Approximating the ARL to Monitor Small Shifts in the Mean of an AR
Fractionally Integrated with an exogenous variable Process
Running on an EWMAControl Chart
WILASINEE PEERAJIT
Department of Applied Statistics, Faculty of Applied Science,
King Mongkut’s University of Technology,
North Bangkok, Bangkok 10800,
THAILAND
Abstract: - Control charts are used to monitor processes and detect changes in a given control scheme. The
Exponential Weighted Moving Average (EWMA) control chart is a well-recognized control chart used to detect
small changes in parameters. The efficiency of the chart studied is usually achieved using ARL. Approximating
ARL using the Gauss-Legendre quadrature method, also known as NIE,. This approach is used to evaluate the ARL
of developments, such as explicit formulas because it provides a robust way to validate their validity and accuracy.
Moreover, it evaluates the performance of control charts for time series under exponential white noise. Exponential
white noise is obtained from a long-memory fractionally integrated AR with exogenous variables or the long-
memory ARFIX process. Under the long-memory ARFIX model, the proposed technique compares the control
chart's performance to an explicit formula using the criterion of percentage accuracy. The results of the
comprehensive numerical study include investigations into a wide range of out-of-control processes and situations.
Specifically, the results from the accuracy percentage in all cases are more than 95%, which means that the
proposed technique is accurate and completely consistent with the well-defined explicit formula. Therefore, it is
recommended that it be used in this situation. There are examples from real data that were found to be consistent
with the research results.
Key-Words: - exponentially weighted moving average (EWMA) control chart, long-memory, fractionally
integrated autoregressive process with an exogenous variable process, Gauss-Legendre quadrature,
explicit formulas, exponential white noise.
Received: January 22, 2024. Revised: February 25, 2024. Accepted: March 13, 2024. Published: May 20, 2024.
1 Introduction
Control charts are statistical tools used to monitor a
process and indicate when it goes out of control.
Since the introduction of the Shewhart control chart,
[1], it has become common practice to make use of
control charts to monitor changes that can occur in
various manufacturing and production processes, [2].
The Shewhart control charts are frequently referred
to as memoryless because information from the past
is not utilized in its derivation, and thus it is only
suitable for monitoring large process parameter shift
sizes (i.e., location and/or dispersion). On the other
hand, memory control charts are extremely useful for
monitoring small-to-moderate changes in a process
parameter. An example of this is the exponentially
weighted moving average (EWMA) control chart,
[3], which is of interest in the present study.
Real-world scenarios often contain serially
correlated data in the underlying observations. One
technique to deal with autocorrelation in the
observations of a process running on a control chart
is to examine the correlation between subsequent
data points. A comprehensive elucidation of long-
memory processes is presented in [4], [5]. According
to [6], long-memory processes require differencing
parameter d in an autoregressive fractionally
integrated moving average order (p, d, q),
abbreviated as (ARFIMA(p, d, q) process to fall
within the range of 0 to 0.5. In addition to the
primary time series data, there can be auxiliary or
exogenous variables that are either already accessible
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or can be easily obtained. These variables can be
significantly correlated with the primary time series.
As stated by [7], the inclusion of these exogenous
variables in time series models enhances their
performance and improves the forecasting accuracy.
The fractionally integrated AR model with an
exogenous variable (ARFIX) is of special interest in
the present study.
In a time series model, the difference between the
actual and estimated values is referred to as the error
(also known as white noise). This should be
minimized to achieve the highest possible[ level of
accuracy with the model. The white noise produced by
autocorrelated data following a normal distribution is
commonly referred to as Gaussian white noise, [8].
Nevertheless, many phenomena such as wind speed,
oxygen content, and water flow rate have been studied
using non-Gaussian white noise, with exponentially
distributed white noise being particularly intriguing,
[9].
The effectiveness of a control chart is frequently
evaluated utilizing the average run length (ARL) to
determine the expected number of observations
before a control chart signals a change in the
monitored process. The ARL consists of two parts:
ARL0 and ARL1. Separate ARL—ARL0, which is the
in-control state, and ARL—ARL1, which is the out-
of-control state. Ideally, ARL0 should be as large as
possible, while ARL1 should be as small as possible.
In the study of techniques for evaluating the
performance of control charts in terms of ARL.
These include the Monte Carlo simulation method,
[10], the Markov chain approach, [11], the explicit
formulas put forth by [12], [13], [14], [15], and the
numerical integration equation (NIE) technique,
which has received support from [16], [17]. The NIE
technique is considered an efficient technique for the
computation of ARL. Since it is a computation
technique to get accurate and precise results for
research, several rules have been studied, including
the trapezoidal rule, the Simpsons rule, the midpoint
rule, and the quadrature rule. This article focuses on
quadrature rules that incorporate weighting and
interpolation. This rule is proposed as a technique
that was chosen from integration for approximating
ARL. Moreover, it has been devised by many
researchers to expand in many different fields.
The author in [18], developed a numerical technique
for approximating the ARL of processes running on an
EWMA control chart. In [19], the author resolved an
integral equation to determine the
ARL
of a process
running on a cumulative sum (CUSUM) chart while in
the in-control process stage. Numerous studies have
been devoted to assessing the performance of the NIE
technique in various scenarios, including detecting
changes to the mean of an autocorrelation process. In
the present research, we utilized the NIE technique to
approximate the ARL through an integral equation
using the Gauss-Legendre quadrature for a long-
memory ARFIX running on an EWMA control chart.
