y
2 2 1
... ) (1 )
t q t q t
Y
For t=1,
1 0 1 0 2 1 1 1 0 2 1
( ... pp
H Y Y Y
1 1 0
... ) (1 )
qq Y
Let
0 1 0 2 1 1 1 0 2 1 1
... ...
p p q q
V Y Y Y
Consider the in-control process, given LCL=0,
UCL=
b
and initial value
0
Y
that is
0t
Hb
10
0 (1 )V Y b
The change-point time is studied. Therefore,
()J
can be expressed by Fredholm integral equation of
the second kind, [22], as follows:
1
0
( ) 1 ( (1 ) ) ( )
bV
J J V y f y dy
(4)
Let
(1 ) ,w V y
then
1.dy dw
After changing the variable in (4), it can be
rewritten as
1
1[]
0
11
( ) 1 ( )
wV
b
J J w e dw
Since
1
is
()Exp
, then
1
( ) .
y
f y e
Thus,
1
0
1
( ) 1 ( )
Vw
b
e
J J w e dw
Setting that
1
()
V
e
O
and
0
1
()
w
b
G J w e dw
,
( ) 1 ( ) .J O G
(5)
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Since
0
1
( ) ,
w
b
G J w e dw
0
1 ( )
bw
G O w G e dw
00
V
bb
ww
Ge
e dw e dw
.
[ 1]
[1 ( 1)]
b
Vb
e
G
ee
Substituting G in (5), we have
(1 )
[ 1]
( ) 1 .
1 ( 1)
bV
Vb
ee
J
ee
0 1 0 2 1 1 1 0 2 1 1
0 1 0 2 1 1 1 0 2 1 1
(1 ) ( ... ... )
( ... ... )
[ 1]
( ) 1 .
1 ( 1)
p p q q
p p q q
Y Y Y
b
Y Y Y b
ee
J
ee
(6)
The Banach Fixed-point Theorem establishes a
mathematical justification for the validity of the
ARL equation by guaranteeing the existence of a
unique solution to the integral equation for
specified formulas. Let
F
be an operation on the
set of all continuous functions, as defined by
1
1[]
0
11
( ) 1 ( )
wV
b
J J w e dw
(7)
where
0 1 0 2 1 1 1 0 2 1 1
... ...
p p q q
V Y Y Y
According to Banach’s Fixed-point Theorem, if an
operator
F
is a contraction, the fixed-point
equation
( ( )) ( )F J J
has a unique solution. To
establish the existence and uniqueness of the
solution for Eq. (6), the following theorem can
potentially be employed.
Theorem 1: Banach’s Fixed-point Theorem
Let
( , )Xd
define on a complete metric space and
:F X X
satisfy the conditions of a contraction
mapping with contraction constant
01r
such
that
1 2 1 2
( ) ( ) ,F J F J r J J
12
,.S S X
Then,
there exists a unique
()JX
such that
( ( )) ( )F J J
, i.e., a unique fixed-point in
.X
Proof: Let
F
define in Eq. (7), which is a
contraction mapping for
12
, [0, ],J J K b
such that
1 2 1 2
( ) ( ) ,F J F J r J J
12
,JJ
[0, ]Kb
with
01r
under the norm
[0, ]
sup ( )
b
JJ
, so
1
1 2 1 2
[0, ] 0
1
( ) ( ) sup ( ( ) ( ))
Vw
b
b
e
F J F J J w J w e dw
1
12
[0, ]
1
sup ( )( 1)
Vb
bJ J e e
1
12
[0, ]
sup 1
Vb
b
J J e e
12
r J J
where
1
[0, ]
sup 1
Vb
b
r e e
;
01r
.
Thus,
1 2 1 2
( ) ( ) F J F J r J J
where a
positive constant
0,1r
and
F
represents the
contraction, such that a mapping of contractions can
have at most one fixed point. By applying the
Banach contraction principle, a unique solution of
the
()J
is thus verified.
3.2 The Numerical Integral Equation for the
ARL of ARMA(p,q) Process
The numerical methods, which consist of
Simpson's rule, Midpoint rule, Trapezoidal rule,
and Gaussian rule, can approximate the ARL by
using numerical integral equation, [16]. The
outcomes demonstrated that the explicit formulas
and numerical integral equation solutions were in
outstanding agreement. In this research, the
Gauss‐Legendre rule is applied to approximate the
ARL on the HWMA control chart for the ARMA
process as follows:
0 1 0 2 1 1 1 0 2 1 1
0
1 ( ... ... )
1
( ) 1 ( ) ( )
bp p q q
w Y Y Y
J J w f dw
The approximation for an integral is evaluated by
the quadrature rule as follows:
1
0
bn
kk
k
f x dx w f a
where
k
a
is a point and
k
w
is a weight that is
determined by the different rules.
