Approximating the ARL of Changes in the Mean of a Seasonal Time
Series Model with Exponential White Noise Running
on a CUSUM Control 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 comprise an excellent statistical process control tool for monitoring industrial processes.
Especially, the CUSUM control chart is very sensitive to small-to-moderate process parameter changes. The
proposed approach utilizes the numerical integral equation (NIE) method to approximate the average run length
(ARL) of changes in the mean of a seasonal time series model with underlying exponential white noise running on
a CUSUM control chart. This was achieved by solving a system of linear equations and integration through
partitioning and summation using the area under the curve of a function obtained by applying the Gauss-Legendre
quadrature. A numerical study was conducted to compare the capabilities of the ARL derivations obtained using
the NIE method and explicit formulas to detect changes in the mean of a long-memory
,SARFIMA( , )( ), s
P D Qpd
model with exponential white noise running on a CUSUM control chart. The results reveal that the performances of
both were comparable in terms of the accuracy percentage, which was greater than 95%, meaning that the ARL
values were highly consistent. Thus, the NIE method can be used to validate ARL results for this situation.
Key-Words: - CUSUM control chart, average run length (ARL), exponential white noise,
,SARFIMA( , )( ), s
P D Qpd
process, numerical integral equation (NIE).
Received: January 8, 2023. Revised: September 11, 2023. Accepted: October 14, 2023. Published: November 15, 2023.
1 Introduction
Statistical process control (SPC) has been extensively
employed for monitoring processes and services to
avoid the occurrence of problems in industrial
processes. One of the most commonly utilized SPC
tools is the control chart, which is robust, reliable,
and powerful for monitoring industrial processes.
Moreover, they are easy to implement and
interpretation of their output is unambiguous. Control
charts are designed to detect changes in a process
parameter from the in-control state to the out-of-
control state. Information about a process
characteristic is plotted against time in conjunction
with so-called control limits. A signal is transmitted
once the plotted statistic exceeds the predetermined
control limit and indicates that the process is possibly
out-of-control, [1]. There are two categories of
control charts: memoryless (e.g., Shewharts) and
memory-type (e.g., the cumulative sum (CUSUM)
and exponentially weighted moving average
(EWMA)). The CUSUM, [2], and EWMA, [3],
control charts are extensively utilized to detect small-
to-moderate changes in the parameter of a process
while the Shewhart control chart is better at detecting
large changes. Much research has been performed to
evaluate the efficacy of the CUSUM control chart,
[4], [5], [6]. Our research is mainly concentrated on
the upper one-sided CUSUM control chart, [7].
An important control chart evaluation criterion is
the average run length (ARL). It is the average
number of observations taken until a control chart
signals that the process is out of control. It comprises
two parts: ARL0 when the process is in control,
which should be as large as possible, and ARL1 when
the process is out of control, which should be as
small as possible. Various methods have been
developed to derive the ARL for changes in the mean
of a normal process running on a CUSUM control
chart. For example, [8], derived the approximate
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ARL for changes in the parameters of processes
running on a CUSUM control chart with a zero head-
start as the ratio of numerical solutions to two
integral equations. Likewise, [9], provided a solution
for the ARL under similar circumstances but with a
non-zero head-start using the Markov chain and
integral equation methods for normally distributed
observations.
Autocorrelation can have a substantial impact on
the effectiveness of a CUSUM control chart, [10].
Nevertheless, since it is often an inherent part of a
process, it must be modeled and monitored
appropriately. Econometric data has been used to
develop some of these models since it often fits
autoregressive (AR), moving-average (MA), ARMA,
or AR fractionally integrated MA (ARFIMA)
models. Measuring errors (the difference between the
actual and estimated values) is crucial when creating
a model: the lower the number of errors, the higher
the efficiency of the model. A time series model with
autocorrelated data often contains errors indicated as
white noise, which in certain situations, is
exponentially distributed, [11], [12], [13].
A time series has long-memory properties when
the differencing parameter d in a
ARFIMA( ,,)p d q
model lies within the range (0, ½), [14 ], [15 ]. This
characteristic is exemplified either by the hyperbolic
decline of the autocorrelation function or the lack of
bounds for the spectral density function. In contrast,
an ARMA model shows a geometric rate of reduction
in the correlation between the observations. Our
primary interest lies in long-memory
SARFIMA( ,,)p d q
(,,)S
P D Q
models with the added
complication of seasonality, [16 ], [17], [18], which
often appears empirically. The study, [17], adopted
the Kalman filter methodology to deduce the values
of parameters
d
and
D
for a
,S 0ARFIMA( )(0 00, , ), S
Dd
process. In the present
study, we explore this specific scenario under the
presumption that the white noise follows an
exponential distribution. Several control charts have
been adapted to run processes with the fractional
integration element, [18], [19]. The present research
is centered on identifying shifts in the mean of a
long-memory
,S ,,R ,A FIMA( )( ) S
PDd Qpq
process
running on a CUSUM control chart.
