Accurate Average Run Length Analysis for Detecting Changes in a
Long-Memory Fractionally Integrated MAX Process Running
on EWMA Control Chart
WILASINEE PEERAJIT
Department of Applied Statistics, Faculty of Applied Science,
King Mongkut’s University of Technology North Bangkok,
1518 Pracharat 1 Road,Wongsawang, Bangsue, Bangkok 10800,
THAILAND
Abstract: - Numerical evaluation of the average run length (ARL) when detecting changes in the mean of an
autocorrelated process running on an exponentially weighted moving average (EWMA) control chart has
received considerable attention. However, accurate computation of the ARL of changes in the mean of a long-
memory model with an exogenous (X) variable, which often occurs in practice, is challenging. Herein, we
provide an accurate determination of the ARL for long-memory models such as the fractionally integrated
MAX processes (FIMAX) with exponential white noise running on an EWMA control chart by using an
analytical formula based on an integral equation. From a computational perspective, the analytical formula
approach is accomplished by solving the solution for the integral equation obtained via the Fredholm integral
equation of the second kind. Moreover, the existence and uniqueness of the solution for the analytical formula
were confirmed via Banach’s fixed-point theorem. Its efficacy was compared with that of the ARL derived by
using the well-known numerical integral equation (NIE) technique under the same circumstances in terms of
the ARL percentage accuracy and computational processing time. The percentage accuracy was 100%, which
indicates excellent agreement between the two methods, and the analytical formula also required much less
computational processing time. An example to illustrate the effectiveness of the proposed approach with a
process involving real data running on an EWMA control chart is also provided herein. The explicit formula
method offers an accurate determination of the ARL and a new approach for validating its computation,
especially for long-memory scenarios running on EWMA control charts.
Key-Words: - Fractionally Integrated Moving Average with Exogenous Variable, Exponential white noise, Numerical
Integral Equation Method.
Received: December 8, 2022. Revised: May 25, 2023. Accepted: June 17, 2023. Published: July 24, 2023.
1 Introduction
Control charts are critical for monitoring processes
in the production and manufacturing sectors. They
are divided into two categories: memory-less and
memory-type charts. The first and most well-known
memory-less control chart is the Shewhart control
chart introduced in the 1920s, [1], which relies
entirely on the present observations without
consideration of past ones. This is why the Shewhart
control chart is only sensitive to detecting large
shifts in a process parameter. On the other hand,
both the current and past data are used in the
plotting statistic of memory-type charts, of which
the exponentially weighted moving average
(EWMA) control chart, [2], and the cumulative sum
(CUSUM) control chart, [3], are the most well-
known. This feature helps them to be sensitive for
detecting small-to-moderate shifts in a process
parameter. The CUSUM control chart is used to
monitor process dispersion while the EWMA
control chart is used to monitor changes in the
process mean. The EWMA control chart has been
widely utilized in a wide range of fields and
operations, including healthcare, manufacturing,
credit card fraud detection, weather monitoring, and
stock exchange trading where the small process
shifts may inflict significant financial penalties. For
more related works on an EWMA chart, we refer to
[4], [5], [6], and therein cited references.
Monitoring the performance of a process is based
on a control chart and the distribution of the
observations from both simple and complex
processes. However, phenomena such as
autocorrelation, which often occur in real situations,
violate the assumption that the observations are
independently and normally distributed. Thus,
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normal and non-normal distributions must be
considered for effective process monitoring.
Autocorrelation in processes can be captured
using time series models. An important category of
these is the stationary process model, in which it is
assumed that the process is stable around a constant
mean. This type of model provides a foundation for
monitoring processes involving autocorrelation. In
the present research, we considered the following
fundamental time series models. The conventional
Box-Jenkins autoregressive (AR) integrated moving-
average (MA) (ARIMA) model can be generalized as
the AR fractionally integrated MA (ARFIMA) model,
which enables non-integer (fractional) differencing
parameter values. The ARFIMA processes contain a
fractional differencing parameter (d) that is used to
determine whether the model is stationary and
invertible, [7], [8], [9], [10], [11]. A complete
explanation of long-memory processes is provided in
[12].
Numerous applications in fields such as
economics, finance, environmental research, and
engineering involve long-memory processes.
In [13], the study used ARFIMAX models for
estimating the realized volatilities in a Dow Jones
Industrial Average portfolio. Although there is a
relationship between econometric models and
economic indicators (variables affecting economic
forecasting), the exogenous variable is not affected
by other variables in the system, only by external
influences such as exchange, interest, and inflation
rates, among others. Exogenous variables affect an
econometric model when forecasting economic
situations. If the forecasting model includes an
exogenous variable for economic forecasting and
other fields, the model is usually more accurate than
the one without it. The EWMA control chart has
often been used with long-memory processes
involving time series, [14], [15].
The error in a time series model (also called
white noise) is defined as the difference between the
actual and estimated values. This should be
minimized to maximize the accuracy of the model.
It is not always the case that the white noise (also
called Gaussian white noise) created by
autocorrelated data follows a normal distribution.
Considering non-Gaussian white noise has been
effective in studying many phenomena, such as
wind speed, oxygen concentration, and water flow
rate. Numerous academicians have concentrated on
time series models using non-Gaussian white noise,
with exponentially distributed white noise being
especially interesting, [16], [17].
Evaluating the performance of the EWMA
control chart can be made based on the average run
length (ARL), which is the average number of
consecutive points in a process that falls within the
control limits before an out-of-control signal is
given. ARL0 denotes the in-control ARL value,
whereas ARL1 denotes the out-of-control ARL
value. ARL0 should be the largest value, while
ARL1 should be the smallest value for measuring
the performance of charts. The ARL can be
computed via Monte Carlo simulation, the Markov
Chain approach, or the integral equation technique.
There are two types of integral equation techniques:
using an analytical formula and the numerical
integral equation (NIE) technique. Many researchers
have calculated the ARL through the solution of an
integral equation. In [18], the authors derived
analytical formulas for the ARL for MA(q)
processes with exponential white noise running on
EWMA and CUSUM control charts. Recently, [19],
the authors used the integral equation technique to
provide an analytical formula for the ARL of a
stationary MAX process running on an EWMA
control chart. Finally, in [20], the author derived the
ARL for a long-memory seasonality SFIMAX
model with exponential white noise running on a
CUSUM control chart using analytical formulas.
The existence and uniqueness of a solution for the
analytical formula of the ARL can be proved by
using Banach’s fixed-point theorem, [21], [22]. As
mentioned above, the research has applied the NIE
technique to verify the accuracy of an analytical
formula, which is an accepted method for evaluating
the performance of control charts.
The main aim of the present study is to derive an
analytical formula to accurately compute the ARL
for a long-memory FIMA model focusing on an
exogenous (X) variable with exponential white noise
running on an EWMA control chart and compare its
efficacy with that using the well-established NIE
method. In addition, the analytical formula for
detecting changes in the mean is applied to
processes involving real data.
The rest of the article is as follows. In Section 2,
we provide brief outlines of the
FIMAX( ),,d q r
model with exponential white noise and the EWMA
control chart. The ARLs obtained by using the
analytical formula and NIE techniques are also
provided. In Section 3, a performance comparison
of the proposed analytical formula with the NIE
technique is provided. An example of a process
involving real data is also presented to illustrate the
effectiveness of the proposed technique. Section 4
offers conclusions on the study. Finally, the
existence and uniqueness of the ARL computation
were confirmed via Banach’s fixed-point theorem,
the details of which are shown in Appendix A.
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2 Materials and Methods
Here, brief outlines of the
FIMAX( ),,d q r
model with
exponential white noise and the EWMA control
chart along with ARL computations derived by
using the analytical formula and NIE techniques are
provided.