Using the quadrature formula, we obtain
0 1 0 2 1 1 1 0 2 1 1
1
1 ( ... ... )
1
1 ( ) ( )
np p q q
b k k
k
w Y Y Y
J a w J a f
2, . 1, ,.. ,bn
The system of
n
linear equations is as follows:
0 1 0 1 1 0 1
1
1 ( ... ... )
1
1 ( ) ( )
np p q q
b k k
k
w Y Y
J a w J a f
0 1 0 1 1 0 1
1
1
1 ( ... ... )
1
1 ( ) ( )
nk p p q q
kk
k
a Y Y
J a w J a f
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0 1 0 1 1 0 1
2
1
1 ( ... ... )
1
1 ( ) ( )
nk p p q q
kk
k
a Y Y
J a w J a f
0 1 0 1 1 0 1
1
1 ( ... ... )
1
1 ( ) ( )
nk p p q q
n k k
k
a Y Y
J a w J a f
This system can be shown as:
1
11n n n n n
J I R 1
,where
1
2
1, 1,1,...,1
nn
n
Ja
Ja diag
Ja
JI
and
1
1
1.
1
n
1
Let
nn
R
be a matrix and define the
n
to
th
n
as
an element of the matrix
R
as follows
0 1 0 1 1 0 1
1 ( ... ... )
1()
p p q q
bk k
w Y Y
wf
R
If
1
I- R
exists, the numerical approximation for
the integral equation is the term of the matrix,
1
1 1 1n n n n n
J I R 1
Finally, we substitute
b
a
by
in
()
b
Ja
and the
approximation of numerical integral for the
function
()J
is:
0 1 0 1 1 0 1
1
1 ( ... ... )
1
1 ( ) ( )
nk p p q q
kk
k
a Y Y
J w J a f
(8)
In this study, we compare the ARL0 and ARL1
through the use of explicit formulas and the NIE
method for ARMA(p,q) process carried out on a
HWMA control chart. The accuracy of the ARL is
compared with the accuracy percentage which can
be obtained from
( ) ( )
% 100 - 100%.
()
JJ
Accuracy J
Furthermore, performance metrics such as the
Median Run Length (MRL) and Standard
Deviation Run Length (SDRL) are employed to
measure the efficacy of control charts, [23]. The
calculation for SDRL and MRL for the in-control
process is as follows:
0
0 0 0
2
00
0
1
1 log(0.5)
, , ,
log(1 )
ARL SDRL MRL
(9)
where
0
denotes an error of type I. The present
investigation established ARL0 at 370, and it can be
computed using Eq.(9) as SDRL0 and MRL0 at
approximations 370 and 256, correspondingly.
Conversely, SDRL1 and MRL1 are computed by
replacing
0
with
1
, where
1
signifies type II
error. The lowest values of the Average Run
Length for out of control (ARL1), SDRL1, and
MRL1 demonstrate a better capacity to quickly
identify changes in the process mean. To carry out
a comparative analysis of the HWMA and CUSUM
control charts for the ARMA(p,q) model, the
relative mean index (RMI) value is calculated
according to the approach described in [24].
,,
1,
[]
1
[]
nshift i shift i
ishift i
ARL Min ARL
RMI n Min ARL
where
,shift i
ARL
denotes the ARL of the control
chart corresponding to the shift size of row i, while
,
[]
shift i
Min ARL
indicates the ARL at the same level
that is the smallest among all control charts.
In addition, the performance measurements can be
used to assess a control chart's success throughout a
variety of changes
min max
( ).
Moreover, the
average extra quadratic loss (AEQL) may refer to
the average extra loss incurred due to an out-of-
control condition. This comparison might involve
different control chart types to find the most
effective approach for a particular process. AEQL
can be calculated as follows; [25]:
max
min
2
1()
i
ii
AEQL ARL
(10)
where
denotes the specific change in the
process, and
denotes the aggregate of number of
divisions from
min
to
max
. In this study,
10
is determined from
min 0.001
to
max 1.00.