The principal methods for calculating the ARL
have been based on utilizing Monte Carlo simulation,
the Markov chain technique, explicit formulas, and
integral equations. The study, [20], employed the
Markov chain methodology to determine the ARL of
a process running on a CUSUM control chart while
presupposing that the observations are independently
and identically distributed (i.i.d.). The author in, [21],
refined this approach through the incorporation of
Richardson extrapolation for observations from a
comprehensive array of distributions, including the
Chi-squared distribution. Integral equations such as
the Fredholm integral equation of the second kind
have been utilized within the numerical integral
equation (NIE) methodology to compute the ARL,
[22], [23], [24], [25], [26], [27], [28]. Alternatively,
the Gauss-Legendre quadrature has been employed in
the integral equation approach, [22], in which it is
noteworthy that the sample variance adheres to a
right-skewed Chi-squared distribution restricted to
the half-real line. The study, [29], used a piecewise
collocation technique as an alternative to the Gauss-
Legendre quadrature for ARL approximation.
Numerical integration (or quadrature) is a commonly
utilized method for approximating integrals. As well
as the Gauss-Legendre quadrature, other examples
include using the midpoint, composite trapezoidal,
composite Simpson’s, and Gaussian rules. In the
present study, we obtained approximated the ARL for
changes in the mean of a long-memory
,SARFIMA( , )( ), S
P D Qpd
process with underlying
exponential white noise running on a CUSUM control
chart by using the NIE method based on an integral
equation using the Gauss-Legendre quadrature.
The remainder of the paper Is organized as
follows. Section 2 provides a concise overview of
the upper-sided CUSUM control chart and the
generalized long-memory
,SARFIMA( , )( ), S
P D Qpd
model with underlying exponential white noise.
Similarly, the design of the upper-sided CUSUM
control chart is presented in Section 3. Approximating
the ARL for changes in the mean of a long-memory
,SARFIMA( , )( ), S
P D Qpd
with underlying exponential
white noise running on a CUSUM control chart by
using the NIE method is covered in Section 4. The
ARL numerical results obtained using the NIE method
and the explicit formulas are compared in Section 5.
Last, conclusions and future recommendations are set
out in Section 7.
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2 Preliminaries
Here, we overview the upper-sided CUSUM control
chart and the generalized long-memory
,SARFIMA( , )( ), S
P D Qpd
model with underlying
exponential white noise.
2.1 The Upper-Sided CUSUM Control Chart
CUSUM statistic
,
t
Z
at time
t
is defined as follows:
1
max , 0 ,
t t t
Z Z Y k
for
1,2,... ,t
(1)
where
t
Y
is the sequence of the generalized
,SARFIMA( , )( ), S
P D Qpd
process with exponential
white noise and
k
is a reference value. The starting
value
0
Z
is set to
in this study whereby
,h
where
h
is either the decision parameter or upper
control limit of the CUSUM control chart.
2.2 The Generalized Long-Memory
SARFIMA(p, d, q)(P, D, Q)S Model
With Underlying Exponential White
Noise
A time series has long-term dependence or long
memory if its autocorrelation coefficient does not
decay. If the coefficient of autocorrelation of order
,,
k
k
satisfies the condition
1
,
k
k
then such a
time series is called a long-memory process. As the
latter has often been observed in many economic
time series, several models for describing it have
been developed. Analysis of long-term dependency
on the volatility of exchange rates has often been
performed using the ARFIMA model, [14], [15].
Nevertheless, the utilization of fractional differencing
(or integration) alone does not cover the
characteristics of seasonality. Consequently, the
SARFIMA model, which is the ARFIMA model with
a seasonality component, has been devised. The
parameters of the
,S ,,R ,A FIMA( )( ) S
PDd Qpq
model can be described in terms of a seasonal time
series
()
t
Y
as follows:
( ) ( )(1 ) (1 ) ( ) ( ) ,
s d s D s
p P t q Q t
B B B B Y B B
(2)
where
t
is a white noise process assumed to be
exponentially distributed with
()~
tExp
when
shift parameter
0.
2
12
1
( ) (1 ... ) 1 ,
p
pi
p p i
i
B B B B B
and
2
12
1
( ) (1 ... ) 1 ,
q
qj
q q j
j
B B B B B
are non-seasonal AR and MA polynomials in
B
of
order p and q respectively;
2
12
1
( ) (1 ... ) 1 ,
P
s s s Ps ks
P P k
k
B B B B B
and
2
12
1
( ) (1 ... ) 1 ,
Q
s s s Qs ls
Q Q l
l
B B B B B
are seasonal AR and MA polynomials in
B
of order
P and Q, respectively;
B
is the backshift operator
satisfying
1,
tt
BY Y
and
,
st t s
B Y Y
; d and D are
the annual and seasonal fractionally differencing
parameter, respectively, and
s
is the number of time
periods utilized in a year (e.g.,
12s
is a monthly
time series). In particular, the
,S ,,R ,A FIMA( )( ) S
PDd Qpq
process is when
0.p q P Q
This process is a non-seasonal
and seasonal fractionally integrated
(SARFIMA(0, ),0d
),,(0 0) S
D
model, which can be
defined as
(1 ) (1 ) ,
d S D tt
B B Y
where d and D are the non-seasonal and seasonal
differencing parameters, respectively. All real values
of
, 1,dD
can be expressed in terms of their
binomial expansion as follows:
2
0
( 1)
(1 ) ( ) 1 ...., ;
2
xv
v
B B B B
v
1, , and , , s d D
(3)
where
( 1) , and (.)