2.1 Preliminaries
The long memory fractional integration
MAX( ), (or FIMAX( )), , , ,d q r d q r
model was
chosen for this study because it is stationary (as most
processes are in practice) and contains both
fractionally integrated and MA components with an
exogenous (X) variable. Hence, the effect of each
parameter can be examined. In addition, we
consider white noise with an exponential
distribution.
2.1.1 The Long-Memory
FIMAX( ),,d q r
Model
with Exponential White Noise
Let
t
Y
be a sequence of a long-memory
FIMAX( ),,d q r
model where
d
is the fractional
integration parameter,
q
is the order of the MA
process, and
r
is the explanatory variable order in
the model, [13]. The latter can be written as
1
(1 ) ( ) ( ) ,
r
dt t j jt
j
B Y B X
(1)
where
is the mean of
t
Y
2
12
( ) 1 ...B B B
q
qB
comprising MA
polynomials on backward-shift operator B (
12
, ,..., q
are the coefficients for the MA
polynomials),
jt
X
are explanatory variables,
j
are
unknown parameters, and
t
is a white noise
process assumed to be exponentially distributed as
~()
tExp v
when shift parameter
0.v
To
determine whether the process is long-memory, the
fractional
()d
can take on non-integer values in the
range
(0,0.5);
this fractional order of integration
gives rise to the long-memory FIMAX model, [11].
Since the fractional difference operator
(1 )d
B
is defined by the expansion
0
(1 ) ( )
dk
k
d
BB
k
23
(1 ) (1 )(2 )
1 ...,
26
d d d d d
dB B B
(2)
for any real value of
d
, the fractionally integrated
white noise process can be defined as
(1 ) ,
dtt
BY
1 2 3
(1 ) (1 )(2 ) ... ,
26
t t t t t
d d d d d
Y dY Y Y
(3)
Note that
kt t k
B Y Y
for order k.
Therefore, equations (1) and (3) can be
rearranged to satisfy the generalized form of the
FIMAX model as follows:
1 1 2 2
1 1 2 2
1 2 3
...
...
(1 ) (1 )(2 ) ... ,
26
t t t t q t q
t t r rt
t t t
Y
X X X
d d d d d
dY Y Y
or
11
qr
t t i t j j jt
ij
YX
1 2 3
(1 ) (1 )(2 ) ... ,
26
t t t
d d d d d
dY Y Y
(4)
where
1; 1,2,...,
iiq
are MA coefficients and
; 1,2,...,
jjr
are coefficients depending on
variable
.r
The initial value of a long-memory
FIMAX( ),,d q r
model must satisfy
1 2 3
, , ,...,
t t t
Y Y Y
and
12
, ,...,
t t rt
X X X
= 1. For
exponential white noise, the initial value of
t
is 1.
By using this fact, we can apply the generalized
form of the
FIMAX( ),,d q r
model in equation (4) to
the EWMA control chart.
2.1.2 The EWMA Control Charts for Long-
Memory
d q rF AX( ,M ,I)
Model with
Exponential White Noise
The EWMA control chart is exceptional at rapidly
detecting small-to-moderate shifts in a process
parameter it suitably assigns weights to both the
current and the past observations. The EWMA
control statistic
()
t
Z
for monitoring a shift in the
process mean is given by
0
1
,0
(1 ) , 1, 2,... ,
ttt
Zt
ZZ Y t
(5)
where the initial value
00
ZY
(the target process
mean),
t
Y
is the sequence of the
FIMAX( ),,d q r
process with exponential white noise and
is the
smoothing parameter (or weighting parameter)
satisfying
(0,1].
In general, a large value (close
to one) of the smoothing constant is suitable for
detecting a large shift while a small value (
[0.05,0.25]
) is recommended for detect a small
shift, [23].
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Note that, when
is large (close to one), a
relatively lower weight is given to older data,
leading to a short-memory process on the EWMA
control chart. Indeed,
1
is equivalent to the
Shewhart control chart. Meanwhile, as the value of
approaches zero, more weight is given to the
older observations than the most recent ones. Thus,
for very small values of
, the EWMA control
chart becomes more like the CUSUM control chart,
in which observations are weighted equally, [24].
The value of
and the control limits of the
EWMA control chart have a strong impact on its
performance. Thus, their values should be carefully
chosen by the user to bestow the chart with
desirable properties for both in-control and out-of-
control situations.
The upper control limit (UCL), control limit
(CL), and lower control limit (LCL) of the EWMA
control chart are respectively defined as
2
0
0
1 (1 ) ,
2
,
t
UCL L
CL
2
01 (1 ) ,
2
t
LCL L
(6)
where
0
and
are the process mean and standard
deviation, respectively, and L is the design
parameter for the EWMA control chart, the value of
which depends on the choice of the smoothing
constant λ and the desired value of the in-control
ARL. For the EWMA control chart statistic
t
Z
, an
out-of-control signal occurs whenever
> UCL
t
Z
or
LCL.
t
Z
2.2 Computation of the ARL for a Long-
Memory FIMAX(d, q, r) Model with
Exponential White Noise on a One-Sided
EWMA Control Chart
To evaluate the performance of the EWMA control
chart in terms of the ARL of a long-memory
FIMAX( ),,d q r
model running on it, we derived it
using both the analytical formula and NIE
techniques based on integral equations while
focusing on the upper-sided EWMA. The successive
values of the EWMA statistic generated by the long-
memory
FIMAX( ),,d q r
process in equation (4)
can be expressed as
111
(1 )
qr
t t t i t j j jt
ij
Z Z X
1 2 3
(1 ) (1 )(2 ) ...,
26
t t t
d d d d d
d Y Y Y
(7)
where the initial value for monitoring with the
EWMA statistic is
0; 0 .ZH
Let
H
be the stopping time for detecting when
the out-of-control process on an upper-sided
EWMA control chart exceeds the given
predetermined threshold for the first time; i.e.,
inf 0; ,
Ht
t Z H
(8)
where
H
is the predetermined UCL of the EWMA
control chart. If
t
Z
is in the range
0t
ZH
, then
the process is in control, which can be defined as
((1 )
1
11
23
(1 ) (1 )(2 ) ...
26
qr
i t j j jt t
ij
tt
X dY
d d d d d
YY
)t
(1 )H
1
11
23
(1 ) (1 )(2 ) ...
26
qr
i t j j jt t
ij
tt
X dY
d d d d d
YY
)
or
.
t
lu
2.2.1 Derivation of the ARL as an Analytical
Formula based on an Integral Equation
Here, the analytical formula is derived as the
solution to an integral equation.
Let
()
L
denote the ARL of a long-memory
FIMAX( ),,d q r
model with initial value
0
Z
running on an EWMA control chart; i.e.,
ARL ( ) ( ).
H
E
L
Function
()
L
can be
written in the form
( ) 1
pP l u
L
(1 ((1 )
u
l
L
1
11
23
(1 ) (1 )(2 ) ...
26
qr
i t j j jt t
ij
tt
X dY
d d d d d
YY
) ( )fd
1 ((1 )
u
l
L
1
11
23
(1 ) (1 )(2 ) ...
26
qr
i t j j jt t
ij
tt
X dY
d d d d d
YY
) ( ) .fd
(1 )
1
11
23
(1 ) (1 )(2 ) ...
26
qr
i t j j jt t
ij
tt
X dY
d d d d d
YY
can be
used to change the integral variable. Thereby, we
can obtain the integral equation as
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1
0
1 2 3
1
1 (1 )
( ) 1 + ( ) (
(1 ) (1 )(2 ) ... )
26
Hq
p i t j
i
r
j jt t t t
j
f
d d d d d
X dY Y Y d
LL
(9)
0
11
( ) 1 + ( ) exp
H
pvv
LL
(1 )
exp v
1
11
23
(1 ) (1 )(2 ) ...
26
qr
i t j j jt t
ij
tt
X dY
d d d d d
YY
1
d
Accordingly, the integral equation is derived from
the Fredholm integral equation of the second kind as
follows:
0
11
( ) 1 + ( ) exp
H
pvv
LL
(1 )
exp v
1
11
23
(1 ) (1 )(2 ) ...