The
most effective control chart is the one with the
lowest AEQL value. Furthermore, the assessment
of control chart performance can be conducted by
using performance evaluation criteria such as the
Performance Comparison Index (PCI). The
calculation of the PCI value involves determining
the ratio between the AEQL of a given control
chart and the AEQL of the control chart with the
lowest value, which signifies the control chart that
shows the most efficient control chart. The
computational model that describes the PCI is:
lowest
AEQL
PCI AEQL
(11)
4 Numerical Results
A comparison of the ARL1 values obtained from
the explicit formula and the NIE method on the
HWMA control chart for ARMA(p,q) processes is
presented in Table 1 (Appendix) with
00.5,
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10.1,
10.1,
20.2,
and
30.3,
0.1,0.2,
0
ARL 370
is implemented so that the computation
time (CPU time) and percentage accuracy are
utilized to compare the two methods. The results
suggest that the ARL of the two methods are very
similar with a one hundred percent accuracy rate,
which is employed to validate this explicit precise
formula. In addition, the explicit formula requires
a CPU time of less than 0.01 seconds, which is
significant in comparison to the NIE method. For
instance, if we consider ARMA(1,1) with
0.1
and
0.02, ARL1 will decrease to 301.32377,
which is the same value obtained from both
approaches. Table 2 (Appendix) displays the results
of ARL of the HWMA control chart using an
explicit formula against CUSUM control chart for
the ARMA(2,1) process. As an illustration, when
00.5
10.1,
20.2,
10.1,
the HWMA control
chart is given
0.05,0.10,
0.2,0.3
and CUSUM
control chart is given a = 4 and 6. According to the
findings, the HWMA control chart performed better
than the CUSUM control chart in terms of ARL,
SDRL, and MRL. Furthermore, the results indicate
that the HWMA control chart demonstrates the
most minimum values for RMI, AEQL, and PCI.
The comparison between the ARL on the HWMA
control chart for the ARMA(2,3) model and the
CUSUM control chart given
00.5,
20.2,
10.1,
10.1,
20.2,
and
30.3,
0.1,
and
0
ARL
370
is presented in Table 3 (Appendix).
The average run length (ARL) of the HWMA
control chart is consistently lower than that of the
CUSUM control chart for all values of
.
Hence,
it can be concluded that the HWMA control chart
demonstrates superior performance in comparison
to the CUSUM control chart. Furthermore, the
RMI, AEQL, and PCI values obtained from each
control chart are employed to evaluate the efficacy
of the mentioned control charts. The HWMA
control chart demonstrated that it displayed the best
results, as seen by the lowest values for RMI,
AEQL, and PCI, all of which were equal to 1.
Application
Using the count of nonconformities identified in 26
consecutive samples of 100 printed circuit boards
(PCBs), [26]. The following coefficient parameters
are estimated for the ARMA(1,1) model, based on
the model estimation using maximum likelihood
estimation:
10.978,
10.482,
and the in-
control parameter equals to 6.69666, as shown in
Table 4 (Appendix). By applying the parameters of
this forecasting model, the ARMA(1,1) model can
be represented as:
11
ˆ0.978 0.482
t t t
YY
Subsequently, employing the explicit formula
approach, a comparison is made between the ARL
values of the ARMA(1,1) model on the HWMA
and CUSUM control charts about their efficiency in
terms of ARL, SDRL, and MRL. The findings are
succinctly shown in Table 5 (Appendix) and Figure
1 (Appendix), indicating an obvious agreement
with the results seen in Table 2 (Appendix) and
Table 3 (Appendix). Figure 2 (Appendix) presents
a comparison of the RMI, AEQL, and PCI values
obtained from each control chart, aiming to
determine the effectiveness of the control charts.
Furthermore, the HWMA (Ht) statistic, which
represents the count of nonconformities detected in
26 consecutive samples of 100 printed circuit
boards, has been analyzed using the ARMA(1,1)
model. The results are illustrated in Figure 3
(Appendix). The findings indicated that the
HWMA control chart can identify a shift in the
process starting from the sixth observation. It has
been determined that the quantity of
nonconformities per board remains at an
unsatisfactory level. Further measures are required
to enhance the procedure.