( 1) ( 1)
vvv
is a
gamma function.
For
0,q
the
,S 0ARFIMA( ),,(),S
P D Qpd
or
,SARFIMA( , )( ), S
P D Qpd
model can be defined as
( ) ( )(1 ) (1 ) ( ) ,
s d s D s
p P t Q t
B B B B Y B
(4)
,SARFIMA( , )( ), S
P D Qpd
models are commonly
used to model time series with long-memory
behavior. They have the same characteristics as the
corresponding ARFIMA model (i.e. stationarity and
invertibility) when
12dD
and
andd D
are
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less than
12
, which indicates a long-memory
process.
The equation
(4)
can be rearranged in favor of
()
t
Y
for the generalized
,SARFIMA( , )( ), S
P D Qpd
model on the CUSUM control chart as follows:
() , or
( ) ( )(1 ) (1 )
s
Qt
ts d s D
pP
B
YB B B B
11
11
(1 ) (1 ) (1 ) (1 )
pP
i is d s D
t i j
ij
Y B B B B
1 2 2
.( ... )
t t s t s Q t Qs
(5)
where
( ).~
tExp
The initial value is normally the
process mean (i.e.,
2
, ,..., 1.
t s t s t Qs
); the
coefficient parameters are
,1 ,1 ,1
, , ,
i i p j j P k k Q
;
and the initial value of the long-memory
,SARFIMA( , )( ), S
P D Qpd
process is 1.
3 The Design of Upper-Sided CUSUM
Control Chart
Here, we discuss the design of the CUSUM control
chart running a generalized long-memory
,SARFIMA( , )( ), S
P D Qpd
model with underlying
exponential white noise.
Let
, 1,2,...,
tt
represent a sequence of
continuous i.i.d. random variables from an
exponential distribution with parameter
.
The
process is considered in-control when
0,
whereas
it is out-of-control when
1.
The following are the
change points for
t
:
0
10
no ch
(
ange)( ), 1, 2, ..., 1 (
(ch), , 1, ... ange),
t
Exp t m
Exp t m m
(6)
The ARFIMA process in Equation (5) can be
substituted into Equation (1), so the CUSUM statistic
becomes
1
11
1
11
(1 ) (1 ) (1 ) (1 )
t t t
pP
i is d s D
ij
ij
Z Z Y k
Z B B B B
1 2 2
.( ... )
t t s t s Q t Qs k
(7)
where
1.
t
Z
Hence, the CUSUM stopping time
(
h
) can be written as
inf 0; ,f ,or
ht
Zh ht
(8)
Note that
0,
t
Zh
indicates that the process is in
control whereas
t
Zh
indicates that the process is
out of control.
In the context of the in-control process,
modifying the CUSUM statistic by reorganizing the
error term
()
t
is possible, resulting in
t
being
between 0 and
.h
Subsequently,
t
can be rearranged
as follows:
11
11
1 1 2 1 2 1
11
1
11
1 1 2 1 2 1
(1 ) (1 ) (1 ) (1 )
.( ... )
(1 ) (1 ) (1 ) (1 )
.( ... ),
pP
i is d s D
ij
ij
s s Q Qs
pP
i is d s D
ij
ij
s s Q Qs
k B B B B
h k B B B B
4 Approximating the ARL for Changes
in the Mean of a Process Running on
a CUSUM Control Chart Via the
NIE Method
The ARL can be approximated by utilizing the NIE
method based on Fredholm's integral equation of the
second kind, [23]. Subsequently, the NIE method
was used in conjunction with the CUSUM statistic to
approximate the ARL. In this section, the application
of the Gauss-Legendre rule technique for the
numerical calculation of the integral equations of the
NIE method is proposed.
To evaluate the performance of the CUSUM
control chart, it is necessary to determine the
stochastic properties of the corresponding stopping
time
()
h
. Assuming there is a change point in
Equation (6), it is possible to establish a rigorous
definition of the ARL using
(.)
m
E
under the
assumption that the change point occurs at time
.m
Consequently,
1 1
00
0
( ), ; in-control (ARL )
At
s
-
ta
e
RL ( ), ; ou of l
te
stat-contro ( ARL ).
h
h
E
E
Let
()
L
be the ARL of a change in the mean of
a long-memory
,S ,,R ,A FIMA( )( ) S
PDd Qpq
model
conditioned on the initial value of the CUSUM
statistic
0;Z
0.h
This structure utilizes the
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stopping time
()
h
for the process such that the ARL
is given as
( ) E ( ) ,
h
L
where
E ( )
h
is the expectation under the density
function
( , ).
t
f
The solution of the integral equation becomes
1 1 1
( ) 1 P { 0} (0) E [ {0 } ( )],
zz
Z I Z h Z
L L L
(9)
where
1
Z
is the first observation and
1
{0 }I Z h
represents the indicator function.