26
qr
i t j j jt t
ij
tt
X dY
d d d d d
YY
1d
v
0
1
( ) 1 + ( ) exp
H
pv
LL
(1 )
exp v
1
11
23
(1 ) (1 )(2 ) ...
26
qr
i t j j jt t
ij
tt
X dY
d d d d d
YY
1d
v
(10)
Equation
10
corresponds to the analytical formula,
which is defined as
0
1
( ) ( ) exp
H
pv
LL
(1 )
exp
v
1
11
23
(1 ) (1 )(2 ) ...
26
qr
i t j j jt t
ij
tt
X dY
d d d d d
YY
11d
v
(11)
where
0
1 (1 )
( , ) ( ) exp exp
H
kg vv
L
1
11
23
(1 ) (1 )(2 ) ...
26
qr
i t j j jt t
ij
tt
X dY
d d d d d
YY
1d
v
is a kernel
function,
,:[0 ]HL
as an unknown function,
and the mapping
T
is defined as
0
1
( ) 1 + ( ) exp
H
p
Tv
LL
(1 )
exp v
1
11
23
(1 ) (1 )(2 ) ...
26
qr
i t j j jt t
ij
tt
X dY
d d d d d
YY
1d
v
(12)
This existence and uniqueness of the ARL
computation were confirmed via Banach’s fixed-
point theorem, the details of which are shown in
Appendix A.
Theorem 1 (Banach’s fixed-point theorem, see,
[25]).
Let
,MM
.
be a complete metric space, then
the mapping
:T M M
is said to be a contraction
mapping on
M
if there exists real number
; 0 1.
, such that
1 2 1 2
( ), ( ) ,TT
L L L L
for
12
, .MLL
Subsequently,
T
has a precisely unique fixed point
(e.g. unique
().ML
such that
TLL
). Let
(1 )
( ) exp
Qv
1
11
23
(1 ) (1 )(2 ) ...
26
qr
i t j j jt t
ij
tt
X dY
d d d d d
YY
1,
d
v
0.
H
Consequently,
0
()
( ) 1 + ( ).exp , 0 .
H
pQdH
vv
LL
Let so we have
()
( ) 1 + .
pQ
v
L
By solving constant we
obtain
0
()
1 + .exp
HQd
vv
00
exp ( )exp
HH
d Q d
v v v
0
exp
Hd
vv
0
(1 )
exp
H
v
+
1
11
23
(1 ) (1 )(2 ) ...
26
qr
i t j j jt t
ij
tt
X dY
d d d d d
YY
1.exp
d
vv
1 exp
H
vv
1
exp
v
1
11
23
(1 ) (1 )(2 ) ...
26
qr
i t j j jt t
ij
tt
X dY
d d d d d
YY
. 1 exp
H
v
1
exp
v
1
11
23
(1 ) (1 )(2 ) ...
26
qr
i t j j jt t
ij
tt
X dY
d d d d d
YY
. 1 exp
H
v
1 exp
H
vv
0
( ).exp ,
Hd
v
L
0
( ).exp ,
Hd
v
L
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Hence,
1
1 exp 1
H
vv
1
exp .
v
1
11
23
(1 ) (1 )(2 ) ...
26
qr
i t j j jt t
ij
tt
X dY
d d d d d
YY
. 1 exp
H
v
(13)
Substitute a constant
into the Equation
()
( ) 1 + .
pQ
v
L
, then
()
p
L
as
(1 )
( ) exp
Qv
1
11
23
(1 ) (1 )(2 ) ...
26
qr
i t j j jt t
ij
tt
X dY
d d d d d
YY
1
v
where
0.
H
(1 )
( ) 1 + exp
pv
L
1
11
23
(1 ) (1 )(2 ) ...
26
qr
i t j j jt t
ij
tt
X dY
d d d d d
YY
1
.
v
1
1 exp 1 1 exp
HH
vv
exp
1
11
23
(1 ) (1 )(2 ) ...
26
qr
i t j j jt t
ij
tt
X dY
d d d d d
YY
1
1
.
v
Thereby,
()
p
L
can be written in the form
(1 )
( ) 1 1 exp .exp
pH
vv
L
1
1 exp exp
H
vv
1
11
23
(1 ) (1 )(2 ) ...
26
qr
i t j j jt t
ij
tt
X dY
d d d d d
YY
1
(14)
According to equation (14), when the process is
in control, the parameter
v
can be replaced with
0
v
.
Subsequently, the analytical formula for the in-
control ARL becomes
0
00
(1 )
ARL 1 1 exp .exp
H
vv
00
1
1 exp exp
H
vv
1
11
23
(1 ) (1 )(2 ) ...
26
qr
i t j j jt t
ij
tt
X dY
d d d d d
YY
1
(15)
On the contrary, for the out-of-control process,
the parameter
v
can be replaced with
1.v
Therefore,
the analytical formula for the out-of-control ARL
can be written as
1
11
(1 )
ARL 1 1 exp .exp
H
vv
11
1
1 exp exp
H
vv
1
11
23
(1 ) (1 )(2 ) ...
26
qr
i t j j jt t
ij
tt
X dY
d d d d d
YY
1
(16)
This ARL derived from the analytical formula
shows that the calculation scheme can be easily
performed.
2.2.1 The Approximate ARL by using the NIE
Technique based on an Integral Equation
The NIE technique is usually used to verify the
accuracy of an analytical formula. It is based on the
solution for the integral equation, [26], in equation (9).
The composite midpoint Rule is applied to divide
domain interval
[0, ]H
into
m
sub-grids of equal
length; i.e.,
1
11
0
(1 )
( ) (
Hqr
i t j j jt t
ij
f X dY
L
23
1
(1 ) (1 )(2 ) ...) ( )
26
m
t t j j
j
d d d d d
Y Y d w f a
(17)
We can then substitute equation (17) into equation
(9) to obtain a linear system of equations. Thereby,
the approximate ARL calculated by using the NIE
technique can be written in the form
1
(1 )
1
( ) 1 + ()
m
j
j
j
jNN
wa
fa
LL
1
11
23
(1 ) (1 )(2 ) ...
26
qr
i t j j jt t
ij
tt
X dY
d d d d d
YY
(18)
with a set of constant weights
,
jH
wm
and
1; 1,2, , .
2
jH
a j j m
m
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2.3 Algorithms to establish the in-control and
out-of-control ARL values
Algorithms were constructed to determine the
control limits and obtain results for the out-of-
control ARL.
2.3.1 Construction of the control limits
Algorithm 1: The analytical formula derived by
using the Mathematica program to establish the in-
control ARL value
Step-1: Solve the generalized form of the long-
memory FIMAX(0.05, 1, 1) model with
exponential white noise defined as
t
Y
:
(1.1) MA coefficients
10.8, 0.4, 0.2.
(1.2) Exogenous variable coefficient
11.
(1.3) The mean of the exponential parameter
values
( ( ))
tExp
for the in-control
process
0
( = )
= 1.
Step-2: Compute the EWMA statistic:
(2.1) Smoothing parameter
= 0.01.
(2.2) Compute the proposed EWMA statistic
()
t
Z
for the long-memory
FIMAX
mode given in equation (5).
Step-3: Compute decision interval H in conjunction
with
by utilizing equation (15) so that the
attained in-control ARL is close to or equal to
500 corresponding to the specific shift size
( ) 0.
Step-4: Repeat Steps 2 and 3 for
= 0.05 and 0.10
Step-5: Repeat Steps 1–4 for long-memory
FIMAX(0.2, 1, 1) and FIMAX(0.40, 1, 1).
2.3.2 Computation of the out-of-control ARL
Algorithm 2: Analytical formula derived from
Mathematica program for a shift in the process
mean from
0
to
1
, where
10
(1 )
Step-1: Repeat Algorithm 1, Steps 1 and 2 to solve
the generalized form of a long-memory
FIMAX(0.05, 1, 1) process running on an
EWMA control chart.
