Double Moving Average Control Chart for Time Series Data with Poisson
INARCH(1)
SUGANYA PHANTU1, YUPAPORN AREEPONG2, SAOWANIT SUKPARUNGSEE2,*
1Faulty of Science, Energy, and Environment,
King’s Mongkut University of Technology North Bangkok,
Rayong 21120,
THAILAND
2Department of Applied Statistics, Faculty of Applied Science,
King Mongkut’s University of Technology, North Bangkok,
Bangkok,
THAILAND
*Corresponding Author
Abstract: - The objectives of this research are to find the explicit formulas of the average run length (ARL) of a
double moving average (DMA) control chart for first-order integer-valued autoregressive conditional
heteroscedasticity (INARCH1))) of Poisson count data. In addition, the numerical results obtained from the
proposed explicit formulas are compared with those obtained from Monte Carlo simulations (MC) for the Poisson
INARCH(1) counting process. An out-of-control ARL (ARL1) is the criteria for measuring the performance of
control charts. The numerical results found that the values of both ARL0 and ARL1 obtained from explicit formulas
agree with the numerical results obtained from the Monte Carlo simulation (MC), but the latter is very time-
consuming.
Key-Words: - Poison Process, Average Run Length, Explicit Formulas, Optimal Design, Volatility, Time series.
Received: November 14, 2023. Revised: January 7, 2024. Accepted: January 29, 2024. Published: February 23, 2024.
1 Introduction
Statistical Quality Control (SQC) is critically
important in various industries, [1], and business, [2],
sectors. It plays a crucial role in ensuring that
products and processes meet specified standards and
customer expectations. Here are some reasons
highlighting the importance of statistical quality
control: consistency and uniformity, defect reduction,
cost reduction, customer satisfaction, process
improvement, and early detection of issues. In
summary, Statistical Quality Control is a
fundamental aspect of quality management in various
industries. It enables organizations to produce high-
quality products, reduce costs, meet customer
expectations, and stay competitive in the market. The
SQC relies on several key assumptions to effectively
apply statistical methods and tools to monitor and
control processes. Many statistical methods in SQC
assume that the data follows a normal distribution.
This assumption is particularly important when using
control charts and other statistical tools. While
normality is not always strictly required, deviations
from normal distribution may affect the accuracy and
interpretation of results. It's important for
practitioners to be aware of these assumptions and to
assess whether they hold in a particular context.
Deviations from these assumptions may necessitate
adjustments to the statistical approach or additional
considerations in the interpretation of results.
In real-world applications, time-series data which
may be found in a variety of fields including
communication engineering, epidemiology, [3],
monetary economics [4], financial and insurance [5],
environmental science [6], and so on—may be tied to
time. The number of incidents and accident rates,
multiple crimes, the identification of communication
errors, the number of customers using the internet in
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an hour, the volume of business phone calls, the
number of customers from different organizations
who have used the service in the previous year, and
the amount of time spent waiting for the plane to take
off are other scenarios that may give rise to it.
Consequently, utilizing the thinning operator
concepts described by, [7], [8], [9], several
researchers have created a model appropriate for
these data, dubbed the first order of Integer-valued
Autoregressive (INAR(1)), proposed by, [10], [11].
They took advantage of a discrete distribution time
series model created by, [12].
The discrete probability distribution known as
the Poisson distribution, which represents the number
of occurrences that take place within a specific time
or location, is especially well-suited for describing
counting operations. A sort of time series model
intended for count data that displays conditional
heteroscedasticity and autocorrelation is an integer-
valued autoregressive conditional heteroscedasticity
(INARCH) model with a Poisson process. The term
"autoregressive conditional heteroscedasticity"
(ARCH) in this context denotes that the conditional
variance of the process is not constant but rather
fluctuates with time, and "integer-valued" refers to
the type of data. It can also happen in other
circumstances, like the quantity and rate of accidents,
multiple crimes, the identification of communication
breakdowns, the number of users accessing the
internet in a single hour, the number of users from
different organizations who have used the service in
the previous year, and the amount of time spent
waiting for an aircraft to take off from the airport.