The remainder of the paper is organized as
follows. Section 2 provides brief derivations of a
long-memory ARFIX(p, d, k) process with
underlying exponential white noise and an EWMA
control chart. The numerical approximation of the
integral equation utilizing the NIE method through
the application of the Gauss-Legendre rule technique
is presented in Section 3. The numerical results for the
NIE technique and explicit formula are compared in
Section 4. To illustrate the efficacy of the proposed
technique, an example process involving real data is
also provided in Section 5. Finally, conclusions and
future recommendations are offered in Section 6.
2 The Long-Memory ARFIX(p, d, k)
Model With Underlying Exponential
White Noise and the EWMA Control
Chart
The following subsections provide brief derivations
of the EWMA chart and the model of interest.
2.1 The ARFIX(p, d, k) Model
Let
; 1, 2, ...
t
Yt
be a sequence of fractionally
integrated AR models with exogenous variables of
order
( , , ),p d k
where p is the order of the AR
process,
d
is the fractional integration parameter, and
k
is the exogenous variable in the model. The
ARFIX( , , )p d k
model with exponential white noise
can be written as
11
(1 )(1 ) ,
pk
id
i t j jt t
ij
B B Y X
(1)
where
i
is the i-th AR coefficient,
j
the j-th
coefficients corresponding to
,k
t
is a white noise
process assumed to follow exponential distribution
~ ( )
tExp
when shift parameter
0,
and
(1 )d
B-
is the fractional differencing operator in which B is
the backward-shift operator and
d
is the degree of
the differencing parameter. Since the focus of the
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current investigation is on long-term memory
processes,
d
was limited to between 0 and 0.5.
Operator
(1 )d
B
can be expanded using a
binomial series expansion in the following manner:
0
(1 ) ( )
dr
r
d
BB
r
23
11
1 (1 ) (1 )(2 ) ... ,
26
dB d d B d d d B
(2)
For any real value of
d
, the fractionally integrated
white noise process
(1 ) ,
dtt
BY
can be defined as
1 2 3
11
(1 ) (1 )(2 ) ... ,
26
t t t t t
Y dY d d Y d d d Y
(3)
Note that
rt t r
B Y Y
for order r.
Equation (1) can be reformulated to solve for
t
Y
in
the generalized long-memory
ARFIX( , , )p d k
process running on an EWMA control chart as
follows:
1
11
(1 ) (1 ) ,
pk
id
t i j jt t
ij
Y B B X
or
1 1 2 2
1 1 2 2 3 1
2 1 3 2 4 2
...
( ... )
1( 1)( ... )
2
t t t p t p
t t t p t p
t t t p t p
Y Y Y Y
d Y Y Y Y
d d Y Y Y Y
1 1 2 2
... ...
t t k kt t
X X X
(4)
where the initial value of
1 2 ( 1)
, ,..., , ,...,
t t t p t p
Y Y Y Y
12
, ,...,
t t kt
X X X
are equal to 1 and the initial value of
exponential white noise
1.
t
2.2 The EWMA Control Chart
The EWMA control chart is highly effective for
rapidly detecting small changes in a process
parameter by appropriately assigning weights to both
the current and previous observations. According to
,
t
M
the plotting statistic for the EWMA control
chart is defined in the following form:
1
(1 ) ,
t t t
M M Y
for
1,2,... ,t
(5)
where
t
Y
is the sequence of the long-memory
ARFIX( , , )p d k
process with underlying exponential
white noise
,t
M
represents the MA at time
1
,t
tM
represents the past values, and
0
M
represents the
initial value. If the process parameters are known, the
target or in-control mean
0
()Y
is assumed to be
0
M
while the smoothing parameter (or weighting
parameter)
is constrained by
0 1.
Note that although the value of
can range
from zero to one inclusively, it is typically selected to
be between 0.01 and 0.05 because the EWMA
control chart is designed to detect small changes in a
process parameter. When
1,
it becomes the same
as the Shewhart chart. In addition, the smoothing
parameter has an inverse relationship with the chart's
sensitivity to slight shifts.
Assuming that observations
t
Y
are independent
random variables with mean
0
()
and variance
2
( ),
the respective mean and variance of the
EWMA statistic for the in-control process are
0
()
t
EM
and
22
( ) ( / (2 )) 1 (1 ) .
t
t
VM
(5)
The upper control limit (UCL) and lower control
limit (LCL) of the EWMA chart can be derived as
2
0
( , ) 1 (1 ) ,
2
t
UCL LCL L
(6)
where constant
L
determines the control limits'
width, the value of which is determined by the
desired in-control ARL (ARL0). For sufficiently large
values of
,t
the control limits can be expressed as
0
( , ) ,
2
UCL LCL L
(7)
where
L
is the coefficient of the control chart for a
predetermined rate of false alarms. EWMA statistic
t
M
is plotted to fall between the UCL and the LCL
when the process is in control. On the contrary, the
process is considered to be out of control when
t
M
is
less than the LCL or more than the UCL.
The long-memory
ARFIX( , , )p d k
process in
Equation (4) can be replaced with Equation (5).
Consequently, the EWMA statistic can be expressed
as
1 1 1 2 2
1 1 2 2 3 1
2 1 3 2
1 1 2 2
(1 ) ...
( ... )
1( 1)( ... )
2
... ...
t t t t p t p
t t t p t p
t t p t p
t t k kt t
M M Y Y Y
d Y Y Y Y
d d Y Y Y
X X X
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1 1 1 2 2
1 1 2 2 3 1
2 1 3 2
(1 ) ...