5 Summary and Conclusions
This work presents the exact solution of the ARL
for using the ARMA(p,q) model on the HWMA
scheme. The dataset of the count of
nonconformities of printed circuit boards is applied
to show the efficiency of the exact formula. The
exact solution obtained accurate results and can
reduce computational time. The ARL values were
also subjected to a comparison of the accuracy
percentage between the explicit formula and the
numerical integral equation approach. As a result,
the finding did not reveal a statistically significant
difference in the ARL values. Moreover, the study's
findings indicate that the HWMA chart consistently
outperforms the CUSUM chart across all levels.
Consequently, future research can derive the
explicit formulas for Average Run Length (ARL)
on alternative control charts, including but not
limited to DHWMA, THWMA schemes, or other
interesting models. In general, control charts are an
important tool in statistical process control (SPC)
for detecting changes in manufacturing industries
to improve quality control. However, modern
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technologies, such as the Internet of Things,
artificial intelligence (AI), and smart
manufacturing, are an important concept that
expresses the ultimate goal of digitization in
manufacturing. Conventional control charts may
fail to create processes for pattern detection and
interpretation, while smart manufacturing needs
automated processes that can handle big data from
simultaneous processes. To solve these issues,
machine learning (ML) algorithms which are
excellent tools for analysis and can be used with
SPC control charts should be thoroughly
investigated in future studies.
Acknowledgement:
This research was funded by Thailand Science
Research and Innovation Fund (NSRF), and King
Mongkut’s University of Technology North
Bangkok with Contract no. KMUTNB-FF-67-B-12.
References:
[1] M. Aslam, G.S. Rao, A. Shafqat, L. Ahmad,
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[13] C. W. Champ, S. E. Rigdon, A comparison of
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[14] K. Petcharat, The effectiveness of CUSUM
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[15] W. Peerajit, Approximating the ARL of
Changes in the Mean of a Seasonal Time
Series Model with Exponential White Noise
Running on a CUSUM Control Chart,
WSEAS Transactions on Systems and
Control, Vol. 18, 2023, pp. 370 – 381.
[16] S. Phanyaem, Explicit formulas and
numerical integral equation of ARL for
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[17] W. Suriyakat and K. Petcharat, Exact run
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[18] C. Chananet and S. Phanyaem, Improving
CUSUM control chart for monitoring a
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Applied Mathematics, Vol. 52, No. 3, 2022,
pp.1-8.
WSEAS TRANSACTIONS on SYSTEMS
DOI: 10.37394/23202.2024.23.15
Yupaporn Areepong,
Saowanit Sukparungsee, Tanapat Anusas-Amornkul
E-ISSN: 2224-2678
134
Volume 23, 2024
[19] W. Peerajit, Accurate Average Run Length
analysis for detecting changes in a long-
memory fractionally Integrated MAX process
running on EWMA control chart. WSEAS
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[20] Y. Areepong, S. Sukparungsee, Capability
process on a Two-Sided Extended EWMA
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Exogenous Factors Model, IAENG
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[21] Y. Supharakonsakun, Y. Areepong, ARL
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– 1786.
[22] C.W. Champ and S.E. Rigdon, A comparison
of the Markov chain and the integral equation
approaches for evaluating the run length
distribution of quality control charts.
Communications in Statistics-Simulation and
Computation, Vol. 20, 1991, pp.191–204.
[23] A. Fonseca, PH. Ferreira, DC. Nascimento,
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[24] Tang, P. Castagliola, J. Sun, X. Hu, Optimal
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[25] V. Alevizakos, K. Chatterjee, C.
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[26] D.C. Montgomery, Introduction to Statistical
Quality Control, John Wiley& Sons, 2009.
Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
- Yupaporn Areepong has organized the
conceptualization and simulation, writing-
original draf.
- Saowanit Sukparungsee has implemented the
methodology and software.
- Tanapat Anusas-Amornkul has implemented the
methodology and validation.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
This research was funded by Thailand Science
Research and Innovation Fund (NSRF), and King
Mongkut’s University of Technology North
Bangkok with Contract no. KMUTNB-FF-67-B-12.