The integral equation utilized to determine the
ARL of a change in the process parameter running on
a one-sided CUSUM control chart is obtained by
employing Equation (9) and applying a Fredholm
integral equation of the second kind. This allows us
to rewrite for
()
L
as follows:
11
11
1
11
11
0
1
( ) 1 (0) ( (1 ) (1 )
(1 ) (1 ) ( ... ))
( ) ( (1 ) (1 )
(1 ) (1 ) ( ... )) ,
pP
i is
ij
ij
d s D t t s Q t Qs
hpP
i is
ij
ij
d s D t t s Q t Qs
F k B B
BB
u f u k B B
B B du
LL
L
(10)
where
( ) 1Fe
and
( ) .fe
When
applying the final term in Equation (10) to the
quadrature rule, the integral can be estimated by
summation of the rectangles.
The Gauss-Legendre quadrature rule can be
utilized in numerical solutions based on integral
equations in the final term in Equation (10). Clearly,
integral
0()
hf u du
can be approximated by summing
the areas of the rectangles with bases
/hm
and heights
maintained as the values of
f
at the zero-based
midpoints of the intervals of length
/.hm
Interval
0, h
is partitioned into
1
0 ...
m
a a h
partitions
while
/0
j
w h m
is a set of weights. Thus, the
integral can be approximated in summation form as
1
0
( ) ( ) ( )
hm
jj
j
W u f u du w f a
where
()Wu
is a weight function,
1
= ,
2
jh
aj
m
and
/ ; 1,2,..., .
j
w h m j m
Solving the system of m linear equations in mm
unknowns can be used to approximate the solution
for
()
L
on interval
0, h
by substituting
with
i
a
in Equation (10) as follows:
11
11
1 1 1
11
1 1 1
1
1
11
ˆˆ
( ) 1 ( (1 ) (1 )
.(1 ) (1 ) ( ... ))
ˆ( (1 ) (1 )
.(1 ) (
)
1 ) ( ... )), 1, 2
(
,..., .
(
)
pP
i is
ij
i
P
ij
d s D s Q Qs
p
mi is
j j i i j
j i j
d s D s Q Qs
j
F k B B
BB
w f a k a B
i
aa
a B
B B m
LL
L
Let
ˆ()
L
denote the approximated ARL using the
NIE method when applying the Gauss-Legendre rule
on interval
0, .h
Therefore, the integral equation in
Equation (10) comprises the set
12
( ), ( ), ..., ( )
ˆ ˆ ˆ ˆ
() m
a a a
L L L L
, which can be
approximated as
11
1
11
1 1 1
11
1
11
1
1
1
1
1
2
(
( (1 ) (1 ) (1 )
.(1 ) ( ... ) )
ˆ)
ˆ
1
( (1
)
(
) (1 )
.(1 (1 ) ( ... )
(
)
ˆ
)
pP
i is d
ij
ij
sD s Q Qs t
pP
i is
ij
ij
d s D s Q Qs t
m
j
j
F k a B B B
B
w f k B B
BB
w
aa
LL
L11
1
11
1
( (1 ) (1 )
.(1 ) (1 ) ( ... )),
)
pP
i is
j i j
ij
d s D t s Q t Qs
jf a k a B B
BB
a
11
2
11
1 1 1
11
1
2
12
11
11
1
1
( (1 ) (1 )
.(1 ) (1 ) ( ... ))
ˆˆ
1
( (1 ) (1 )
.(1 ) (1 ) ( .
( ) ( )
.. ))
pP
i is
ij
ij
d s D t s Q Qs
pP
i is
ij
ij
d s D s Q Qs
j
j
F k a B
B
a
B
BB
w
a
f a k a B
BB
w
LL
11
2
2 1 1
1 1 1
ˆ( (1 ) (1
1
() )
.(1 ) ( ) ( ... )),
p
mP
i is
j i j
ij
d s D s Q Qs
jf a k a B
B
a B
B
L
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11
11
1 1 1
11
11
11
1 1 1
1
( (1 ) (1 )
.(1 ) (1 ) ( ... ) )
ˆˆ
1
( (1 ) (1 )
.(1 )
(
(1 ) ( ... )
( ) )
)
pP
i is
m i j
ij
d s D t s Q Qs t
pP
i is
m i j
ij
d s D t s Q Q
m
s
F k a B B
BB
w f a k a B B
BB
w
aa
LL
11
2 1 1
1 1 1
ˆ( (1 ) (1 )
.(1 ) (1 ) ( ... )),
()
p
mP
i is
j j m i j
j i j
d s D t s Q
j
Qs
faa k a B B
BB
L
The above set of m equations in m unknowns can
be expressed in matrix form:
1 1 1m m m m m
L 1 C L
which is equivalent to
11
( ) .
m m m m m
I C L 1
(11)
where
1 1 2
( ), ( ),..., ( ) ,
ˆ ˆ ˆ
mm
a a a
LL L L
11,1,...,1
m
1
is a column vector of ones, and
(1,1, ,1)
mdiagI
is
the unit matrix order
.m
Matrix
C
with dimensions
mm
becomes
11 12 1
21 22 2
12
...