Step-2: Compute the out-of-control ARL for
changes in the process mean:
(2.1) Take the value of the control
coefficient
( , )H
from the output of
Algorithm 1.
(2.2) Computation of the out-of-control ARL
corresponding to
( , )H
a shift size of
0.01 by utilizing equation (16).
Step-3: Record the computational time for the first
out-of-control ARL signal from the control
limits.
Step-4: Repeat Steps 2 and 3 for
= 0.05, 0.25,
0.50, 0.75, or 1.00.
Step-5: Repeat Steps 1-4 for long-memory
FIMAX(0.2, 1, 1) and FIMAX(0.40, 1, 1).
3 Results and Discussion
The details and results of a comparative study of the
performances of the proposed analytical formula with
the
NIE
technique are provided in this section. An
example of a process involving real data to illustrate the
effectiveness of the proposed technique is also offered.
Table 1. The values of H for various
FIMAX( ),,qrd
models and values
1
for in-control
ARL
= 500.
Coefficient parameters
λ
d
1
1
0.01
0.05
0.10
0.05
0.80
0.10
2.50211521E-13
2.57482300E-07
1.12643400E-02
0.40
0.10
1.67720900E-13
1.72595400E-07
7.41667000E-03
0.20
0.10
1.37319000E-13
1.41309200E-07
6.03307000E-03
-0.20
0.10
9.20490000E-14
9.47223000E-08
4.00579000E-03
-0.40
0.10
7.53640000E-14
7.75521000E-08
3.26830000E-03
-0.80
0.10
5.05179300E-14
5.19847000E-08
2.17959000E-03
0.20
0.80
0.10
1.97060000E-13
2.02783300E-07
8.76905000E-03
0.40
0.10
1.32090000E-13
1.35929700E-07
5.79698000E-03
0.20
0.10
1.08148495E-13
1.11289800E-07
4.72230000E-03
-0.20
0.10
7.24938000E-14
7.45998000E-08
3.14201000E-03
-0.40
0.10
5.93510000E-14
6.10771000E-08
2.56548000E-03
-0.80
0.10
3.97871000E-14
4.09412000E-08
1.71278400E-03
0.40
0.80
0.10
1.52550000E-13
1.56983000E-07
6.72397000E-03
0.40
0.10
1.02258000E-13
1.05228900E-07
4.45961000E-03
0.20
0.10
8.37213700E-14
8.61542000E-08
3.63712200E-03
-0.20
0.10
5.61205000E-14
5.77508000E-08
2.42414100E-03
-0.40
0.10
4.59500000E-14
4.72823000E-08
1.98057000E-03
-0.80
0.10
3.08010000E-14
3.16943000E-08
1.32350000E-03
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Volume 22, 2023
The performance metric for the comparison is
( ) ( )
%Accuracy 100 100%,
()
pN
p
LL
Lp
(19)
where
()
p
L
and
()
N
L
are the
ARL
values
obtained by using the analytical formula and
NIE
techniques, respectively. A value greater than 95%
means that the proposed formula provided an out-of-
control ARL value close to that for the
NIE
technique, which indicates good agreement between
them.
The first task was to compute the value of
decision interval H in conjunction with the selection
of
so that the in-control ARL is close to the target
value (500 in this case, which is commonly used in
the statistical process monitoring) The values for H
for various models and values of the coefficient
parameter using Algorithm 1 are reported in Table
1.
3.1 Performance Comparison
Using the models and parameter values in Table 1,
we computed the out-of-control ARL obtained by
using the analytical formula and NIE techniques for
FIMAX(0.05, 1, 1), FIMAX(0.20, 1, 1), and
FIMAX(0.40, 1, 1) models running on an EWMA
control chart, the results for which are reported in
Table 2, Table 3, and Table 4. The first row of each
cell in the tables shows the out-of-control ARL
values using the analytical formula and NIE
techniques corresponding to shift magnitudes of
0.01, 0.05, 0.25, 0.50, 0.75, or 1.00 (in that order)
and the second row shows the computational time
(seconds). For each value of the smoothing
parameter
()
(see first column) and MA
coefficient (see second column), the best-
performing technique is indicated in bold.
Moreover, similar performances of the two
techniques are indicated in the percentage accuracy
(Acc%) column by gray shading.
The results suggest that the out-of-control ARL
values calculated by using the analytical formula are
close to those approximated by using the NIE
technique. As expected, as the shift size was
increased, the sensitivity (i.e., the out-of-control ARL
values) of both techniques also increased. In
particular, both techniques showed great sensitivity
by quickly detecting small-to-moderate shifts in the
process mean
(0< 0.5)
but not moderate-to-
large and large shifts
(0.50 1.00).
The precision values of the proposed analytical
formula compared to the NIE technique in terms of
percentage accuracy are reported in Table 2, Table 3,
and Table 4.
It can be seen that the percentage accuracy
results were 100% in all cases, implying good
agreement between the two methods and that the
proposed analytical formula is very accurate.
The computational times for calculating the out-
of-control ARL values only took a fraction of a
second with the analytical formula compared to 3–
120 seconds with the NIE technique. As
was
decreased, the computational time increased
inversely with the out-of-control ARL value.
It can be seen that the lowest out-of-control ARL
values occurred with the following long-memory
models: FIMAX(0.40, 2, 1), FIMAX (0.20, 1, 1), and
FIMAX(0.05, 1, 1).
The out-of-control ARL values for FIMAX(0.4, 1, 1)
with different values of coefficient parameter
1
are shown in Figure 1. The results reveal that out-
of-control ARL values tended to decrease rapidly
when the magnitude of the shift was small
( 0.25)
, followed by small-to-moderate shifts
(0.25 0.50)
for all cases. The green line for
= 0.01 indicates the lowest out-of-control ARL
value. Meanwhile, the out-of-control ARL values
were for all shift sizes and levels of
.
In summary, the analytical formula performed
exceptionally well in detecting small-to-moderate
changes in the mean of a long-memory FIMAX
model running on the EWMA control chart. Its
accuracy was confirmed by comparison with the
well-established NIE technique. Moreover, it could
compute out-of-control ARL values much more
quickly than with the NIE technique.
3.2 Application of the Proposed Technique to
Processes Involving Real Data
For this demonstration, we used movements in the
gold futures price, [27], with the UDS/THB
exchange rate, [28], as the exogenous variable. As
the USD/THB exchange rate increased (i.e., the
Thai baht depreciates), the price of gold decreased,
and vice versa. The dataset covers the period from
September 1 , 2001, to January 1 , 2023, and
comprises 2 5 7 daily observations. We tested
whether the dataset can fit a long-memory process
and the distribution of the white noise by utilizing
the statistical software packages Eviews and SPSS,
respectively.
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DOI: 10.37394/23206.2023.22.58
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Volume 22, 2023
Table 2. Out-of-control ARL values are computed by using the analytical formula and NIE technique for
FIMAX( 0 ),.05 1, 1d
running on an EWMA control chart when the in-control ARL is 500.