( ... )
1( 1)( ... )
2
t t t p t p
t t t p t p
t t p t p
M Y Y Y
d Y Y Y Y
d d Y Y Y
1 1 2 2
... ...
t t k kt t
X X X
(8)
Hence, the stopping time for the EWMA control
chart
()
b
can be written as
inf 0; ,for ,
bt
t M b b
(9)
where
b
is a constant equivalent to the UCL.
Assume that the process is in control at time
t
if
the EWMA statistics
t
M
is in range
0
t
Mb
and
the process is out-of-control if
.
t
Mb
Let us
establish that the lower limit is
0,L
and the upper
limit is
.Ub
Concerning the EWMA statistics
1
M
while the process is in an in-control state:
1 1 1 2 2
1 1 2 2 3 1
2 1 3 2
1 1 2 2 1
0 (1 ) ...
( ... )
1( 1)( ... )
2
... ...
t t t p t p
t t t p t p
t t p t p
rr
M Y Y Y
d Y Y Y Y
d d Y Y Y
X X X b
Let
()
denote the ARL to monitor small shifts
in the mean of the long-memory
ARFIX( , , )p d k
process with an initial value
0
()M
. Now, we
define the function
()
as follows:
.AL 0R ( ) ( ) ,
b
Eb
(10)
where
()
b
E
is the expectation under the density
function
( , ).
t
f
Let
1 0 0 1
(1 ) ... ( ... )
p t p p t p
L Y Y d Y Y
2 2 1 1
1( 1)( ... ) ... ...
2t p t p k k
d d Y Y X X
1 0 0 1
(1 ) ... ( ... )
p t p p t p
U b Y Y d Y Y
2 2 1 1
1( 1)( ... ) ... ...
2t p t p k k
d d Y Y X X
For a probability distribution function
1,
, the
probability that
1
()
f
satisfies the constraints in the
previous equation can be rewritten as follows.
1
( ) ( ) ,
U
L
P L H f z dz
where
()fz
is the probability density function of
.z
3 Approximating the ARL to Monitor
Small Shifts in the Mean of an
ARFIX Process Running on an
EWMAControl Chart
The numerical approximation of the integral equation
utilizing the NIE method through the application of
the Gauss-Legendre rule technique is proposed in this
section.
Let
0
M
represent an initial value of the
EWMA statistics, and replace
z
with
t
where
()
tExp
represents white noise error terms.
According to [20], we propose a method where in the
function
()
can be reformulated as follows:
10
0 1 2 1
2 1 3 2
11
1 1 0
0 1 2 1
2 1 3
( ) 1 (1 ) ...
( ... )
( 1)( ... )
2
... ...
(1 ) ...
( ... )
( 1)( ...
2
ptp
t p t p
t t p t p
kk
ptp
t p t p
t t p
P Y Y
d Y Y Y
d d Y Y Y
XX
b Y Y
d Y Y Y
d d Y Y
2
11
)
... ...
tp
Y
X
10
0 1 2 1
2 1 3 1
1 1 1
(1 (1 ) ...
( ... )
( 1)( ... )
2
... ... ( )
U
ptp
L
t p t p
t t p t p
rr
YY
d Y Y Y
d d Y Y Y
X X f y dy
10
0 1 2 1
1 (1 ) ...
( ... )
U
p t p
L
t p t p
YY
d Y Y Y
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2 1 3 2
1 1 1
1( 1)( ... )
2
... ... ( )
t t p t p
kk
d d Y Y Y
X X f y dy
As a result of changing the integral variable, the
integral equation can be expressed:
11
0
1 1 2 1
2 1 3 2
11
1 (1 )
( ) 1 ( ) ( ...
( ... )
1( 1)( ... )
2
... ...
b
t p t p
t t p t p
t t p t p
t k kt
z
z f Y Y
d Y Y Y
d d Y Y Y
X X dz
0
1 1 1
1 ( )( exp
bz
11
1 1 2 1
22
11
(1 )
( ...
( ... )
1( 1)( .. )
2
... ...
t p t p
t t p t p
t p t p
t k kt
zYY
d Y Y Y
d d Y Y
XX
)dz
Thus, we get the integral equation in the following
manner:
0
1 (1 )
( ) 1 ( ).exp .exp
bz
z
.exp
1 1 2 2
1 1 2 1
2 1 3 2
11
1...
( ... )
1( 1)( ... )
2
... ...
t t p t p
t t p t p
t t p t p
t k kt
Y Y Y
d Y Y Y
d d Y Y Y
XX
dz
(11)
Equation (11) is represented as a linear Fredholm
integral equation of the second kind, [21]. By using
the Quadrature Rule, we can approximate the integral
0()
bf z dz
as the sum of the areas of rectangles.
These rectangles have bases of length
/bm
and
heights determined by the values of
f
at the
midpoints of intervals with a length of
/,bm
starting
from zero. The interval [0, b] is partitioned into a
sequence of points
12
0 .... ,
mb
where
/ 0.
j
w b m
represents a set of constant weights.
An integral equation from Equation (11) can be
approximated using the quadrature rule as follows:
1
0
( ) ( ) ( )
bm
jj
j
W z f z dz w f
(12)
where
()Wz
is a weight function,
( 1 2)
jb m j
and
/ ; 1,2,..., .
j
w b m j m
Solving a system of algebraic linear equations
with
m
equations and
m
unknowns, can be used to
approximate the solution for
()
by replacing
with
i
in Equation (11) as follows:
11
1
1 1 2 1
2 1 3 2
11
(1 )
1
ˆˆ
( ) 1 ( ) ...