Consequently, utilizing the thinning operator
concepts described by, [7], [8], [9], several
researchers have created a model appropriate for
these data, dubbed the first order of Integer-valued
Autoregressive (INAR(1)), proposed by, [10], [11].
The first-order numerical correlation Poisson model
with unstable variance Poisson Integer-valued
Autoregressive Conditional Heteroscedasticity
(INARCH(1)) was developed from many research
fields such as application use in pharmaceutical
science by, [13], infectious rates by, [14], [15],
studied the number of insurance claims from
insurance companies, [16], and applied to the
queueing of internet access claims data, [17].
With its Poisson distribution and constant
variance, the Poisson INARCH(1) model bears a
resemblance to the AR(1) model. By multiplying the
random operator by a numerical random variable to
construct an INAR(1) and INARCH(1) by, [18], [19],
[20], [21], [22], these use a random operator known
as the Binomial Thinning Operator to assist in the
creation of an integer value model.
Counting data has been the subject of numerous
studies, many of which have taken dependent data
characteristics and variable variance into account. To
effectively identify the nature of this type of data,
such as when it is highly correlated, they have
created a control chart. The autocorrelation model for
Poisson counting first-order unstable variance
(INARCH(1)) is a model that works well with this
kind of data. Its control chart performance has been
examined by several researchers using various
control charts examined the Poisson INAR(1)
model's cumulative combined control chart, [23],
[24], examined the Poisson INAR(1) model of the
counting data's two-sided cumulative combined
control chart, and [25], examined the correlated
Poisson model's quality control.
The expected number of samples or observations
collected from a process before a signal indicating a
shift or change in the process mean or variability is
detected is referred to as the average run length
(ARL) in statistical process control. A statistical
process control chart is a graphical tool used to
monitor and control a process over time. The ARL
evaluates the performance of these charts. It can be
broken down into two categories: processes that are
under control (in-control processes) are represented
by the symbol ARL0, and processes that are not under
control are represented by the symbol ARL1. In
mathematical notation, they can be expressed as
follows:
0()ARL E T
and
11
Minimize ( 1)ARL E
where
(.)
E
is the expected time.
is the first passage time (stopping
times)
T
is a constant (usually given
T
=370)
is the change-point time process has
changed from
1
,Fx
to
0
,Fx
.
When comparing the capabilities of various
control charts, numerical techniques for calculating
the ARLs are frequently employed. There are
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multiple methods for figuring out this number. Its
benefits and drawbacks are as follows: conventional
techniques, like Monte Carlo Simulation (MC), are
frequently employed. It is frequently used to confirm
accuracy in comparison to alternative methods, but
processing times may occasionally be lengthy. The
inverse of a matrix is found using the Markov chain
approach (MCA), [26], [27], but the theory of
convergence properties still needs to be backed up.
To estimate, the Numerical Integral Equations (NIE)
method makes use of sophisticated mathematical
computations, [28]. The Martingale Approach is a
contemporary technique that is quick, simple to
calculate, and convenient, but it could take some time
for the simulation method to verify accuracy, [29],
[30].
Small changes in the mean or variability can be
more easily detected with a control chart that has a
short ARL since it is more responsive to process
variations. A control chart with a longer ARL, on the
other hand, is less sensitive and might be more
suitable for processes that naturally fluctuate.
Therefore, the purpose of this research is to use
moving average control charts in conjunction with
the Poisson INARCH(1) model to design an
appropriate control for detecting the process's rapid
dynamics. to identify abrupt average shifts in the
manufacturing process. By comparing the detection
efficiency of the MA control chart with the DMA
chart while taking the average run length into
account, and by verifying the accuracy of the results
obtained from the formula with the simulation
results.
2 Research Methodology
The Poisson INARCH(1) model, moving average
control charts, double-moving average control charts,
and their properties are briefly reviewed in this
section.