1
δ
0.01
Acc
%
0.05
Acc
%
0.25
Acc
%
0.50
Acc
%
0.75
Acc
%
1.00
Acc
%
()
puL
()
NuL
()
puL
()
NuL
()
puL
()
NuL
()
puL
()
NuL
()
puL
()
NuL
()
puL
()
NuL
0.01
0.8
361.892
361.892
100%
105.922
105.922
100%
1.701
1.701
100%
1.009
1.009
100%
1.000
1.000
100%
1.000
1.000
100%
(0.001)
(3.95)
(0.001)
(4.25)
(0.001)
(5.96)
(0.001)
(7.22)
(0.001)
(9.10)
(0.001)
(11.53)
0.4
360.462
360.462
100%
103.943
103.943
100%
1.647
1.647
100%
1.007
1.007
100%
1.000
1.000
100%
1.000
1.000
100%
(0.001)
(14.02)
(0.001)
(15.58 )
(0.001)
(17.42 )
(0.001)
(19.12)
(0.001)
(21.45)
(0.001)
(22.88)
0.2
359.749
359.749
100%
102.966
102.966
100%
1.622
1.622
100%
1.007
1.007
100%
1.000
1.000
100%
1.000
1.000
100%
(0.001)
(24.11)
(0.001)
(25.69 )
(0.001)
(27.45 )
(0.001)
30.05 )
(0.001)
(31.11)
(0.001)
(32.98)
-0.2
358.338
358.338
100%
101.046
101.046
100%
1.574
1.574
100%
1.006
(1.006
100%
1.000
1.000
100%
1.000
1.000
100%
(0.001)
(34.05)
(0.001)
(36.48)
(0.001)
(39.81)
(0.001)
41.5)
(0.001)
(43.14)
(0.001)
(44.80)
-0.4
357.632
357.632
100%
100.098
100.098
100%
1.551
1.551
100%
1.005
(1.005
100%
1.000
1.000
100%
1.000
1.000
100%
(0.001)
(46.85)
(0.001)
(49.13)
(0.001)
(52.1)
(0.001)
(53.98)
(0.001)
(54.22)
(0.001)
(57.68)
-0.8
356.218
356.218
100%
98.225
98.225
100%
1.509
1.509
100%
1.004
1.004
100%
1.000
1.000
100%
1.000
1.000
100%
(0.001)
(58.46)
(0.001)
(60.22)
(0.001)
(63.59)
(0.001)
(65.87)
(0.001)
(67.23)
(0.001)
(69.98)
0.05
0.8
412.819
412.819
100%
198.975
198.975
100%
11.0908
11.0908
100%
1.7243
1.7243
100%
1.108
1.108
100%
1.025
1.03
100%
(0.001)
)4.11)
(0.001)
(5.782)
(0.001)
(7.42)
(0.001)
(9.094)
(0.001)
(10.79)
(0.001)
(12.55)
0.4
411.191
411.191
100%
195.239
195.239
100%
10.3149
10.3149
100%
1.6339
1.6339
100%
1.091
1.091
100%
1.021
1.021
100%
(0.001)
(14.23)
(0.001)
(15.89)
(0.001)
(17.61)
(0.001)
(19.42)
(0.001)
(21.11)
(0.001)
(22.87)
0.2
410.379
410.379
100%
193.398
193.398
100%
9.9598
9.9598
100%
1.593
1.593
100%
1.083
1.083
100%
1.019
1.019
100%
(0.001)
(24.68)
(0.001)
(26.36)
(0.001)
(27.98)
(0.001)
(29.65)
(0.001)
(31.45)
(0.001)
(33.15)
-0.2
408.762
408.762
100%
193.398
189.768
100%
9.262
9.262
100%
1.519
1.519
100%
1.070
1.070
100%
1.015
1.015
100%
(0.001)
(34.81)
(0.001)
(36.48)
(0.001)
(39.81)
(0.001)
(41.5)
(0.001)
(43.14)
(0.001)
(44.8)
-0.4
407.955
407.955
100%
187.979
187.979
100%
8.938
8.938
100%
1.486
1.486
100%
1.064
1.064
100%
1.014
1.014
100%
(0.001)
(48.23)
(0.001)
(49.93)
(0.001)
(51.56)
(0.001)
(53.2)
(0.001)
(54.87)
(0.001)
(56.56)
-0.8
406.347
406.347
100%
184.451
184.451
100%
8.327
8.327
100%
1.425
1.425
100%
1.054
1.054
100%
1.011
1.011
100%
(0.001)
(58.39)
(0.001)
(60.05)
(0.001)
(65.01)
(0.001)
(66.67)
(0.001)
(68.36)
(0.001)
(70.00)
0.10
0.8
455.816
455.816
100%
320.169
320.169
100%
75.9308
75.9308
100%
21.441
21.441
100%
8.885
8.885
100%
4.789
4.789
100%
(0.001)
(71.74)
(0.001)
(73.45)
(0.001)
(75.11)
(0.001)
(78.47)
(0.001)
(80.12)
(0.001)
(81.87)
0.4
453.823
453.823
100%
313.505
313.505
100%
69.6153
69.615
100%
18.668
18.668
100%
7.543
7.543
100%
4.051
4.051
100%
(0.001)
(83.53)
(0.001)
(85.25)
(0.001)
(86.9)
(0.001)
(88.56)
(0.001)
(90.19)
(0.001)
(91.89)
0.2
452.842
452.842
100%
310.269
310.269
100%
66.7045
66.7045
100%
17.445
17.445
100%
6.970
6.970
100%
3.743
3.743
100%
(0.001)
(93.64)
(0.001)
(95.30)
(0.001)
(96.97)
(0.001)
(98.8)
(0.001)
(100.47)
(0.001)
(102.11)
-0.2
450.903
450.903
100%
303.957
303.957
100%
61.3078
61.3078
100%
15.274
15.274
100%
5.982
5.982
100%
3.223
3.223
100%
(0.001)
(103.75)
(0.001)
(105.50)
(0.001)
(107.14)
(0.001)
(110.45)
(0.001)
(112.28)
(0.001)
(113.97)
-0.4
449.944
449.944
100%
300.872
300.872
100%
58.8005
58.8005
100%
14.307
14.307
100%
5.555
5.555
100%
3.003
3.003
100%
(0.001)
(115.69)
(0.001)
(119.11)
(0.001)
(120.83)
(0.001)
(122.5)
(0.001)
(124.15)
(0.001)
(125.84)
-0.8
448.033
448.033
100%
294.827
294.827
100%
54.1259
54.1259
100%
12.579
12.579
100%
4.813
4.813
100%
2.629
2.629
100%
(0.001)
(127.62)
(0.001)
(129.26)
(0.001)
(130.9)
(0.001)
(132.59)
(0.001)
(134.41)
(0.001)
(136.11)
Note: The numerical results in parentheses are computational times in seconds
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DOI: 10.37394/23206.2023.22.58
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Volume 22, 2023
Table 3. Out-of-control ARL values are computed by using the analytical formula and NIE technique for
FIMAX( 0 ),.2 1, 1d
running on an EWMA control chart when the in-control ARL is 500.