( ... )
1( 1)( ... )
2
... ... ,
mji
i j j t
j
p t p t t p t p
t t p t p
t k kt t
w f Y
Y d Y Y Y
d d Y Y Y
XX
for
1,2,..., .im
Let
ˆ()
denote the NIE method for ARL when
using the interval
0,b
to apply the Gauss-Legendre
rule. Hence, the integral equation represented by
Equation (11) consists of the set
12
( ), ( ), ..., ( ),
ˆ ˆ ˆ ˆ
() m
which can be
approximated as
1
1 1 1
1
1 1 2 1
2 1 3 2
11
(1 )
1
ˆˆ
( ) 1 ( ) ...
( ... )
1( 1)( ... )
2
... ...
mj
j j t
j
p t p t t p t p
t t p t p
t k kt t
w f Y
Y d Y Y Y
d d Y Y Y
XX
2
2 1 1
1
1 1 2 1
2 1 3 2
11
(1 )
1
ˆˆ
( ) 1 ( ) ...
( ... )
1( 1)( ... )
2
... ...
mj
j j t
j
p t p t t p t p
t t p t p
t k kt t
w f Y
Y d Y Y Y
d d Y Y Y
XX
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11
1
1 1 2 1
2 1 3 2
11
(1 )
1
ˆˆ
( ) 1 ( ) ...
( ... )
1( 1)( ... )
2
... ...
mjm
m j j t
j
p t p t t p t p
t t p t p
t k kt t
w f Y
Y d Y Y Y
d d Y Y Y
XX
The above set of m equations in m unknowns can
be rewritten in matrix form.
Let
1 1 2 (
ˆˆ
( ), ( ),. ., )
ˆ
.
mm
L
be a column
vector of
ˆ( ); 1, 2, ..., ,
iim
11,1,...,1
m
1
is a
column vector of ones, and
1(1,1, ,1)
mdiag
I
is the
unit matrix order
.m
Let matrix
mm
R
be a matrix
with dimension
mm
with element can be expanded
as
11
1 1 2 1
2 1 3 2
11
(1 )
1...
( ... )
1( 1)( ... )
2
... ...
ji
ij j t
p t p t t p t p
t t p t p
t k kt t
R w f Y
Y d Y Y Y
d d Y Y Y
XX
where
; , 1,2,....,
ij i j mR
If the inverse of
1
()
m m m
IR
exists and is invertible,
then the ARL approximation for the NIE can be
expressed in a system of linear equations in matrix
form as follows:
1
11
( ) ,
m m m m m
L I C 1
(13)
Finally, the function
12
( ), ( ), ..., ( ),
ˆ ˆ ˆ m
is
obtained by replacing
i
with
.
Therefore, NIE
technique for approximating the ARL of a long-
memory
ARFIX( , , )p d k
process running on the
EWMA control chart is
11
1
1 1 2 1
2 1 3 2
(1 )
1
ˆˆ
( ) 1 ( ) ...
( ... )
1( 1)( ... )
2
mj
j j t
j
p t p t t p t p
t t p t p
w f Y
Y d Y Y Y
d d Y Y Y
11
... ...
t k kt t
XX
(14)
where
( 1 2)
jb m j
and
/ ; 1,2,..., .
j
w b m j m
The NIE technique is used to detect small shifts in the
mean process and the accuracy of explicit formulas.
Therefore, the explicit formula for the in-control ARL is
0 0 0
1
11
1 1 2 1
22
1 1 0
1
0
ARL 1 exp (1 ) exp 1
...
( ...
. exp 1( 1)( ... )
2
... ... /
exp / 1 .
t p t p
t t p t p
t p t p
t k kt t
b
YY
d Y Y Y
d d Y Y
XX
b
(15)
On the contrary, the explicit formula for the out-of-
control ARL is:
1 1 1
1
11
1 1 2 1
22
1 1 1
1
1
ARL 1 exp (1 ) exp 1
...
( ...
. exp 1( 1)( ... )
2
... ... /
exp / 1 .
t p t p
t t p t p
t p t p
t k kt t
b
YY
d Y Y Y
d d Y Y
XX
b
(16)
4 The Numerical Study
To assess the efficacy of the proposed NIE technique
in comparison with deriving the ARL using explicit
formulas, the accuracy percentage between them can
be expressed as
ˆ
( ) ( )
%Accuracy 100 100%,
( ))
(17)
where
ˆ()
and
()
are the
ARL
results for the
NIE technique and explicit formulas, respectively. An
accuracy percentage of greater than 95% means that
the ARL results of the two methods are close to each
other (i.e., the results are highly consistent).
The performances of the NIE technique and
explicit formulas were assessed using
= 0.03, 0.05,
or 0.10 to compute the UCL (b) from Equation (14)
to obtain ARL0 = 370. In the experiment, we
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generated several long-memory ARFIX(p, d, k)
processes with exponential white noise running on a
EWMA
control chart using Equation (8) and
employed a wide range of possible changes and
autocorrelation coefficient values. The white noise of
the process in this investigation was exponentially
distributed
( ( )).
tExp
The in-control process
0
()
has an exponential mean parameter of 1
whereas the out-of-control processes was assigned
changes in the mean
1
()
of 1.025,1.05, 1.075,
1.100, 1.125, 1.15, 1.20, 1.30, or 1.50. The
autocorrelation coefficients were assigned values of
12
0.1, 0.2,
31
0.3, 0.1.