... ,
...
m
m
mm
m m mm
c c c
c c c
c c c
C
where
; , 1,2,....,
ij
c i j m
can be expanded as
11
11
1 1 1
11
11
1 1 1
( (1 ) (1 )
.(1 ) (1 ) ( ... ) )
( (1 ) (1 )
.(1 ) (1 ) ( ... ))
pP
i is
i i j
ij
d s D s Q Qs t
pP
i is
j j i i j
ij
d s D sQ
ij
Qs
c F k a B B
BB
w f a k a B B
BB
If
1
()
m m m
IC
is invertible and exists, then the ARL
approximation for the NIE can be reformed into a
system of linear equations in matrix form as follows:
1
11
( ) ,
m m m m m
L I C 1
(12)
After the computation,
12
( ), ( ), ..., ( )
ˆ ˆ ˆ m
a a aL L L
is
substituted for
i
a
by
as follows:
11
11
1 1 1
11
11
1
1
1
11
ˆˆ
( ) 1 ( (1 ) (1 )
.(1 ) (1 ) ( ... ))
ˆ( (1 ) (1 )
.(1 ) (1 ) ( ... )),
()
()
pP
i is
ij
ij
d s D s Q Qs
p
mP
i is
j j i j
j i j
d s D Qs
j
sQ
F k B B
BB
w f a k B B
BB
a
a
LL
L
(13)
with
1
, and ; 1, 2, , .
2
jj
w h m a h m j j m
This is the proposed approximation of the ARL
for changes in the mean of a long-memory
,S 0ARFIMA( ),,(),S
P D Qpd
process with
underlying exponential white noise on a CUSUM
control chart using the NIE method. The Gauss-
Legendre quadrature rule technique can be used to
approximate the ARL quite accurately, as shown in
the next section.
5 The Performance Evaluation Results
of the Numerical Study
Here, we present the numerical results of a
comparative analysis conducted to evaluate the
performances of the proposed NIE method utilizing
explicit formulas.
5.1 Derivation of the ARL using Explicit
Formulas
To evaluate the performance of the proposed ARL
for a long-memory
,SARFIMA( , )( ), S
P D Qpd
model running on a CUSUM control chart, we used
the ARL derived by using explicit formulas denoted
as
()
L
, which can be written in the form
11
11
1 1 1
( ) exp 1 exp ( (1 ) (1 )
.(1 ) (1 ) ( ... )) exp
pP
i is
ij
ij
d s D s Q Qs
h h k B B
BB
L
(14)
where
is replaced by
0
for the in-control ARL
(ARL0) process and
is replaced by
1
for the out-
of-control ARL (ARL1).
5.2 The Standard Deviation of the Run
Length (SDRL) Performance Measure
As well as the ARL, we also computed the SDRL
used as a performance measure for the CUSUM
control chart for the situation described earlier. The
in-control SDRL (SDRL0) is defined as
02
1
SDRL =
(15)
where
1 (0 )
t
P Z h
is a false alarm (or type I
error); the probability at which a false alarm occurs is
known as the false alarm rate (FAR). Thus, the
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probability of a type I error or FAR is 0.0027. Hence,
0
l
ARL = 370,
and
0
SDRL 370.
On the contrary, the out-of-control SDRL (SDRL1)
can be characterized as
12
SDRL = ,
(1 )
(16)
where
1
(0 )
t
P Z h
is the probability of a type
II error occurring.
1
1
ARL = 1
(1 )
corresponds to
ARL0 = 370 for large changes in the process mean.
5.3 The Percentage Accuracy Performance
Measure
Let
ˆ()
L
and
()
L
be the ARL values obtained by
using the NIE and explicit formula methods,
respectively. Thus,
ˆ
( ) ( )
%Accuracy 100 100%,
()
LL
L
(17)
An accuracy percentage of greater than 95%
implies that the ARL results using both methods are
close to each other.
This work aims to numerically approximate the
average run lengths (ARLs) for long memory with
,SARFIMA( , )( ), S
P D Qpd
a model underly
exponential white noise when implemented on a
CUSUM control chart using the NIE method and
explicit formulas. The white noise in the long-
memory
,SARFIMA( , )( ), S
P D Qpd
process was
distributed exponentially where the mean parameter
of the exponential is
in the study situation. In
addition, the value of
0
is equal to 1 for the in-
control process, whereas
10
(1 )
represents the
value for the out-of-control process, where
is the
magnitude of shift size;
= 0.025, 0.05, 0.10, 0.25,
0.50, 0.75, 1.0, 1.5, 2.0, 2.5, 3.0, 4.0, or 5.0
respectively. In the long-memory process with
4
SARFIMA(1, 0.1)(1,0.1, 1)
the model, coefficient
parameters
1
= 0.1, 0.3, 0.5, or 0.7,
1
= 0.10, and
1
= 0.10 are employed and compared. The NIE
method employs 800 division points, denoted as m,
to solve systems of linear equations in calculations.