2.1 Binomial Thinning Operator
The probabilistic operation of binomial thinning is
introduced by, [31]. If
X
is a discrete random
variable with range
00,1,...N
and if
0,1 ,
then the random variable
1
:N
i
i
NX
is said to
arise from
X
by binomial thinning, and the
i
X
are
referred to as the counting series. The thinning
operation, when operated on by a parameter proven
to be an adequate alternative to scalar multiplication,
is defined as
1
N
i
i
NX
where
i
X
are independent and identically distributed
(i.i.d.) Bernoulli random variables with success
probability
.
The operator is a random operator, and
the random variable
N
has a binomial
distribution with parameters
,N
and counts the
number of survivors from the count
N
remaining
after thinning.
The variance and expectations of
N
can be
obtained with ease by using the following well-
known rules for a conditional moment
[ ] [ ]Eα N αE No
and
2
[ ] [ ] (1 ) [ ].Vα N α V N α α E N o
2.2 Poisson INARCH (1) Model
A particular kind of time series model called an
INARCH (1) process is used to model the number of
events that transpire within a given time interval to
explain the behavior of a counting process.
Specifically, an integrated autoregressive conditional
heteroskedasticity model that assumes the counting
process is driven by its past values and a stochastic
component is called an INARCH(1) Poisson
counting process. The following details relate to the
INARCH(1) process:
1t t t
NN
where is a random operator and at time t is
independent of
()
t
and
()
SSt
N
t
is an innovation-independent counts of
()
SSt
N
the
distribution
1t
Pois N
t
N
is counting observations at time t.
If the initial count
0
N
is distributed as
()
1
β
Pois
α
then
t
N
is stationary and distributed as
( ).
1
β
Pois
α
According to the above situation, it can be modeled
as a Poisson INARCH(1) model. The expectation and
variance of the Poisson INARCH(1) model are:
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[]1
t
EN
and
2
[ ] .
11
t
VN
2.3 Moving Average Control Chart for
Poisson INARCH(1) Model
A time-varying control chart with unequal weights,
known as a moving average control chart (MA chart),
[32], was developed to count variables such as the
quantity of nonconformities in a product's inspection
unit. Assume that discrete observations are obtained
from a sequence of identically independent
distributions and the Poisson INARCH(1) model.
The definition of the width at a time moving average
is:
1
1
1 ;
.
1 ;
t
j
j
tt
j
j t k
N i k
i
MA
N i k
k
The expectation of the MA statistics for the
Poisson INARCH(1) model when
ik
and
ik
is:
.
1
t
E MA
The variance of the MA statistics for the Poisson
INARCH(1) model for both cases of
ik
and
ik
is:
2
2
,
11
.
,
11
i
ik
i
Var MA
ik
k
The upper and lower control limits of the MA
statistics are given as follows.
2
2
1
1
,
111
,
111
H i k
H i k
i
k
where
1
H
refers to a coefficient of control limit of
the MA chart.
2.4 Double Moving Average Control Chart
for Poisson INARCH(1) Model
A double-moving average control chart (DMA chart)
was proposed by, [33]. The observations of DMA
statistics are the collected double moving average of
the MA statistics. The DMA of span
k
at the time
t
is defined as:
12
11
11
... ;
... ; 2 1
... ; 2 1
i i i
i i i w
i
i i i w
MA MA MA ik
i
MA MA MA
DMA k i k
k
MA MA MA ik
k
(1)
where
i
MA
refers to the statistic of the MA chart.
It is a time-weighted moving control chart based on a
simple, unweighted moving average. Assume N1, N2,
… are obtained from a Poisson INARCH(1) process.
The MA statistic of span
w
at a time
i
defined as,
[34]
11
... ;
i i i k
iX X X
MA k
for
ik
.
For the period
ik
we do not have
k
measurements to compute a moving average of span
.k
For these periods, the average of all measurements
up to periods
i
defines the MA. The mean based on
an in-control process of the DMA chart are:
()
1
i
E DMA
(2)
and variance based on an in-control process of the
DMA chart are:
22
1
1
22
1
22
1;
11
11
( ) ( 1) ; 2 1
11
; 2 1.