Eight hundred
division points (m) were utilized in the system of
linear equations. The calculations for the numerical
results for the ARL derived by using both techniques
were performed using Wolfram Mathematica.
The principal findings using the suggested NIE
technique for approximating the ARL of the long-
memory ARFIX processes running on a EWMA
chart for each scenario are reported in Table 1
(Appendix). The smoothing parameter
()
of the
control chart was utilized to determine the optimal
value for
to compute the UCL
( ).b
For each
coefficient parameter combination in each long-
memory ARFIX process, we found that the values of
and
b
increased. Furthermore, upon examination
of the coefficient parameters, contrasting results were
obtained for the positive and negative values of
1.
Table 2 (Appendix) and Table 3 (Appendix)
report the numerical results for the out-of-control
ARL
10
()
computed using the process and
parameter values in Table 1. The
1
ARL
was then
calculated using the NIE technique in Equation (14)
and explicit formulas in Equation (16) for the long-
memory ARFIX(p, d = 0.1, k = 1) process when p
was varied as 1, 2, or 3 running on an EWMA control
chart. To accomplish this, the
NIE
search algorithm
was utilized to identify the corresponding values of b.
The results indicate that the ARL efficacies derived
from both techniques were similar for detecting small
changes in the process mean. The
1
ARL
results for
the NIE and the explicit formulas methods decreased
rapidly as the mean change magnitude was increased.
When analyzing the chart's properties, it is evident
that an increase in the
value resulted in a
proportional rise in
1
ARL
. This demonstrates that the
sensitivity of the EWMA chart decreased as
was
increased.
Figure 1 shows the
1
ARL
results for the
NIE
technique where several processes were assigned
various positive and negative coefficient values. It
was found that positive coefficient values resulted in
a reduction in
1
ARL
at every change level, which
resulted in increased detection efficacy. In addition,
the percentage change results were computed for
various changes in mean magnitude for each
scenario. The results of the calculations were greater
than 95%, which indicates that the suggested
technique is accurate and fully consistent with the
explicit formulas method.
5 An Illustration of the Efficacy of the
NIE Technique with Real Data
For this part of the study, we utilized the weekly
stock market price data for iron ore futures 62% Fe
CFR-(TIOc1) from January 5, 2020, to November
26, 2023, obtained from https://th.investing.com/. In
addition, daily UDS/THB exchange rate data were
also included as the exogenous variable. The datasets
consisted of 204 observations each.
Estimation of the parameters and testing of the
distribution of the white noise were performed using
the statistical software packages Eviews and SPSS,
respectively (Table 4 and Table 5). The p-values of
all of the parameters were found to be less than 0.05
indicating that they were all statistically significant.
Moreover, the value of
d
(0.163219, p-value < 0.5)
means that this model is a long-memory process.
Table 4. Parameter estimates for the TIOc1 dataset
including the UDS/THB exchange rates as the
exogenous variable
Parameters:
Coefficient
t-Statistic
Prob.
UDS/THB
-3.4336
-2.9599
0.0034*
d
0.1632
2.7586
0.0063*
AR(1)
0.9998
177.3717
0.0000*
R-squared
0.96493
Adjusted R-squared 0.96458
*A significance level of 0.05.
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Table 5. The results of testing the distribution of the
white noise of the TIOc1 dataset
Testing exponential white noise.
Exponential Parameter (
0
)
3.6471
Kolmogorov-Smirnov
0.8572
Asymptotic Significance (2-Sided)
0.4544ns
ns non-significance level of 0.05.
The residuals (white noise) of the long-memory
ARFIX model were tested to see whether they
followed an exponential distribution by using a
Kolmogorov-Smirnov test, which was the case (p-
value > 0.05). The exponential parameter (
) was
3.6471 (Table 5). The model is defined as
1 2 3 4
01. 0136 0.0949 . 265 0.0418
t t t t t
Y Y Y Y Y
14;)3.4 (336 3.6 71
tt
X Exp
(18)
Subsequently, the NIE technique to solve the
integral equation in Equation (14) for the ARL of the
process running on an EWMA control chart becomes
1
(1 )
1
ˆˆ
( ) 1 + () j
j
m
j
j
wf
1 2 3
41
0.0949 0.0265
0.04
1.136
3.1 38 4 36
t t t
tt
Y Y Y
YX
(19)
with a set of constant weights
,
j
w b m
and
( 1 2) ; 1,2, , .
jb m j j m
The results of the calculations and a comparison
with those using the explicit formulas are provided in
Table 6 (Appendix). They reveal that there was no
difference in the
ARL
values obtained by using the
two techniques when ARL0 = 370 with various
smoothing parameter values (0.05, 0.10, or 0.20) for
the ARFIX(1, 0.1632,1) process running on an
EWMA control chart. As the mean change
magnitude was increased, the
ARL
calculated via
both methods decreased, yielding the same findings
as those in Table 2 (Appendix) and Table 3
(Appendix). Moreover, the percentage accuracy was
1 0 0 % in all cases. This indicates high consistency
between the two techniques. Moreover, for the same
mean change magnitude, the out-of-control
ARL
increased as the value of the smoothing parameter
was increased from 0.01 to 0.05. These results are in
agreement with the numerical results in Section 4. The
efficacy of both techniques concerning out-of-control
processes were assessed by comparing the EWMA
control chart to the CUSUM control chart.