1
δ
0.01
Acc
%
0.05
Acc
%
0.25
Acc
%
0.50
Acc
%
0.75
Acc
%
1.00
Acc
%
()
puL
()
NuL
()
puL
()
NuL
()
puL
()
NuL
()
puL
()
NuL
()
puL
()
NuL
()
puL
()
NuL
0.01
0.8
361.039
361.039
100%
104.738
104.738
100%
1.668
1.668
100%
1.007
1.007
100%
1.000
1.000
100%
1.000
1.000
100%
(0.001)
(5.44)
(0.001)
(7.15)
(0.001)
(9.25)
(0.001)
(11.25)
(0.001)
(12.15)
(0.001)
(13.25)
0.4
359.611
359.611
100%
102.777
102.777
100%
1.617
1.617
100%
1.006
1.006
100%
1.000
1.000
100%
1.000
1.000
100%
(0.001)
(15.22)
(0.001)
(16.25)
(0.001)
(19.56)
(0.001)
(20.15)
(0.001)
(21.55)
(0.001)
(24.12)
0.2
358.909
358.909
100%
101.814
101.814
100%
1.592
1.592
100%
1.006
1.006
100%
1.000
1.000
100%
1.000
1.000
100%
(0.001)
(25.2)
(0.001)
(27.55)
(0.001)
(29.1)
(0.001)
(30.56)
(0.001)
(31.22)
(0.001)
(32.96)
-0.2
357.493
357.493
100%
99.912
99.912
100%
1.547
1.547
100%
1.005
1.005
100%
1.000
1.000
100%
1.000
1.000
100%
(0.001)
(34.96)
(0.001)
(37.15)
(0.001)
(38.2)
(0.001)
(39.68)
(0.001)
(41.24)
(0.001)
(42.56)
-0.4
356.775
356.775
100%
98.972
98.972
100%
1.525
1.525
100%
1.005
1.005
100%
1.000
1.000
100%
1.000
1.000
100%
(0.001)
(44.15)
(0.001)
(45.89)
(0.001)
(48.05)
(0.001)
(49.78)
(0.001)
(51.13)
(0.001)
(52.98)
-0.8
355.385
355.385
100%
97.132
97.132
100%
1.485
1.485
100%
1.004
1.004
100%
1.000
1.000
100%
1.000
1.000
100%
(0.001)
(54.15)
(0.001)
(55.39)
(0.001)
(57.15)
(0.001)
(59.45)
(0.001)
(62.78)
(0.001)
(63.85)
0.05
0.8
411.846
411.846
100%
196.736
196.736
100%
10.620
10.620
100%
1.669
1.669
100%
1.097
1.097
100%
1.023
1.023
100%
(0.001)
(5.79)
(0.001)
(7.48)
(0.001)
(9.07)
(0.001)
(10.68)
(0.001)
(12.34)
(0.001)
(13.95)
0.4
410.223
410.223
100%
193.043
193.043
100%
9.881
9.881
100%
1.585
1.585
100%
1.082
1.082
100%
1.018
1.018
100%
(0.001)
(15.54)
(0.001)
(17.15)
(0.001)
(18.79)
(0.001)
(20.43)
(0.001)
(22.04)
(0.001)
(23.65)
0.2
409.413
409.413
100%
191.223
191.223
100%
9.532
9.532
100%
1.548
1.548
100%
1.075
1.075
100%
1.017
1.017
100%
(0.001)
(25.23)
(0.001)
(26.84)
(0.001)
(28.45)
(0.001)
(30.15)
(0.001)
(31.79)
(0.001)
(33.54)
-0.2
407.799
407.799
100%
187.634
187.634
100%
8.876
8.876
100%
1.479
1.479
100%
1.063
1.063
100%
1.014
1.014
100%
(0.001)
(35.2)
(0.001)
(36.79)
(0.001)
(38.39)
(0.001)
(40.11)
(0.001)
(41.75)
(0.001)
(43.55)
-0.4
406.994
406.994
100%
185.865
185.865
100%
8.568
8.568
100%
1.448
1.448
100%
1.058
1.058
100%
1.012
1.012
100%
(0.001)
(44.96)
(0.001)
(46.56)
(0.001)
(48.26)
(0.001)
(49.85)
(0.001)
(51.45)
(0.001)
(53.14)
-0.8
405.389
405.389
100%
182.377
182.377
100%
7.986
7.986
100%
1.392
1.392
100%
1.049
1.049
100%
1.010
1.010
100%
(0.001)
(54.74)
(0.001)
(56.34)
(0.001)
(57.95)
(0.001)
(59.57)
(0.001)
(61.18)
(0.001)
(62.84)
0.10
0.8
454.619
454.619
100%
316.157
316.157
100%
72.076
72.076
100%
19.729
19.729
100%
8.050
8.050
100%
4.327
4.327
100%
(0.001)
(64.43)
(0.001)
(66.02)
(0.001)
(67.63)
(0.001)
(69.34)
(0.001)
(70.92)
(0.001)
(72.52)
0.4
452.652
452.652
100%
309.648
309.648
100%
66.157
66.157
100%
17.219
17.219
100%
6.865
6.865
100%
3.687
3.687
100%
(0.001)
(74.12)
(0.001)
(75.71)
(0.001)
(77.31)
(0.001)
(78.88)
(0.001)
(80.56)
(0.001)
(82.15)
0.2
451.681
451.681
100%
306.477
306.477
100%
63.418
63.418
100%
16.108
16.108
100%
6.357
6.357
100%
3.418
3.418
100%
(0.001)
(83.76)
(0.001)
(85.35)
(0.001)
(86.96)
(0.001)
(88.56)
(0.001)
(90.23)
(0.001)
(91.82)
-0.2
449.758
449.758
100%
300.278
300.278
100%
58.328
58.328
100%
14.128
14.128
100%
5.477
5.477
100%
2.963
2.963
100%
(0.001)
(93.45)
(0.001)
(95.05)
(0.001)
(96.65)
(0.001)
(98.24)
(0.001)
(99.85)
(0.001)
(101.48)
-0.4
448.805
448.805
100%
297.244
297.244
100%
55.958
55.958
100%
13.245
13.245
100%
5.095
5.095
100%
2.770
2.770
100%
(0.001)
(103.09)
(0.001)
(104.66)
(0.001)
(106.37)
(0.001)
(108.09)
(0.001)
(109.84)
(0.001)
(111.41)
-0.8
446.910
446.910
100%
291.291
291.291
100%
51.533
51.533
100%
11.662
11.662
100%
4.431
4.431
100%
2.441
2.441
100%
(0.001)
(113.02)
(0.001)
(114.63)
(0.001)
(116.29)
(0.001)
(118.02)
(0.001)
(119.63)
(0.001)
(121.23)
Note: The numerical results in parentheses are computational times in seconds
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Table 4. Out-of-control ARL values are computed by using the analytical formula and NIE technique for
FIMAX( 0 ),.4 1, 1d
running on an EWMA control chart when the in-control ARL is 500.
1
δ
0.01
Acc
%
0.05
Acc
%
0.25
Acc
%
0.50
Acc
%
0.75
Acc
%
1.00
Acc
%
()
puL
()
NuL
()
puL
()
NuL
()
puL
()
NuL
()
puL
()
NuL
()
puL
()
NuL
()
puL
()
NuL
0.01
0.8
360.125
360.125
100%
103.479
103.479
100%
1.635
1.635
100%
1.007
1.007
100%
1.000
1.000
100%
1.000
1.000
100%
(0.001)
(4.05)
(0.001)
(5.11)
(0.001)
(7.56)
(0.001)
(9.15)
(0.001)
(11.23)
(0.001)
(13.25)
0.4
358.709
358.709
100%
101.546
101.546
100%
1.586
1.586
100%
1.006
1.006
100%
1.000
1.000
100%
1.000
1.000
100%
(0.001)
(14.53)
(0.001)
(16.32)
(0.001)
(18.23)
(0.001)
(20.54)
(0.001)
(21.31)
(0.001)
(24.90)
0.2
358.005
358.005
100%
100.595
100.595
100%
1.563
1.563
100%
1.005
1.005
100%
1.000
1.000
100%
1.000
1.000
100%
(0.001)
(25.65)
(0.001)
(28.78)
(0.001)
(29.15)
(0.001)
(32.56)
(0.001)
(33.25)
(0.001)
(34.25)
-0.2
356.595
356.595
100%
98.712
98.712
100%
1.519
1.519
100%
1.005
1.005
100%
1.000
1.000
100%
1.000
1.000
100%
(0.001)
(36.44)
(0.001)
(37.56)
(0.001)
(38.98)
(0.001)
(40.25)
(0.001)
(42.56)
(0.001)
(43.98)
-0.4
355.895
355.895
100%
97.792
97.792
100%
1.499
1.499
100%
1.004
1.004
100%
1.000
1.000
100%
1.000
1.000
100%
(0.001)
(45.35)
(0.001)
(46.89)