The ARL results using the NIE method were
compared to the ARL results derived from explicit
formulas. The results showed comparable
performance of ARL obtained from both methods in
detecting changes in the mean process.
Table 1 presents the chief findings of our proposed
CUSUM control chart. We have used the sensitivity
parameter of the control chart k = 2.5, 3.0, and 5.0,
which are the optimal choices for calculating the
upper control limit (h) from (13) such that the ARL0
is fixed at 370.
Table 1. Values of CUSUM control limit with
corresponding values of k for the desired ARL0 = 370
of long-memory
4
SARFIMA(1, 0.1)(1,0.1, 1)
models.
SARFIMA
(1, 0.1) (0, 0.1, 1)4
In-control ARL0 = 370
1
1
1
k = 2.5
k = 3.0
k = 3.5
0.1
0.1
0.1
4.020943
3.303497
1.149340
0.3
0.1
0.1
3.937120
3.240665
1.098567
0.5
0.1
0.1
3.856962
3.178866
1.047840
0.7
0.1
0.1
3.779928
3.11,8004
0.997154
Table 1 contains reports on the upper control limits
(h) and k for every scenario in the control process
( 0).
The study revealed the value of h decreased
as k was systematically increased for every
coefficient parameter combination in each
4
SARFIMA(1, 0.1)(1,0.1, 1)
model. Moreover, if we
consider the coefficient parameters, it is found that as
1
increases, the value of h decreases for each value
of k as
1
changes between 0.1 and 0.7.
Consequently, we propose a design structure based
on the CUSUM statistic to detect changes in the
process mean.
Numerical results of ARL1 were obtained using the
NIE method and explicit formulas for out-of-control
ARL1
10
( ),
which can be calculated through the
Wolfram Mathematica. Both methods of detecting
changes in the process mean are reported in Table 2,
Table 3, Table 4, and Table 5. We also proposed
SDRL to compare the performance of ARL in this
scenario. According to these findings, the ARL1
results calculated using the NIE method in (13) and
explicit formulas in (14) tended to decrease
sensitively as the shift size increased for small to
moderate shift magnitude shifts in the process. The
ARL1 is more effective at detecting small
(0 0.5)
to moderate
(0.5 1.0)
shifts than
large
(1.0 5.0)
ones. In case of a small shift in the
process mean, NIE methods for k = 2.5 (see Figure
1(a)) produced lower ARL1 values compared to k =
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3.0 and 5.0, respectively (see Figure 1(b)-(c)). In case
k = 2.5, ARFIMA processes with small AR
coefficient values
1
( ) 0.1
(see Figure 1(a)) were
more sensitive to process mean shift detection by
both methods than those with large AR coefficients
of
1
= 0.3, 0.5, and 0.7, respectively (see Figure 1(b)
– (c)). However, both methods produced the same
ARL1 value for all values of k for a large shift.
Similar to the ARL1 results, the SDRL1 results also
demonstrate a decreasing pattern as the shift size
increased for all scenarios (see Table 2, Table 3,
Table 4, and Table 5).
Furthermore, percentage change results calculated
at various magnitudes of process mean shifts for the
long-memory processes were greater than 95%,
indicating that the proposed method is accurate and
highly consistent with the explicit formulas.
The summary of the solution of the Integral
equation can be approximated ARL using the Gauss-
Legendre quadrature rule. The results demonstrate
that the NIE method is a simpler alternative to ARL
calculations, which approximate the accuracy of the
ARL, [27], [28]. However, the Gauss-Legendre rule
yielded the most straightforward ARL calculation
and the highest accuracy for the given number of
nodes. Lasty, the graphical displays of approximating
the ARL1 accurately using NIE method on the
CUSUM control is presented in Figure 2.