Calculation of the UCL (b) parameter when the
reference value (a) is set to 6 on the CUSUM control
chart and ARL0 is set according to the EWMA
control chart. For detecting small changes in process
parameters on both control charts, consistent results
are obtained. It was later found that the presented
results were consistent with the previous. Overall, the
NIE technique is performed as a accomplishing
choice.
6 Conclusions and Recommendations
The research presented here is an innovative
approach for detecting mean changes on EWMA
control charts of the ARFIX time series. The Gauss-
Legend method is applied to an approximation of
ARL with IE interpolation. The results of a numerical
study comprising the proposed technique and the
ARL derived using explicit formulas showed
excellent agreement between the two methods
(accuracy percentage > 95%), and it was found that
the out-of-control ARL results decreased rapidly and
in the same direction for both techniques. Therefore,
the NIE technique is a suitable choice for
determining the ARL for this specific situation.
Moreover, the method could be modified for other
control charts and used in practical scenarios that
include other time series models.
Acknowledgments:
The author gratefully acknowledges the editor and
referees for their valuable comments and suggestions
which greatly improve this paper. This research was
funded by King Mongkut’s University of Technology
North Bangkok, Contract No.KMUTNB-67-BASIC-
21.
References:
[1] W. A. Shewhart, "Economic quality control of
manufactured product," The Bell System
Technical Journal, 9, 2, (1930).
[2] D. C. Montgomery, Introduction to statistical
quality control, edited by N.J. Hoboken (John
Wiley, 2005).
[3] S.W. Robert, “Control Chart Test Based on
Geometric Moving Averages,” Technometrics.
1, 239-250 (1959).
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[4] C. W. J. Granger and R. Joyeux, “An
Introduction to Long Memory Time Series
Models and Fractional Differencing,” J. Time
Ser. Anal. 1, 1, 15-29 (1980).
[5] W. Palma, Long-Memory Time Series —
Theory and Methods, New Jersey (John
Wiley, 2007).
[6] J. R. M. Hosking, “Fractional differencing,”
Biometrika. 68, 1, 165-176 (1981).
[7] R.K. Paul, B. Gurung, and A.K. Paul,
“Modelling and forecasting of retail price of
arhar dal in Karnal,” Haryana. Ind. J. Agril.
Sci., 85,1, 69-72 (2014).
[8] L. Rabyk and W. Schmid, “EWMA control
charts for detecting changes in the mean of a
long-memory process,” Metrika. 79, 267–301
(2016).
[9] S. Suparman, “A new estimation procedure
using a reversible jump MCMC algorithm for
AR models of exponential white noise,” Int. J.
GEOMATE, 15, 49, 85-91 (2018).
[10] A. Ridwan Sanusi, Muhammad Riaz,
Nurudeen A. Adegoke, Min Xie, An EWMA
monitoring scheme with a single auxiliary
variable for industrial processes, Computers
and Industrial Engineering, 114, 1-10 (2017).
[11] 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,” Commun Stat-Simul C. 20, 191-204
(1991).
[12] Y. Areepong and W. Peerajit, “Integral
equation solutions for the average run length
for monitoring shifts in the mean of a
generalized seasonal ARFIMAX(P, D, Q, r)s
process running on a CUSUM control chart,”
PLoS ONE. 17, 2, e0264283 (2022).
[13] W. Peerajit, “Accurate Average Run Length
Analysis for Detecting Changes in a Long-
Memory Fractionally Integrated MAX
Process Running on EWMA Control Chart,”
WSEAS Transactions on Mathematics. 22,
514–530 (2023),
https://doi.org/10.37394/23206.2023.22.58.
[14] K. Karoon and Y. Areepong, “Improving
Sensitivity of the DEWMA Chart with Exact
ARL Solution under the Trend AR(p) Model
and Its Applications,” Emerging Science
Journal, 7, 6, 1875-1891 (2023).
[15] R. Sunthornwat, Y. Areepong and S.
Saowanit, “Performance Evaluation of
HWMA Control Chart based on AR(p) with
Trend Model to Detect Shift Process Mean,”
WSEAS Transactions on Business and
Economics. 21, 50, 603–616 (2024),
https://doi.org/10.37394/23207.2024.21.50.
[16] R. Sunthornwat, Y. Areepong and S.
Saowanit, “The Development and Evaluation
of Homogenously Weighted Moving Average
Control Chart based on an Autoregressive
Process,” HighTech Innov. J. 5, 1, 16-35
(2024).
[17] S.V. Crowder, “A Simple Method for
Studying Run Length Distributions of Expo-
nentially Weighted Moving Average Charts.”
Technometrics. 29, 401-407 (1987).
[18] P. Saesuntia, Y. Areepong and S. Saowanit,
“On the Average Run Lengths of Quality
Control Schemes Using a Numerical Integral
Equation Approach,” Applied Science and
Engineering Progress, 16, 2, 5881, (2023).
[19] D. M. Hawkins, “A CUSUM for a Scale
Parameter,” J. Qual. Technol. 13, 228–31
(1981).
[20] 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,” Commun Stat-Simul C. 20, 191-204
(1991).
[21] A. Kharab and R. B. Guenther, An
Introduction to Numerical methods A
MATLAB Approach, 3rd edition, USA: CRC
Press, (2012).