11
i
j
k
i
j i k
ik
ji
Var DMA j k k i k
jk
k
ik
k
(3)
From Eq. (2) and (3), the upper and lower control
limit of the DMA chart can be established as follows
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222
1
1
222
1
222
1;
111
11
( 1) ; 2 1
111
; 2 1
111
i
j
k
j i k
H i k
ji
H j k k i k
jk
k
H i k
k
(4)
when
2
H
is a coefficient of control limit based
on a desired in-control ARL0. The Poisson
INARCH(1) model of the DMA chart will signal the
out-of-control situation when
i
DMA LCL
or
.
i
DMA UCL
3 An Explicit Formulas for Average
Run Length of DMA Control Chart
In quality control and process monitoring, Average
Run Length (ARL) is frequently used to assess how
well a control chart detects shifts or modifications in
a process. Assuming the process is under control, the
ARL value indicates the anticipated number of
observations before a control chart indicating a
process shift. Explicit formulas, integral equations,
Markov chain analysis, simulation, and other
mathematical techniques can all be used to calculate
the ARL value. The results of these techniques can
then be compared with Monte Carlo simulation
results. The latter, which is employed in situations
where ARL's explicit formulas or closed-form
formulas are unavailable, takes a very long time.
Based on the central limit theorem (CLT), this
section presents the derivative analytical ARL of the
DMA chart for the Poisson INARCH(1)
observations. The following formula can be used to
determine the DMA chart's average run length:
Let
,ARL n
then
11
out of control signal at time
1out of control signal at time 2 1
P i k
ARL n
P k i k
n
(2 2) out of control signal at time 2 1
nk P i k
n
.
(5)
According to Eq. (5), the DMA statistics in terms
of out-of-control signals at time
i
state are replaced
as follows:
1
1k
i i k i i k
i
P M UCL P M LCL
n
22
2 1 2 1
1
1k
j k i k j k i k
j i k
P M UCL P M LCL
n
2 1 2 1
(2 2)
i i k i i k
nk P M UCL P M LCL
n
. (6)
Then, substitute the upper and lower control
limits of DMA statistics from Eq. (1) into Eq. (6),
which can be rewritten.
1
1
222
1
11
111
k
i
i
ji
j
j
n
MA
PH
ij
i
1
22
2
1
111
1
i
j
ji
j
PH
ii
MA
j
22
1
1
1
222
1
111
( 1)
111
k
j i k
i
jk
j i k
j i k
n
MA
P H j k
k j k
k
1
22
1
1
111
11
( 1)
i
j
j i k k
j i k
PH
k
MA
jk
jk
k
1
222
(2 2)
111
i
j
j i k
nk
n
MA
PH
kk
1
22
2
111
i
j
j i k
PH
k
MA
k
.
(7)
The central limit theorem is used to derive the
explicit formulas. Then, Eq. (7) can be rewritten as:
1
22
1
11 1
1
11
ktk
Ai
t
j
UCL
PZ
ARL n
ji
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22
1
1
1
11
tk
Ai
j
LCL
PZ
ji
21
22
11
22
1
11
11
( 1)
11
k i k
k
j i k Bk
j i k
n
UCL
PZ
jk
jk
k
21
1
22
1
1
11
( 1)
11
k i k
Bk
j i k
PLCL
Z
jk
jk
k
21
22
(2 2) 1
11
ik
C
UCL
nk PZ
n
k
21
22
1.
11
ik
C
LCL
PZ
k
(8)
Next, the statistics of the DMA chart in Eq. (8)
are transformed to be standardized, then assume that.
1
22
1
1
1
11
i
j
j
i
j
Ai
Z
j
MA
i
,
1
1
22
1
1
11
( 1)
11
i
j
j
w
j i w
Bk
Z
jk
jk
MA
k
and
1
22
1.
11
i
j
j
Ck
Z
MA
k
According to Eq. (8), let
1
22
1
11
kik
A
t
UCL
A P Z
i
22
1
11
ik
A
LCL
PZ
i
21
22
11
22
1
11
11
( 1)
11
k i k
k
j i k Bk
j i k
n
UCL
B P Z
jk
jk
k
21
1
22
1
1
11
( 1)
11
k i k
Bk
j i k
PLCL
Z
jk
jk
k
and
21
22
1
11
ik
C
UCL
C P Z
k
21
22
1.