Conflicts of Interest
The authors declare no conflict of interest.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.e
n_US
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Volume 23, 2024
APPENDIX
Table1. The ARL values of the explicit formula against the NIE method for ARMA(p,q) on the HWMA control
chart with
00.5,
10.1,
12
0.1, 0.2,
30.3
and
01.
under different conditions
Model
ARMA(1,1)
ARMA(1,2)
ARMA(1,3)
0.1
0.2
0.1
0.2
0.1
0.2
b
0.002679
0.11475
0.00328
0.1428
0.00445
0.20005
0.002
()J
362.74860
358.21106
362.72329
358.85457
362.75708
360.71723
CPU time
(<0.01)
(<0.01)
(<0.01)
(<0.01)
(<0.01)
(<0.01)
ˆ()J
362.74860
358.21111
362.72329
358.85465
362.75708
360.71739
CPU time
(1.624
(1.640
(1.656
(1.641
(1.610
(1.625
%Acc
100.00
100.00
100.00
100.00
100.00
100.00
0.004
()J
355.23423
346.38784
355.35876
348.00700
355.61694
351.44955
CPU time
(<0.01)
(<0.01)
(<0.01)
(<0.01)
(<0.01)
(<0.01)
ˆ()J
355.23423
346.38789
355.35876
348.00708
355.61694
351.44971
CPU time
(1.735)
(1.625)
(1.609)
(1.672)
(1.593)
(1.593)
%Acc
100.00
100.00
100.00
100.00
100.00
100.00
0.008
()J
340.75162
324.55890
341.15587
327.82854
341.83351
333.9952
CPU time
(<0.01)
(<0.01)
(<0.01)
(<0.01)
(<0.01)
(<0.01)
ˆ()J
340.75162
324.55895
341.15587
327.82862
341.83351
333.99533
CPU time
(1.671)
(1.766)
(1.641)
(1.656)
(1.640)
(1.641)
%Acc
100.00
100.00
100.00
100.00
100.00
100.00
0.02
()J
301.32377
270.80199
302.42502
277.27916
304.15210
288.99388
CPU time
(<0.01)
(<0.01)
(<0.01)
(<0.01)
(<0.01)
(<0.01)
ˆ()J
301.32377
270.80202
302.42502
277.27922
304.15210
288.99401
CPU time
(1.750)
(1.688)
(1.640)
(1.625)
(1.610)
(1.625)
%Acc
100.00
100.00
100.00
100.00
100.00
100.00
0.04
()J
246.99573
207.82806
248.88340
216.436960
251.80526
232.23925
CPU time
(<0.01)
(<0.01)
(<0.01)
(<0.01)
(<0.01)
(<0.01)
ˆ()J
246.99573
207.82808
248.88340
216.43700
251.80526
232.23935
CPU time
(1.750)
(1.656)
(1.625)
(1.609)
(1.625)
(1.609)
%Acc
100.00
100.00
100.00
100.00
100.00
100.00
0.08
()J
169.57634
134.40835
172.14889
143.10110
176.13609
159.63051
CPU time
(<0.01)
(<0.01)
(<0.01)
(<0.01)
(<0.01)
(<0.01)
()J
169.57634
134.40837
172.14889
143.10112
176.13609
159.63057
CPU time
(1.750)
(1.719)
(1.656)
(1.640)
(1.609)
(1.641)
%Acc
100.00
100.00
100.00
100.00
100.00
100.00
0.2
()J
63.67039
53.41236
65.87056
58.70493
69.35024
69.25782
CPU time
(<0.01)
(<0.01)
(<0.01)
(<0.01)
(<0.01)
(<0.01)
ˆ()J
63.67039
53.41237
65.87056
58.70494
69.35024
69.25784
CPU time
(1.640)
(1.718)
(1.625)
(1.640)
(1.578)
(1.610)
%Acc
100.00
100.00
100.00
100.00
100.00
100.00
0.4
()J
18.24749
20.11960
19.29795
22.58486
21.01000
27.59650
CPU time
(<0.01)
(<0.01)
(<0.01)
(<0.01)
(<0.01)
(<0.01)
ˆ()J
18.24749
20.11960
19.29795
22.58486
21.01000
27.59651
CPU time
(1.625)
(1.656)
(1.640)
(1.641)
(1.594)
(1.609)
%Acc
100.00
100.00
100.00
100.00
100.00
100.00
0.8
()J
3.95253
6.61371
4.23555
7.49603
4.71587
9.29844
CPU time
(<0.01)
(<0.01)
(<0.01)
(<0.01)
(<0.01)
(<0.01)
ˆ()J
3.95253
6.61372
4.23555
7.49603
4.71587
9.29844
CPU time
(1.641)
(1.640)
(1.672)
(1.610)
(1.610)
(1.641)
%Acc
100.00
100.00
100.00
100.00
100.00
100.00
1.0
()J
2.56482
4.68558
2.73422
5.29793
3.02579
6.54870
CPU time
(<0.01)
(<0.01)
(<0.01)
(<0.01)
(<0.01)
(<0.01)
ˆ()J
2.56482
4.68558
2.73422
5.29793
3.02579
6.54870
CPU time
(1.750)
(1.672)
(1.656)
(1.657)
(1.640)
(1.640)
%Acc
100.00
100.00
100.00
100.00
100.00
100.00
Note: The numerical results in parentheses are computational times in seconds
WSEAS TRANSACTIONS on SYSTEMS
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E-ISSN: 2224-2678
136
Volume 23, 2024
Table 2. The ARL of HWMA control chart for ARMA(2,1) using explicit formula against CUSUM control
chart given
00.5,
1 2 1
0.1, 0.2, 0.1
and
01.