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APPENDIX
Table 1. Values of the upper control limit
()b
with optimal values of
for combination long-memory
)ARFIX , 0( .1, 1p
models at ARL0 = 370.
Long-memory
)ARFIX ,0( .1, 1p
process
Coefficient parameters
1
2
3
1
0.03
0.05
0.10
p = 1
0.1
-
-
0.1
2.5881E-14
2.66337E-08
0.0011283
-0.1
-
-
0.1
3.0534E-14
3.14211E-08
0.0013324
p = 2
0.1
0.2
-
0.1
2.1940E-14
2.25758E-08
0.0009556
-0.1
0.2
-
0.1
2.5881E-14
2.66337E-08
0.0011283
p = 3
0.1
0.2
0.3
0.1
1.7125E-14
1.76181E-08
0.000745
-0.1
0.2
0.3
0.1
2.0200E-14
2.07849E-08
0.0008795
Table 2. Comparison of out-of-control ARL results between the NIE technique and explicit formulas for the long-
memory ARFIX processes when
10.1
running on an EWMA control chart
Long-memory
ARFIX(p,0.1,1)
Technique
1
1.00
1.025
1.05
1.075
1.100
1.125
1.15
1.20
1.30
1.50
p = 1
0.03
NIE
370.000
159.284
71.648
33.721
16.686
8.766
4.962
2.120
1.119
1.003
Explicit
370.000
159.284
71.644
33.720
16.686
8.766
4.962
2.120
1.119
1.003
%Accuracy
100.000
100.000
99.997
100.000
100.000
100.000
100.000
100.000
100.000
100.000
0.05
NIE
370.000
220.118
134.308
83.953
53.716
35.165
23.553
11.338
3.588
1.278
Explicit
370.000
220.118
134.308
83.953
53.716
35.165
23.553
11.338
3.588
1.278
%Accuracy
100.000
100.000
100.000
100.000
100.000
100.000
100.000
100.000
100.000
100.000
0.10
NIE
370.000
279.965
214.627
166.595
130.747
103.744
83.156
54.956
26.535
8.608
Explicit
370.000
279.965
214.627
166.595
130.747
103.744
83.156
54.956
26.535
8.608
%Accuracy
100.000
100.000
100.000
100.000
100.000
100.000
100.000
100.000
100.000
100.000
p = 2
0.03
NIE
370.000
158.663
71.102
33.350
16.453
8.625
4.878
2.089
1.115
1.003
Explicit
370.000
158.672
71.107
33.351
16.453
8.625
4.878
2.089
1.115
1.003
%Accuracy
100.000
99.994
99.993
99.997
100.000
100.000
100.000
100.000
100.000
100.000
0.05
NIE
370.000
219.237
133.263
83.002
52.930
34.543
23.072
11.056
3.491
1.263
Explicit
370.000
219.237
133.263
83.002
52.930
34.543
23.072
11.056
3.491
1.263
%Accuracy
100.000
100.000
100.000
100.000
100.000
100.000
100.000
100.000
100.000
100.000
0.10
NIE
370.000
278.789
212.877
164.581
128.733
101.802
81.340
53.440
25.551
8.191
Explicit
370.000
278.789
212.877
164.581
128.733
101.802
81.340
53.440
25.551
8.191
%Accuracy
100.000
100.000
100.000
100.000
100.000
100.000
100.000
100.000
100.000
100.000
p = 3
0.03
NIE
370.000
157.739
70.292
32.800
16.112
8.419
4.755
2.046
1.108
1.003
Explicit
370.000
157.733
70.295
32.801
16.112
8.419
4.755
2.046
1.108
1.003
%Accuracy
100.000
99.996
99.996
99.997
100.000
100.000
100.000
100.000
100.000
100.000
0.05
NIE
370.000
217.921
131.711
81.596
51.773
33.632
22.370
10.650
3.352
1.242
Explicit
370.000
217.921
131.711
81.596
51.773
33.632
22.370
10.650
3.352
1.242
%Accuracy
100.000
100.000
100.000
100.000
100.000
100.000
100.000
100.000
100.000
100.000
0.10
NIE
370.000
277.037
210.281
161.659
125.776
98.962
78.693
51.250
24.149
7.609
Explicit
370.000
277.037
210.281
161.659
125.776
98.962
78.693
51.250
24.149
7.609
%Accuracy
100.000
100.000
100.000
100.000
100.000
100.000
100.000
100.000
100.000
100.000
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Table 3. Comparison of out-of-control ARL results between the NIE technique and explicit formulas for the long-
memory ARFIX processes when
10.1
running on an EWMA control chart
Long-memory
ARFIX(p,0.1,1)
Technique
1
1.00
1.025
1.05
1.075
1.100
1.125
1.15
1.20
1.30
1.50
p = 1
0.03
NIE
370.000
159.928
72.209
34.102
16.924
8.910
5.048
2.151
1.124
1.003
Explicit
370.000
159.929
72.209
34.102
16.924
8.910
5.049
2.152
1.124
1.003
%Accuracy
100.000
99.999
100.000
100.000
100.000
100.000
99.980
99.954
100.000
100.000
0.05
NIE
370.000
221.004
135.362
84.916
54.515
35.799
24.045
11.626
3.689
1.293
Explicit
370.000
221.004
135.362
84.916
54.515
35.799
24.045
11.626
3.689
1.293
%Accuracy
100.000
100.000
100.000
100.000
100.000
100.000
100.000
100.000
100.000
100.000
0.10
NIE
370.000
281.147
216.394
168.566
132.795
105.726
85.017
56.518
27.559
9.050
Explicit
370.000
281.147
216.394
168.566
132.795
105.726
85.017
56.518
27.559
9.050
%Accuracy
100.000