Table 2. Comparison of ARL1 between NIE method and explicit formulas and SDRL1 of CUSUM chart for a long-memory
4
SARFIMA(1, 0.1)(1,0.1, 1)
model where
10.1
at ARL0 = 370
k = 2.5
SDRL1
Acc%
k = 3.0
SDRL1
Acc%
k = 5.0
SDRL1
Acc%
NIE
Explicit
NIE
Explicit
NIE
Explicit
0.025
313.089
313.759
313.259
99.79
316.375
316.968
316.468
99.81
319.751
319.973
319.473
99.93
0.05
267.614
268.156
267.656
99.80
273.019
273.516
273.016
99.82
278.423
278.612
278.112
99.93
0.10
1,99.797
200.173
199.672
99.81
207.401
207.755
207.254
99.83
215.082
215.221
214.720
99.94
0.25
96.020
96.165
95.664
99.85
103.693
103.842
103.341
99.86
112.081
112.144
111.643
99.94
0.50
39.889
39.931
39.428
99.89
44.571
44.621
44.118
99.89
50.366
50.389
49.886
99.95
0.75
21.765
21.782
21.276
99.92
24.551
24.572
24.067
99.91
28.370
28.381
27.877
99.96
1.0
14.060
14.068
13.559
99.94
15.806
15.817
15.309
99.93
18.419
18.425
17.918
99.97
1.5
7.863
7.866
7.349
99.96
8.663
8.667
8.152
99.95
10.046
10.048
9.535
99.98
2.0
5.451
5.453
4.928
99.96
5.877
5.879
5.356
99.97
6.707
6.708
6.188
99.99
2.5
4.237
4.238
3.704
99.98
4.487
4.488
3.957
99.98
5.031
5.032
4.504
99.98
3.0
3.523
3.524
2.982
99.97
3.680
3.681
3.141
99.97
4.060
4.061
3.526
99.98
4.0
2.736
2.737
2.180
99.96
2.806
2.806
2.251
100.00
3.016
3.016
2.466
100.00
5.0
2.317
2.317
1.747
100.00
2.351
2.351
1.782
100.00
2.480
2.480
1.916
100.00
h
4.020943
3.303497
1.149340
Table 3. Comparison of ARL1 between NIE method and explicit formulas and SDRL1 of CUSUM chart for a long-memory
4
SARFIMA(1, 0.1)(1,0.1, 1)
model where
10.3
at ARL0 = 370
k = 2.5
SDRL1
Acc%
k = 3.0
SDRL1
Acc%
k = 5.0
SDRL1
Acc%
NIE
Explicit
NIE
Explicit
NIE
Explicit
0.025
313.570
314.225
313.725
99.79
316.387
317.171
316.671
99.75
319.774
319.987
319.487
99.93
0.05
268.404
268.944
268.444
99.80
273.367
273.857
273.357
99.82
278.456
278.636
278.136
99.94
0.10
200.899
201.275
200.774
99.81
207.894
208.244
207.743
99.83
215.125
215.258
214.757
99.94
0.25
97.103
97.249
96.748
99.85
104.209
104.357
103.856
99.86
112.131
112.191
111.690
99.95
0.50
40.523
40.567
40.064
99.89
44.903
44.952
44.449
99.89
50.405
50.427
49.924
99.96
0.75
22.131
22.148
21.642
99.92
24.756
24.777
24.272
99.92
28.398
28.408
27.904
99.96
1.0
14.283
14.291
13.782
99.94
15.940
15.951
15.443
99.93
18.439
18.445
17.938
99.97
1.5
7.960
7.963
7.446
99.96
8.728
8.732
8.217
99.95
10.058
10.061
9.548
99.97
2.0
5.500
5.502
4.977
99.96
5.913
5.915
5.392
99.97
6.715
6.716
6.196
99.99
2.5
4.264
4.265
3.732
99.98
4.509
4.511
3.980
99.96
5.037
5.038
4.510
99.98
3.0
3.540
3.540
2.999
100.00
3.695
3.696
3.157
99.97
4.064
4.065
3.530
99.98
4.0
2.743
2.743
2.187
100.00
2.814
2.814
2.259
100.00
3.018
3.019
2.469
99.97
5.0
2.320
2.320
1.750
100.00
2.355
2.355
1.786
100.00
2.482
2.482
1.918
100.00
h
3.937120
3.240665
1.098567
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Table 4. Comparison of ARL1 between NIE method and explicit formulas and SDRL1 of CUSUM chart for a long-memory
with
4
SARFIMA(1, 0.1)(1,0.1, 1)
where
10.5
at ARL0 = 370
k = 2.5
SDRL1
Acc%
k = 3.0
SDRL1
Acc%
k = 5.0
SDRL1
Acc%
NIE
Explicit
NIE
Explicit
NIE
Explicit
0.025
314.001
314.651
314.151
99.79
316.784
317.360
316.860
99.82
319.798
320.000
319.500
99.94
0.05
269.114
269.651
269.151
99.80
273.692
274.174
273.674
99.82
278.487
278.659
278.159
99.94
0.10
201.894
202.269
201.768
99.81
208.356
208.702
208.201
99.83
215.167
215.293
214.792
99.94
0.25
98.087
98.236
97.735
99.85
104.695
104.842
104.341
99.86
112.177
112.234
111.733
99.95
0.50
41.108
41.153
40.650
99.89
45.217
45.266
44.763
99.89
50.440
50.462
49.959
99.96
0.75
22.470
22.489
21.983
99.92
24.952
24.973
24.468
99.92
28.424
28.434
27.930
99.96
1.0
14.491
14.501
13.992
99.93
16.068
16.079
15.571
99.93
18.459
18.464
17.957
99.97
1.5
8.052
8.056
7.539
99.95
8.791
8.795
8.280
99.95
10.070
10.072
9.559
99.98
2.0
5.548
5.549
5.024
99.98
5.949
5.951
5.428
99.97
6.724
6.724
6.204
100.00
2.5
4.291
4.292
3.759
99.98
4.531
4.533
4.002