Control
Chart
HWMA
0.05
HWMA
0.1
HWMA
0.2
HWMA
0.3
CUSUM
a
4
CUSUM
a
6
UCL
0.0000000514
0.002189
0.09253
0.15731
2.911
0.821
0.002
ARL1
355.52781
362.72628
357.21243
323.42272
365.50800
366.10200
SDRL1
355.02746
362.22593
356.71208
322.92233
365.00770
365.60170
MRL1
246.08636
251.07597
247.25405
223.83279
253.00410
253.41580
0.004
ARL1
341.23212
355.06350
344.48336
286.57738
361.03200
361.79900
SDRL1
340.73175
354.56315
343.98300
286.07694
360.53170
361.29870
MRL1
236.17734
245.76453
238.43093
198.29353
249.90160
250.43320
0.008
ARL1
314.49668
340.30433
321.14734
232.88106
352.29400
353.39400
SDRL1
313.99628
339.80396
320.64695
232.38053
351.79360
352.89360
MRL1
217.64573
235.53424
222.25562
161.07403
243.84490
244.60730
0.02
ARL1
247.17163
300.19029
264.58406
147.41599
327.69900
329.69300
SDRL1
246.67112
299.68987
264.08359
146.91514
327.19860
329.19260
MRL1
170.97951
207.72929
183.04890
101.83401
226.79690
228.17900
0.04
ARL1
167.53292
245.09590
199.93938
89.42725
291.52500
294.71100
SDRL1
167.03218
244.59539
199.43875
88.92584
291.02460
294.21060
MRL1
115.77805
169.54072
138.24055
61.63902
201.72300
203.93130
0.08
ARL1
80.51023
167.02881
126.77026
47.86978
233.68400
238.41300
SDRL1
80.00867
166.52806
126.26927
47.36714
233.18350
237.91250
MRL1
55.45814
115.42863
87.52342
32.83301
161.63060
164.90850
0.20
ARL1
12.54729
61.54482
48.96062
17.66371
131.35000
137.28200
SDRL1
12.03691
61.04278
48.45804
17.15643
130.84900
136.78110
MRL1
8.34575
42.31210
33.58915
11.89361
90.69787
94.80964
0.40
ARL1
1.94844
17.25926
18.08744
7.55673
62.55190
67.36890
SDRL1
1.35940
16.75180
17.58033
7.03899
62.04989
66.86703
MRL1
0.96275
11.61319
12.18740
4.88316
43.01017
46.34913
0.80
ARL1
1.03235
3.69499
5.89499
3.31628
23.35030
25.94330
SDRL1
0.18274
3.15563
5.37177
2.77154
22.84483
25.43839
MRL1
0.20015
2.19640
3.72879
1.93141
15.83609
17.63368
1.0
ARL1
1.00974
2.41242
4.18820
2.63654
16.59500
18.54760
SDRL1
0.09917
1.84590
3.65415
2.07721
16.08723
18.04067
MRL1
0.14935
1.29481
2.54073
1.45350
11.15261
12.50644
RMI
0.287
2.181
2.341
0.711
8.589
9.344
AEQL
0.339
1.161
1.408
0.720
4.892
5.358
PCI
1.000
3.421
4.148
2.122
14.418
15.790
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Table 3. The ARL of HWMA control chart for ARMA(2,3) using explicit formula against CUSUM control
chart given
00.5,
1 2 1 2 3
0.1, 0.2, 0.1, 0.2, 0.3,
and
01.