11
ik
C
LCL
PZ
k
Then, the explicit formulas of ARL0 and ARL1
for the DMA chart are rewritten by substituting A, B,
and C into Eq. (8).
1 1 1 (2 2)nk
A B C
n n n n
1(2 2)
AB
nk
C
.
As we give
,ARL n
then
1
[ 1 ] (2 2)ARL A B C k
.
(9)
From Eq. (9), proving the explicit formula of the
DMA chart can be divided into two cases:
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Proposition I: Explicit formulas of ARL0 for the DMA
chart.
1
0 0 0
22
1
00
00
0
22
100
0 0 0
222
1
00
00
0
22
100
0
2
1
11
11
1
11
1
11
11
1
1
11
w
i
i
j
i
j
i
j
Ai
j
A
j
PZ
j
Hji
PZ
ji
ARL
Hi
i
22
1
00
2
22
00
00
0
22
00
00
22
000
1
2
1
1
1
1
2
1
11
11
11
111
11
( 1)
11
( 1)
11
( 1)
w
j i w
k
j i k
w
j i w
k
j i k
PZ
PZ
H
H
jk
jk
k
jk
jk
k
jk
jk
k
0
0
0
22
00
1
1
1
11
11
( 1)
k
j i k
jk
jk
k
0 0 0
22
00
00
0
22
00
1
0 0 0
222
00
00
0
22
00
2
11
11
11
11
11
11 (2 2)
C
C
H
PZ
Hk
PZ
kw
k
k
(10)
Proposition II: Explicit formulas of ARL1 for the
DMA chart.
1
00
1
22
01
00
1
22
11
00
1
222
01
00
1
22
11
1
2
11
11
11
11
11
1
11
k
i
A
A
i
PZ
t
Hi
PZ
i
ARL
H
22
1
00 1
22
01
00
1
22
11
00
22
000
1
2
1
1
1
1
2
1
11
11
11
111
11
( 1)
11
( 1)
11
( 1)
k
j i k
k
j i k
Bk
j i k
k
j i k
B
PZ
PZ
H
H
jk
jk
w
jk
jk
w
jk
jk
w
1
1
1
22
11
1
1
1
11
11
( 1)
k
j i k
jk
jk
w
00
1
22
01
00
1
22
11
1
00
1
222
01
00
1
22
11
2
11
11
11
11
11
11 (2 2)
C
C
H
PZ
Hk
PZ
kk
k
k
(11)
4 Numerical Results
This section presents the performance results of the
DMA chart for the INARCH(1) model. It is divided
into two parts: Part 1 presents the average run length
determination of the DMA chart, and Part 2 presents
the performance comparison control chart for actual
data.
4.1 Average Run Length of DMA Chart
The study compares the accuracy and precision of the
results calculated from the explicit formula from Eqs.
(10) and (11). This research presents the results
obtained from the Monte Carlo simulation method.
The in-control parameter was given
01,
00.1
,
and ARL0 = 370. The magnitude of changes of
0
is
1
which set value equal to 0.1, 0.2,…,1. The
magnitude of the changes of
0
is
2
which set
value equal to 0.1, 0.2, …,1. The period for finding
the moving average
k
equals 2, 5, 10, 15, and 20;
for the MC method, the number of iteration cycles is
50,000 replications. The numerical results are shown
in Table 1 and Table 2 for the given
change and
Table 3 and Table 4 for the given
change as
follows.