(0.001)
(48.63)
(0.001)
(50.09)
(0.001)
(53.13)
(0.001)
(55.36)
-0.8
354.496
354.496
100%
95.962
95.962
100%
1.461
1.461
100%
1.004
1.004
100%
1.000
1.000
100%
1.000
1.000
100%
(0.001)
(57.46)
(0.001)
(59.15)
(0.001)
(61.34)
(0.001)
(62.78)
(0.001)
(64.01)
(0.001)
(66.95)
0.05
0.8
410.806
410.806
100%
194.364
194.364
100%
10.139
10.139
100%
1.614
1.614
100%
1.087
1.087
100%
1.020
1.020
100%
(0.001)
(4.12)
(0.001)
(5.76)
(0.001)
(10.48)
(0.001)
(12.12)
(0.001)
(13.72)
(0.001)
(15.26)
0.4
409.187
409.187
100%
190.716
190.716
100%
9.437
9.437
100%
1.538
1.508
100%
1.073
1.073
100%
1.016
1.016
100%
(0.001)
(16.22)
(0.001)
(18.45)
(0.001)
(20.07)
(0.001)
(21.67)
(0.001)
(23.31)
(0.001)
(24.90)
0.2
408.379
408.379
100%
188.920
188.920
100%
9.1065
9.1065
100%
1.5028
1.5028
100%
1.0674
1.0674
100%
1.0147
1.0147
100%
(0.001)
(26.50)
(0.001)
(28.11)
(0.001)
(29.69)
(0.001)
(31.40)
(0.001)
(33.08)
(0.001)
(34.76)
-0.2
406.769
406.769
100%
185.372
185.372
100%
8.483
8.483
100%
1.440
1.440
100%
1.057
1.057
100%
1.012
1.012
100%
(0.001)
(36.78)
(0.001)
(38.01)
(0.001)
(39.61)
(0.001)
(41.29)
(0.001)
(42.98)
(0.001)
(44.62)
-0.4
405.966
405.966
100%
183.625
183.625
100%
8.190
8.190
100%
1.412
1.412
100%
1.052
1.052
100%
1.011
1.011
100%
(0.001)
(45.12)
(0.001)
(47.8)
(0.001)
(49.47)
(0.001)
(51.09)
(0.001)
(52.78)
(0.001)
(54.37)
-0.8
404.366
404.366
100%
180.179
180.179
100%
7.637
7.637
100%
1.360
1.360
100%
1.044
1.044
100%
1.009
1.009
100%
(0.001)
(57.63)
(0.001)
(59.38)
(0.001)
(60.97)
(0.001)
(62.55)
(0.001)
(64.21)
(0.001)
(65.88)
0.10
0.8
453.357
453.357
100%
311.964
311.964
100%
68.217
68.217
100%
18.076
18.076
100%
7.264
7.264
100%
3.900
3.900
100%
(0.001)
(4.34)
(0.001)
(6.03)
(0.001)
(7.68)
(0.001)
(9.31)
(0.001)
(10.95)
(0.001)
(12.76)
0.4
451.410
451.410
100%
305.598
305.598
100%
62.676
62.676
100%
15.812
15.812
100%
6.223
6.223
100%
3.348
3.348
100%
(0.001)
(14.40)
(0.001)
(16.04)
(0.001)
(17.68)
(0.001)
(19.32)
(0.001)
(21.01)
(0.001)
(22.68)
0.2
450.447
450.447
100%
302.489
302.489
100%
60.104
60.104
100%
14.807
14.807
100%
5.774
5.774
100%
3.115
3.115
100%
(0.001)
(26.11)
(0.001)
(27.73)
(0.001)
(29.7)
(0.001)
(31.14)
(0.001)
(32.82)
(0.001)
(34.50)
-0.2
448.538
448.538
100%
296.401
296.401
100%
55.314
55.314
100%
13.009
13.009
100%
4.995
4.995
100%
2.720
2.720
100%
(0.001)
(36.15)
(0.001)
(37.82)
(0.001)
(39.43)
(0.001)
(41.07)
(0.001)
(42.71)
(0.001)
(44.34)
-0.4
447.589
447.589
100%
293.417
293.417
100%
53.080
53.080
100%
12.205
12.205
100%
4.656
4.656
100%
2.552
2.552
100%
(0.001)
(46.03)
(0.001)
(47.65)
(0.001)
(49.34)
(0.001)
(50.96)
(0.001)
(52.62)
(0.001)
(54.29)
-0.8
445.705
445.705
100%
287.558
287.558
100%
48.903
48.903
100%
10.763
10.763
100%
4.066
4.066
100%
2.265
2.265
100%
(0.001)
(56.06)
(0.001)
(57.76)
(0.001)
(59.48)
(0.001)
(61.21)
(0.001)
(63.04)
(0.001)
(64.78)
Note: The numerical results in parentheses are computational times in seconds
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Volume 22, 2023
1
( ) 0.8a
1
( ) 0.4b
1
( ) 0.2c
1
( ) 0.2d
1
( ) 0.4e
1
( ) 0.8f
Fig. 1: Shifts in the mean of the FIMAX(0.4, 1, 1) model with various values of coefficient parameter
1
running on an EWMA control chart calculated using the analytical formula.
Table 5. The statistical results for the gold futures
price dataset with the UDS/THB exchange rate as
the exogenous variable.
Parameters:
Coefficient
t-Statistic
Prob.
MA(1)
-47.2713
-9.15545
0.00*
d
0.499999
722.704
0.00*
UDS/THB
0.495184
9.055184
0.00*
R-squared
0.981843
Adjusted R-squared 0.981700
Testing whether the white noise is exponentially distributed.
Exponential Parameter (v)
39.577325
Kolmogorov-Smirnov
0.692
Asymptotic Significance (2-Sided)
0.725
*A significance level of 0.05.
As reported in Table 5, the dataset is a valid fit
for a long-memory FIMAX model since all of the
parameters had p-values less than 0.05. The
exponential parameter
()v
of the dataset provided a
Kolmogorov-Smirnov value of 0.692. The
corresponding p-values based on asymptotic
significance (2-sided) were 0.725, suggesting that
the long-memory FIMAX(0.499999, 1, 1) model
was a suitable fit. Testing whether the white noise
fits an exponential distribution also yielded a p-
value less than 0.05. Thus, the process running on
an EWMA control chart was long-memory
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FIMAX(0.499999, 1, 1) with coefficients
1
ˆ
= -
47.2713 and
1
ˆ
= 0.495184.
The model is given by
1 1 1
47.2713 0.495184 0.499999
t t t t t
Y X Y
23
,0.124999999 0.062500042
tt
YY
(20)
where
)39.577325(
tExp v
To apply the analytical formula, we fitted the
dataset based on Equation (20). The EWMA control
limit
()H
was computed equal when in-control ARL
= 500 using Algorithm 1 for smoothing parameter
=
0.50, 0.51, 0.53, or 0.55. Thereby, we used
(1 )
( ) 1 1 exp .exp
pH
vv
L
1
1 exp exp
H
vv
11
12
3
47.2713 0.495184
0.499999 0.124999
0.062500042
tt
tt
t
X
YY
Y
1
.
(21)
and
1
(1 )
1
( ) 1 + ()
m
N j N
j
jjfa
w a
LL
11
12
3
47.2713 0.495184
0.499999 0.124999
0.062500042
tt
tt
t
X
YY
Y
,
(22)
to compute the out-of-control ARL on an EWMA
control chart using the analytical formula and NIE
techniques, respectively. The results are reported in
Table 6 and Figure 2.
Table 6. The out-of-control ARL results using the
analytical formula and
NIE
techniques for the
FIMAX(0.499999,1,1) model with exponential
white noise for real data running on an EWMA
control chart when the in-control ARL is 500.