Control
Chart
HWMA
0.05
HWMA
0.1
HWMA
0.2
HWMA
0.3
CUSUM
a
4
CUSUM
a
6
UCL
0.0000000848
0.00363
0.1596
0.27522
3.0299
0.921
0.002
ARL1
356.11506
362.69350
359.99549
334.36080
366.35600
365.89400
SDRL1
355.61471
362.19315
359.49514
333.86043
365.85570
365.39366
MRL1
246.49341
251.05324
249.18312
231.41450
253.59190
253.27166
0.004
ARL1
342.13461
355.40443
349.62240
304.29549
361.84100
361.59300
SDRL1
341.63424
354.90408
349.12204
303.79508
361.34070
361.09265
MRL1
236.80290
246.00084
241.99304
210.57480
250.46230
250.29043
0.008
ARL1
315.95005
341.34256
330.25243
257.48408
353.02700
353.19000
SDRL1
315.44965
340.84219
329.75205
256.98359
352.52660
352.68965
MRL1
218.65313
236.25389
228.56679
178.12757
244.35290
244.46592
0.02
ARL1
249.76034
302.96425
281.29778
174.59651
328.22800
329.49700
SDRL1
249.25984
302.46384
280.79733
174.09579
327.72760
328.99662
MRL1
172.77387
209.65205
194.63398
120.67417
227.16360
228.04317
0.04
ARL1
170.87806
249.82338
221.53266
111.48707
291.77600
294.52400
SDRL1
170.37733
249.32288
221.03209
110.98594
291.27560
294.02357
MRL1
118.09673
172.81757
153.20790
76.92985
201.89690
203.80171
0.08
ARL1
83.56450
173.44537
148.20004
62.22314
233.5550
238.24600
SDRL1
83.06300
172.94465
147.69919
61.72112
233.05450
237.74547
MRL1
57.57523
119.87626
102.37748
42.78229
161.54120
164.79273
0.20
ARL1
13.55905
67.00002
61.83675
23.87320
130.81100
137.15700
SDRL1
13.04948
66.49814
61.33471
23.36785
130.31000
136.65609
MRL1
9.04742
46.09343
42.51445
16.19860
90.32426
94.72299
0.40
ARL1
2.09480
19.84805
24.05207
10.35762
62.04240
67.28600
SDRL1
1.51439
19.34159
23.54676
9.84493
61.54037
66.78413
MRL1
1.06821
13.40806
16.32260
6.82692
42.65700
46.29166
0.80
ARL1
1.04042
4.387499
8.02209
4.45728
23.08000
25.89960
SDRL1
0.20507
3.85521
7.50545
3.92557
22.57446
25.39468
MRL1
0.21340
2.67968
5.20623
2.72832
15.64870
17.60339
1.0
ARL1
1.01252
2.82595
5.66309
3.48040
16.39670
18.51390
SDRL1
0.11259
2.27157
5.13882
2.93816
15.88883
18.00696
MRL1
0.15779
1.58708
3.56757
2.04633
11.01512
12.48308
RMI
0.172
2.176
2.883
1.042
7.954
8.742
AEQL
0.349
1.314
1.856
0.961
4.844
5.349
PCI
1.000
3.766
5.317
2.753
13.877
15.324
Table 4. The coefficients for the ARMA model using the real-world dataset
model
ARMA(1,1)
parameters
SE
p-value
AR
0.978
0.033
0.000
MA
0.482
0.201
0.024
RMSE
8.970
Normalized BIC
4.638
Residual
Residual AR(1) model
Exponential parameter
6.69666
One-sample
Kolmogorov-Smirnov test
0.786
p-value
0.567
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Volume 23, 2024
Table 5. The ARL of HWMA control chart for ARMA(1,1) using explicit formula against CUSUM control
chart given
11
0.978, 0.482
and
06.9666
Control
Chart
HWMA
0.05
HWMA
0.1
HWMA
0.2
HWMA
0.3
CUSUM
a
4
CUSUM
a
6
UCL
0.32306
0.67631
1.4306
2.27532
15.456
13.371
0.002
ARL1
366.36155
354.40458
341.73151
337.61114
369.36600
369.67100
SDRL1
365.86121
353.90422
341.23114
337.11077
368.86566
369.17066
MRL1
253.59574
245.30780