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Table 1. Average run length of DMA chart for the INARCH(1) model where
01,
00.1
and given
change
1
k
2
5
10
15
20
0.0
370.398*
371.560**
(1.455)
370.398
370.589
(1.513)
370.398
370.677
(1.643)
370.398
371.294
(1.545)
370.398
370.499
(1.451)
0.1
318.871
318.566
(0.956)
179.137
179.556
(0.942)
67.596
67.659
(0.933)
42.582
42.355
(0.956)
41.641
41.328
(0.973)
0.2
236.985
236.454
(0.853)
71.536
71.298
(0.835)
25.468
25.436
(0.867)
27.526
27.483
(0.862)
35.219
35.748
(0.845)
0.3
173.99
173.452
(0.760)
36.633
36.806
(0.743)
18.907
18.706
(0.732)
25.343
25.398
(0.763)
32.345
32.472
(0.739)
0.4
131.593
131.839
(0.678)
23.070
23.547
(0.665)
17.016
17.430
(0.645)
23.726
23.094
(0.654)
28.357
28.445
(0.637)
0.5
103.213
103.454
(0.565)
16.822
16.745
(0.549)
16.090
16.375
(0.548)
21.841
21.427
(0.561)
24.151
24.964
(0.535)
0.6
83.696
83.409
(0.453)
13.541
13.528
(0.467)
15.408
15.398
(0.403)
19.820
19.406
(0.438)
20.464
20.673
(0.468)
0.7
69.822
69.328
(0.346)
11.636
11.545
(0.369)
14.787
14.981
(0.347)
17.888
17.059
(0.317)
17.510
17.451
(0.336)
0.8
59.643
59.462
(0.288)
10.439
10.506
(0.243)
14.180
14.227
(0.231)
16.171
16.548
(0.243)
15.249
15.493
(0.269)
0.9
51.961
51.398
(0.185)
9.636
9.782
(0.187)
13.585
13.679
(0.168)
14.703
14.093
(0.189)
13.554
13.603
(0.172)
1.0
46.019
46.475
(0.098)
9.070
9.113
(0.096)
13.008
13.241
(0.093)
13.473
13.326
(0.087)
12.286
12.096
(0.097)
*results from explicit formulas, **results from simulation, the Italic number is minimum ARL1
The number in parentheses is the standard deviation of run length.
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Table 2. Average run length of DMA chart for the INARCH(1) model where
05,
00.1
and given
change
1
k
2
5
10
15
20
0.0
370.398
370.623
(1.451)
370.398
371.558
(1.565)
370.398
370.699
(1.630)
370.398
370.664
(1.597)
370.398
371.745
(1.468)
0.1
318.871
318.452
(0.988)
179.137
179.564
(0.965)
67.596
67.493
(0.934)
42.582
42.389
(0.974)
41.641
41.657
(0.982)
0.2
236.985
236.548
(0.865)
71.536
71.478
(0.837)
25.468
25.489
(0.863)
27.526
27.438
(0.844)
35.219
35.462
(0.847)
0.3
173.99
173.495
(0.744)
36.633
36.291
(0.732)
18.907
18.539
(0.752)
25.343
25.493
(0.716)
32.345
32.367
(0.759)
0.4
131.593
131.522
(0.676)
23.070
23.541
(0.629)
17.016
17.698
(0.645)
23.726
23.578
(0.653)
28.357
28.433
(0.611)
0.5
103.213
104.540
(0.577)
16.822
16.483
(0.509)
16.090
16.493
(0.521)
21.841
21.437
(0.547)
24.151
24.368
(0.508)
0.6
83.696
83.533
(0.409)
13.541
13.276
(0.425)
15.408
15.463
(0.487)
19.820
19.439
(0.489)
20.464
20.433
(0.465)
0.7
69.822
69.789
(0.354)
11.636
11.478
(0.328)
14.787
14.728
(0.369)
17.888
17.439
(0.376)
17.510
17.213
(0.361)
0.8
59.643
59.675
(0.265)
10.439
10.438
(0.230)
14.180
14.287
(0.261)
16.171
16.585
(0.255)
15.249
15.309
(0.254)
0.9
51.961
51.433
(0.196)
9.636
9.456
(0.126)
13.585
13.269
(0.134)
14.703
14.979
(0.183)
13.554
13.271
(0.168)
1.0
46.019
46.767
(0.093)
9.070
9.327
(0.096)
13.009
13.726
(0.089)
13.473
13.896
(0.097)
12.286
12.078
(0.098)
*results from explicit formulas, **results from simulation, the Italic number is minimum ARL1
The number in parentheses is the standard deviation of run length.