0.50
0.51
0.53
0.55
H
46.07293
49.1854
55.64264
62.64577
0.01
()
puL
474.049
475.216
478.525
483.826
(0.001)
(0.001)
(0.001)
(0.001)
()
NuL
474.049
475.216
478.525
483.826
(29.54)
(30.54)
(31.01)
(30.23)
0.05
()
puL
384.181
388.151
399.714
419.384
(0.001)
(0.001)
(0.001)
(0.001)
()
NuL
384.181
388.151
399.714
419.384
(30.74)
(30.89)
(31.34)
(31.56)
0.25
()
puL
148.991
152.218
161.907
180.098
(0.001)
(0.001)
(0.001)
(0.001)
()
NuL
148.991
152.218
161.907
180.098
(33.98)
(34.62)
(34.68)
(35.18)
0.50
()
puL
59.627
60.697
63.769
69.366
(0.001)
(0.001)
(0.001)
(0.001)
()
NuL
59.627
60.697
63.769
69.366
(39.28)
(40.04)
(41.34)
(42.58)
0.75
()
puL
30.222
30.671
31.874
33.5921
(0.001)
(0.001)
(0.001)
(0.001)
()
NuL
30.222
30.671
31.874
33.5921
(40.56)
(42.35)
(42.72)
(42.94)
1.00
()
puL
18.045
18.292
18.907
19.901
(0.001)
(0.001)
(0.001)
(0.001)
()
NuL
18.045
18.292
18.907
19.901
(41.58)
(42.01)
(42.14)
(42.98)
5.00
()
puL
1.916
1.943
1.996
2.054
(0.001)
(0.001)
(0.001)
(0.001)
()
NuL
1.916
1.943
1.996
2.054
(52.91)
(53.24)
(53.77)
(54.56)
Note: The numerical results in parentheses are
computational times in seconds
For the real data, results suggest that the out-of-control
ARL values calculated using the analytical formula
equal those approximated using the NIE technique,
indicating good agreement between the two methods
and that the proposed analytical formula was very
accurate. We compared the out-of-control ARL
results versus
for
= 0.01, 0.05, 0.25, 0.50, 0.75,
1.00, or 5.00 in the long-memory FIMAX model.
The findings indicate that the out-of-control ARL
values tended to decline rapidly when detecting
small-to-moderate shifts in the process mean and
monotonically as
was increased for all smoothing
parameter values.
Fig. 2: Graphical representation of the out-of-
control ARL results for the FIMAX(0.499999,1,1)
model with exponential white noise for real data
running on an EWMA control chart when the in-
control ARL is 500.
Moreover, when the smoothing parameter value
was increased, detection became slower and the out-
of-control ARL was larger. The smallest smoothing
parameter value (
= 0.50) provided the best
detection performance for all values of
considered.
The computational times for calculating the out-of-
control ARL values took a fraction of a second with
the analytical formula compared to 30–55 seconds
with the NIE technique. The result corresponds to the
computational in Table 2, Table 3, and Table 4.
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The EWMA control chart running the in the
FIMAX model with real data is graphically displayed
in Figure 3. It can be seen that in all cases of the
smoothing parameter value tested, the process
remained under statistical control for the first five
observations. The total number of out-of-control
signals for
= 0.50, 0.51,0.53, or 0.55 were 20, 18,
16, and 14 points, respectively. Hence, the model
with the smallest value of
performed the best.
( ) 0.50a
( ) 0.51b
( ) 0.53c
( ) 0.55d
Fig. 3: Graphical representation of the out-of-control ARL results for the FIMAX(0.499999,1,1) model with
exponential white noise for real data running on an EWMA control chart when the in-control ARL is 500 for
various smoothing parameter values:
( ) 0.50, ( ) 0.51,( ) 0.53a b c
, and
( ) 0.55d
.
4 Conclusions
We provided a new technique to accurately compute
the
ARL
for a long-memory
FIMAX( ),,d q r
model
with exponential white noise running on the EWMA
control chart using an analytical formula based on
an integral equation. Its performance was measured
against that of the well-established
NIE
technique.
For all the control chart smoothing parameter values
of the chart and FIMAX scenarios tested, the
proposed analytical formula provided out-of-control
ARL values close to those with the NIE technique.
For clarity, we have verified the accuracy of the
analytical formula with the NIE method as the
percentage accuracy. The percentage accuracy
results were 100% in all cases, implying good
agreement between the two techniques and that the
proposed analytical formula is very accurate and
quick. Therefore, using the analytical formula as an
alternative approach for deriving the ARL for a shift
in the mean of this scenario is plausible.
To demonstrate the practicability of the
proposed analytical formula, we applied it to a
process involving the gold futures price and
exchange rates over a specific time period. The out-
of-control ARL values show that the analytical
formula approached performed very well in all of
the scenarios tested and that it is a good alternative
to using the analytical formula for this endeavor. In
addition, this analytical formula can be extended to
develop commercial packages to evaluate ARL to
analyze and control the manufacturing process or
other aspects. Future work of this study could be
extended to other control charts that have been
developed, such as modified EWMA, modified
CUSUM, and enhanced EWMA.
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.58
Wilasinee Peerajit
E-ISSN: 2224-2880
527
Volume 22, 2023
Acknowledgment:
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-66-BASIC-05.
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Wilasinee Peerajit
E-ISSN: 2224-2880
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Volume 22, 2023
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Appendix A
Theorem 2:
()
p
L
, the
ARL
obtained from the
analytical formula based on an integral equation for a
long-memory FIMAX model on an EWMA control
chart, exists and is unique.
Proof: To prove the existence of the integral
equation applied to derive the ARL. Let
T
be a
contraction in complete metric space
,,M
.
[0, ]HC
be a set of all continuous functions on
interval
[0, ]H
, and the in-control ARL be an
arbitrary but fixed element in
.M
Define a sequence
of iterates
0
nn
L
in
M
that satisfies
1( ),
nn
T
LL
for all
1.n
Consider
11
1
,,
, ,0 1.
n n n n
nn
TT
TT
L L L L
LL
Continuing inductively, we obtain
2
1 1 1 2
10
, , ,
... , .
n n n n n n
n
L L L L L L
LL
By repeatedly applying the triangle inequality to this
formula when
nm
we arrive at
11
, ( , ) ... ( , ),
n m n n m m
L L L L L L
Thus, it follows that
12
10
, ( ... ) ( , ).
n n m
nm
L L L L
Taking the property of the sum of a geometric series
in
,
we obtain
10
, ( , ).
1
n
nm
L L L L
where
1
n
as
.n
So,
0
nn
L
is a Cauchy
sequence and
lim .
n
nT
LL
Hence, the existing continuous function
,:[0 ]HL
satisfies the integral equation.
Proof: To prove the uniqueness of the integral
equation applied to derive the ARL. Let
1
L
and
2
L
be two arbitrary functions for
[0, ].HC
The common
term for the complete metric space is
.[0, ],.H
C
That is to say, a set of continuous
functions of the
ARL
defined on
[0, ],H
and
[0, ]HC
becomes norm space if
[0, ]
0
sup ( , ) ,
b
b
L k g dg
for all functions
[0,, ]() ,k Hg
C
where
( , )kg
is
0
1 (1 )
( , ) ( ) exp exp
H
kg vv
L
1
11
23
(1 ) (1 )(2 ) ...
26
qr
i t j j jt t
ij
tt
X dY
d d d d d
YY
1d
v
The kernel function of the integral equation used to
define the
ARL
is
1 2 [0, ] 1 2
0
( ) ( ) sup ( , ) ( ) ( )
H
H
T T k g g g dg
L L L L
Hence, we obtain
1 2 1 2
( ) ( )TT
L L L L
where
[0, ]
0
sup ( , ) 1.
H
Hk g dg
Applying Banach’s fixed point theorem leads to
contraction mapping. Therefore,
T
is a unique
continuous function that satisfies the integral
equation in equation (12)
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.58
Wilasinee Peerajit
E-ISSN: 2224-2880
529
Volume 22, 2023
Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
The author contributed in the present research, at all
stages from the formulation of the problem to the
final findings and solution.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
This research was funded by King Mongkut’s
University of Technology North Bangkok, Contract
No. KMUTNB-66-BASIC-05.
Conflict of Interest
The author has 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
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WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.58
Wilasinee Peerajit
E-ISSN: 2224-2880
530
Volume 22, 2023