236.52349
233.66746
255.67827
255.88968
0.004
ARL1
362.11162
339.81558
317.14254
310.10342
368.72900
369.03700
SDRL1
361.61127
339.31521
316.64215
309.60302
368.22866
368.53666
MRL1
250.64992
235.19547
219.47970
214.60055
255.23674
255.45023
0.008
ARL1
353.88623
313.95660
277.26128
266.67949
367.45800
367.77300
SDRL1
353.38588
313.45620
276.76083
266.17902
366.95766
367.27266
MRL1
244.94851
217.27137
191.83609
184.50135
254.35575
254.57409
0.02
ARL1
331.20656
255.55045
201.38396
187.90219
363.68100
364.01600
SDRL1
330.70618
255.04996
200.88334
187.40152
363.18066
363.51566
MRL1
229.22814
176.78727
139.24186
129.89699
251.73773
251.96993
0.04
ARL1
298.96501
194.96525
138.41210
126.07860
357.50000
357.86700
SDRL1
298.46459
194.46461
137.91120
125.57760
356.99965
357.36665
MRL1
206.87999
134.79274
95.59297
87.04399
247.45338
247.70777
0.08
ARL1
249.51600
132.08183
85.31215
76.26771
345.55100
345.97900
SDRL1
249.01550
131.58088
84.81068
75.76606
345.05064
345.47864
MRL1
172.60451
91.20514
58.78662
52.51741
239.17096
239.46763
0.20
ARL1
164.21677
66.79234
39.93514
35.27920
312.75100
313.33500
SDRL1
163.71601
66.29045
39.43197
34.77561
312.25060
312.83460
MRL1
113.47946
45.94948
27.33289
24.10544
216.43572
216.84051
0.40
ARL1
101.34598
36.32982
21.42753
18.95657
266.75800
267.52600
SDRL1
100.84474
35.82633
20.92156
18.44980
266.25753
267.02553
MRL1
69.90053
24.83373
14.50310
12.78999
184.55577
185.08810
0.80
ARL1
53.89928
18.78077
11.38671
10.18703
198.85700
199.80800
SDRL1
53.39694
18.27393
10.87522
9.67412
198.35637
199.30737
MRL1
37.01248
12.66811
7.54078
6.70857
137.49030
138.14949
1.0
ARL1
42.65706
15.08122
9.31329
8.38131
173.58900
174.57300
SDRL1
42.15410
14.57265
8.79910
7.86543
173.08828
174.07228
MRL1
29.21968
10.10297
6.10235
5.45557
119.97582
120.65788
RMI
2.137
0.541
0.074
0.000
6.613
6.641
AEQL
10.217
3.687
2.249
2.013
35.900
36.075
PCI
5.074
1.831
1.117
1.000
17.831
17.917
Fig. 1: The ARL1 values on the control charts with real dataset
0
50
100
150
200
250
300
350
400
0,002 0,004 0,008 0,02 0,04 0,08 0,2 0,4 0,8 1
ARL1
SHIFT SIZES
HWMA
1
HWMA
2
HWMA
3
WSEAS TRANSACTIONS on SYSTEMS
DOI: 10.37394/23202.2024.23.15
Yupaporn Areepong,
Saowanit Sukparungsee, Tanapat Anusas-Amornkul
E-ISSN: 2224-2678
139
Volume 23, 2024
Fig. 2: Comparison of the RMI, AEQL, and PCI values among HWMA 1, HWMA 2, HWMA 3, HWMA 4,
CUSUM 1, and CUSUM 2 control charts
Fig. 3: The dataset fitted to ARMA(1,1) process running on HWMA control chart when
0.3
17,917
36,075
6,641
17,831
35,9
6,613
1
2,013
0
1,117
2,249
0,074
1,831
3,687
0,541
5,074
10,217
2,137
0 5 10 15 20 25 30 35 40
PCI
AEQL
RMI
HWMA 1 HWMA 2 HWMA 3 HWMA 4 CUSUM 1 CUSUM 2
0
1
2
3
4
5
6
7
8
12345678910 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
Sample No.
HWMA Control Chart
Ht
WSEAS TRANSACTIONS on SYSTEMS
DOI: 10.37394/23202.2024.23.15
Yupaporn Areepong,
Saowanit Sukparungsee, Tanapat Anusas-Amornkul
E-ISSN: 2224-2678
140
Volume 23, 2024