The results obtained from the proposed explicit
formulas are compared with those obtained from
Monte Carlo simulations. The results showed that the
efficiency of explicit formulas is the same as those
obtained from the Monte Carlo simulation; however,
the former is less time-consuming. When parameter
values were changed, for example, when the process
was under control
01,
00.1,
given in Table 1,
it was found that when the parameter value changed
magnitude,
10.1,
that
20k
value would
minimize ARL1 when
1
0.2 0.5
that
15k
value
makes ARL1 minimum, and when
10.6
that
10k
value makes ARL1, which yields the same findings as
05,
00.1
shown on Table 2.
For the case of changing the parameters
and
the control parameter is given
01,
and
00.1
the numerical results showed that the performance of
explicit formulas for ARL of the DMA chart was
excellent when compared with the Monte Carlo
simulation method. Unfortunately, the latter is very
time-consuming. In Table 3 and Table 4, it was found
that when the magnitude of the parameter change
20.1,
that value
20k
caused the lowest ARL1
value.
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Table 3. Average run length of DMA chart for the INARCH(1)model where
01,
00.1
and given
change
2
k
2
5
10
15
20
0.0
370.398
371.799(
1.406)
370.398
370.573(
1.476)
370.398
370.599
(1.465)
370.398
371.981(
1.462)
370.398
370.568(
1.532)
0.1
370.388
370.572(
0.978)
370.337
370.452(
0.957)
370.159
370.376
(0.954)
369.874
369.756(
0.976)
369.491
369.769(
0.966)
0.2
370.359
370.465(
0.864)
370.154
370.306(
0.837)
369.443
369.392
(0.834)
368.307
368.776(
0.879)
366.793
366.542(
0.854)
0.3
370.309
370.284(
0.776)
369.848
369.243(
0.767)
368.253
368.583
(0.766)
365.72
365.548(
0.716)
362.373
362.452(
0.754)
0.4
370.24
370.199(
0.649)
369.419
369.292(
0.608)
366.569
366.592
(0.628)
362.151
362.434(
0.645)
356.35
356.539(
0.605)
0.5
370.15
370.173(
0.506)
368.869
368.658(
0.536)
364.483
364.591
(0.547)
357.652
357.563(
0.563)
348.878
348.563(
0.564)
0.6
370.041
370.067(
0.452)
368.197
368.549(
0.433)
361.927
361.973
(0.463)
352.289
352.479(
0.462)
340.141
340.568(
0.438)
0.7
369.911
369.985(
0.377)
367.405
367.291(
0.342)
358.945
358.607
(0.328)
346.136
346.511(
0.342)
330.34
330.522(
0.387)
0.8
369.762
369.675(
0.265)
366.494
366.485(
0.299)
355.555
355.376
(0.265)
339.277
339.678(
0.257)
319.686
319.382(
0.203)
0.9
369.592
369.489(
0.183)
365.464
365.752
(0.124)
351.779
(351.288(
0.189)
331.799
331.575
(0.176)
308.385
308.531(
0.183)
1.0
369.403
369.254(
0.096)
364.318
364.589(
0.074)
347.64
347.547
(0.087)
323.7953
23.609(0.