100.000
100.000
100.000
100.000
100.000
100.000
100.000
100.000
100.000
p = 2
0.03
NIE
370.000
159.285
71.650
33.721
16.686
8.765
4.962
2.120
1.119
1.003
Explicit
370.000
159.284
71.654
33.720
16.686
8.765
4.962
2.120
1.119
1.003
%Accuracy
100.000
99.999
99.994
99.997
100.000
100.000
100.000
100.000
100.000
100.000
0.05
NIE
370.000
220.118
134.308
83.953
53.716
35.165
23.553
11.338
3.588
1.278
Explicit
370.000
220.118
134.308
83.953
53.716
35.165
23.553
11.338
3.588
1.278
%Accuracy
100.000
100.000
100.000
100.000
100.000
100.000
100.000
100.000
100.000
100.000
0.10
NIE
370.000
279.965
214.627
166.559
130.747
103.744
83.156
54.956
26.535
8.608
Explicit
370.000
279.965
214.627
166.559
130.747
103.744
83.156
54.956
26.535
8.608
%Accuracy
100.000
100.000
100.000
100.000
100.000
100.000
100.000
100.000
100.000
100.000
p = 3
0.03
NIE
370.000
158.350
70.826
33.166
16.337
8.556
4.836
2.075
1.113
1.003
Explicit
370.000
158.349
70.828
33.166
16.337
8.556
4.836
2.075
1.113
1.003
%Accuracy
100.000
99.999
99.997
100.000
100.000
100.000
100.000
100.000
100.000
100.000
0.05
NIE
370.000
218.797
132.744
82.531
52.541
34.237
22.835
10.919
3.444
1.256
Explicit
370.000
218.797
132.744
82.531
52.541
34.237
22.835
10.919
3.444
1.256
%Accuracy
100.000
100.000
100.000
100.000
100.000
100.000
100.000
100.000
100.000
100.000
0.10
NIE
370.000
278.204
212.008
163.601
127.739
100.846
80.446
52.699
25.074
7.991
Explicit
370.000
278.204
212.008
163.601
127.739
100.846
80.446
52.699
25.074
7.991
%Accuracy
100.000
100.000
100.000
100.000
100.000
100.000
100.000
100.000
100.000
100.000
Table 6. Comparison of out-of-control ARL results between the NIE technique and explicit formulas for the TIOc1
dataset-based long-memory ARFIX process running on an EWMA and CUSUM control chart
Control
chart
b
ARL
techniques
1
1.025
1.05
1.075
1.100
1.125
1.15
1.20
1.30
1.50
EWMA
0.01
3.242210-11
NIE
188.471
99.309
54.095
30.474
17.787
10.793
4.56032
1.5915
1.0330
Explicit
188.471
99.309
54.095
30.474
17.787
10.793
4.56032
1.5915
1.0330
%Accuracy
100.000
100.000
100.000
100.000
100.000
100.000
100.000
100.000
100.000
0.03
0.008320382
NIE
293.823
235.804
191.116
156.333
128.992
107.302
75.945
41.184
15.598
Explicit
293.823
235.804
191.116
156.333
128.992
107.302
75.945
41.184
15.598
%Accuracy
100.000
100.000
100.000
100.000
100.000
100.000
100.000
100.000
100.000
0.05
0.213871
NIE
310.138
262.247
223.55
191.994
166.039
144.523
111.442
70.408
33.818
Explicit
310.138
262.247
223.55
191.994
166.039
144.523
111.442
70.408
33.818
%Accuracy
100.000
100.000
100.000
100.000
100.000
100.000
100.000
100.000
100.000
Control
chart
a
b
ARL techniques
1.025
1.05
1.075
1.100
1.125
1.15
1.20
1.30
1.50
CUSUM
6
16.35156
NIE
310.012
262.582
224.211
192.904
167.156
145.820
113.057
72.541
36.565
Explicit
310.674
263.115
224.643
193.258
167.448
146.063
113.228
72.631
36.597
%Accuracy
99.787
99.797
99.808
99.817
99.826
99.834
99.849
99.876
99.913
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Fig. 1: Graphical displays of ARL1 results using NIE method running on an EWMA control chart for the long-
memory ARFIX processes with coefficient value
Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
Conceptualization: Wilasinee Peerajit.
Data curation: Wilasinee Peerajit.
Formal analysis: Wilasinee Peerajit.
Funding acquisition: Wilasinee Peerajit.
Investigation: Wilasinee Peerajit.
Methodology: Wilasinee Peerajit.
Software: Wilasinee Peerajit.
Validation: Wilasinee Peerajit.
Writing – original draft: Wilasinee Peerajit.
Writing – review and editing: Wilasinee Peerajit
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
The author gratefully acknowledges the editor and
referees for their valuable comments and suggestions
which greatly improve this paper. This research was
funded by King Mongkut’s University of Technology
North Bangkok, Contract No.KMUTNB-67-BASIC-
21.
Conflict of Interest
The authors have no conflict of interest to declare.
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.en_
US
WSEAS TRANSACTIONS on SYSTEMS
DOI: 10.37394/23202.2024.23.20
Wilasinee Peerajit
E-ISSN: 2224-2678
187
Volume 23, 2024