99.96
5.042
5.043
4.515
99.98
3.0
3.556
3.557
3.016
99.97
3.710
3.711
3.172
99.97
4.068
4.069
3.534
99.98
4.0
2.749
2.749
2.193
100.00
2.821
2.821
2.267
100.00
3.021
3.021
2.471
100.00
5.0
2.322
2.322
1.752
100.00
2.359
2.359
1.790
100.00
2.484
2.484
1.920
100.00
h
3.856962
3.178866
1.047840
Table 5. Comparison of ARL1 between NIE method and explicit formulas and SDRL1 of CUSUM chart for a long-memory
4
SARFIMA(1, 0.1)(1,0.1, 1)
model where
10.7
at ARL0 = 370
k = 2.5
SDRL1
Acc%
k = 3.0
SDRL1
Acc%
k = 5.0
SDRL1
Acc%
NIE
Explicit
NIE
Explicit
NIE
Explicit
0.025
314.394
315.038
314.538
99.80
316.970
317.537
317.037
99.82
319.820
320.013
319.513
99.94
0.05
269.760
270.293
269.793
99.80
273.994
274.472
273.972
99.83
278.516
278.680
278.180
99.94
0.10
202.799
203.172
202.671
99.82
208.789
209.131
208.630
99.84
215.205
215.325
214.824
99.94
0.25
98.990
99.139
98.638
99.85
105.153
105.298
104.797
99.86
112.219
112.274
111.773
99.95
0.50
41.650
41.696
41.193
99.89
45.516
45.565
45.062
99.89
50.474
50.494
49.991
99.96
0.75
22.788
22.807
22.301
99.92
25.139
25.161
24.656
99.91
28.448
28.458
27.954
99.96
1.0
14.688
14.698
14.189
99.93
16.190
16.202
15.694
99.93
18.477
18.482
17.975
99.97
1.5
8.141
8.144
7.628
99.96
8.851
8.856
8.341
99.94
10.081
10.083
9.570
99.98
2.0
5.594
5.595
5.070
99.98
5.983
5.985
5.462
99.97
6.730
6.731
6.211
99.99
2.5
4.318
4.319
3.786
99.98
4.553
4.554
4.023
99.98
5.047
5.048
4.520
99.98
3.0
3.572
3.573
3.032
99.97
3.725
3.725
3.186
100.00
4.072
4.073
3.538
99.98
4.0
2.756
2.756
2.200
100.00
2.828
2.828
2.274
100.00
3.023
3.023
2.473
100.00
5.0
2.325
2.325
1.755
100.00
2.363
2.363
1.795
100.00
2.485
2.485
1.921
100.00
h
3.779928
3.11,8004
0.997154
(a)
2.5k
(b)
3.0k
(c)
5.0k
Fig. 1: Graphical displays of approximating the ARL1 accurately using the NIE method on the CUSUM control
chart for a long-memory
4
ARFIMA 1, 0.1 0,0.1, 1
model with coefficient value
10.1
: (a)
2.5,k
(b)
3.0k
and (c)
5.0k
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Volume 18, 2023
(a)
10.1
(b)
10.3
(c)
10.5
(d)
10.7
Fig. 2: Graphical displays of approximating the ARL1 accurately using NIE method on the CUSUM control
chart:(a)
10.1,
(b)
10.3,
(c)
10.5
and (d)
10.7
6 Conclusions and Recommendations
The ARL can be derived by using the NIE approach
initially introduced by [2]. The ARL can be used to
compare the performances of different control charts.
The CUSUM control chart performs well at detecting
small-to-moderate changes in the process mean. In
this study, we applied the Gauss-Legendre quadrature
to solve the integral equations for the NIE approach
used to derive the ARL for changes in the mean of a
long-memory
ARFIMA( , )( , , )s
p d P D Q
ARFIMA( , )( , , )s
p d P D Q
model with underlying
exponential white noise running on a CUSUM
control chart. In addition to calculating the
ARL
using both the NIE and explicit formula approaches, we
also calculated the
SDRL.
It was found that the
ARL1 and SDRL1 values decreased rapidly and in the
same direction.
The results indicate that the proposed NIE method
is a good candidate for ARL determination in future
research in this scenario. The method could be
adapted for new memory-type control charts. In
addition, the NIE method could be applied to real-life
applications involving time series models.
Acknowledgments:
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-66-62.
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Volume 18, 2023
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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 would like to express her gratitude to the
Faculty of Applied Science, King Mongkut’s
University of Technology North Bangkok, Thailand
for support with research grant No. 662130.
Conflict of Interest
Please declare anything related to this study.
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 and CONTROL
DOI: 10.37394/23203.2023.18.39
Wilasinee Peerajit
E-ISSN: 2224-2856
381
Volume 18, 2023