096)
296.635
296.546(
0.092)
*results from explicit formulas, **results from simulation, , Italic number is minimum ARL1
The number in parentheses is the standard deviation of run length
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Table 4. Average run length of DMA chart for the INARCH(1)model where
05,
00.1
and given
change
2
k
2
5
10
15
20
0.0
370.398
370.571
(1.406)
370.398
371.292
(1.392)
370.398
371.746
(1.434)
370.398
370.579
(1.419)
370.398
371.856
(1.333)
0.1
370.349
370.426
(0.968)
370.093
370.569
(0.924)
369.206
369.760
(0.966)
367.791
367.309
(0.933)
365.906
365.940
(0.954)
0.2
370.2
379.765
(0.843)
369.178
369.675
(0.846)
365.666
365.478
(0.862)
360.164
361.430
(0.873)
353.034
353.403
(0.879)
0.3
369.953
369.675
(0.784)
367.659
367.565
(0.709)
359.897
359.420
(0.774)
348.087
348.466
(0.761)
333.424
333.219
(0.791)
0.4
369.606
369.496
(0.655)
365.549
365.354
(0.611)
352.088
352.218
(0.647)
332.404
332.085
(0.679)
309.286
309.739
(0.675)
0.5
369.161
369.217
(0.579)
362.864
362.948
(0.554)
342.486
342.392
(0.561)
314.096
314.204
(0.566)
282.862
(282.571
(0.546)
0.6
368.617
368.452
(0.463)
359.627
359.650
(0.483)
331.376
331.084
(0.435)
294.157
295.641
(0.463)
256.039
256.659
(0.441)
0.7
367.975
367.290
(0.354)
355.599
355.642
(0.343)
319.064
319.302
(0.372)
273.482
273.439
(0.395)
230.184
230.798
(0.365)
0.8
367.236
367.928
(0.275)
351.599
351.203
(0.265)
305.855
306.549
(0.228)
252.813
253.074
(0.245)
206.151
206.454
(0.277)
0.9
366.4
366.097
(0.187)
346.872
346.087
(0.125)
292.043
292.438
(0.167)
232.713
232.669
(0.173)
184.372
185.078
(0.169)
1.0
365.468
365.289
(0.092)
341.716
341.685
(0.096)
277.894
277.439
(0.085)
213.577
213.094
(0.091)
164.984
165.938
(0.087)
*results from explicit formulas, **results from simulation, the Italic number is minimum ARL1
The number in parentheses is the standard deviation of run length.
4.2 Real Application
This section presents the performance of the DMA
chart with the Shewhart and MA chart. The data
consists of n = 108 monthly work stoppage count
displayed in Figure 1, originally published in, [35].
The mean of INARCH(1) model is 1.173, and the
variance of INARCH(1) model is 0.766. The result
shows that the first sample outside the control limit is
no. 4 for the Shewhart, MA, and DMA charts in
Figure 2, Figure 3 and Figure 4, respectively. It can
be concluded that all three charts are equally
effective in detecting such data change.
Fig. 1: Strikes counts data
0
5
10
15
111 21 31 41 51 61 71 81 91 101
strike count
time
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Fig. 2: Shewhart chart of strike count data
Fig. 3: MA chart (k=5) of strike count data
Fig. 4: DMA chart (k=5) of strike count data
5 Conclusion and Future Work
The derivative proof of explicit ARL formulas of the
DMA control chart for a Poisson INARCH(1)
process was presented. The numerical results were
obtained from the explicit formulas and compared
with the Monte Carlo simulation. They show that the
explicit formulas' accuracy is in excellent agreement
with the MC. Furthermore, the results found that
when a parameter both
and
increases, the DMA
chart will perform better as the value of
k
decreases
for all case studies. In addition, these explicit
formulas are simple and easy to implement with
reduced computation times. For future work, this
work can extend to other time series models and use
for the other control charts that practitioners may
apply in several fields.
Acknowledgment:
The authors would like to express their sincere
gratitude to the Department of Applied Statistics for
allowance and financial support to the Faculty of
Applied Science, King Mongkut's University of
Technology North Bangkok, Thailand for supporting
research grants with contract no. 672158.
References:
[1] H. Kim, Y. Lin, and T.-L.B. Tseng, “A review
on quality control in additive manufacturing,”
Rapid Prototyping Journal 24, 645-669 (2018).
[2] V. Golosnoy and W. Schmid, “EWMA control
charts for monitoring optimal portfolio
weights,” Sequential Analysis 26, 195-224
(2006).
[3] V. Golosnoy and W. Schmid, “EWMA control
charts for monitoring optimal portfolio
weights,” Sequential Analysis 26, 195-224
(2006).
[4] M. Frisén, “Evaluations of methods for
statistical surveillance,” Statistics in Medicine
11, 1489-1502 (1992).
[5] C. C. Torng, P. H. Lee and N. Y. Liao, “An
economic-statistical design of double sampling
X
control chart,” International Journal of
Production Economics 120, 495-500 (2009).
[6] M. Basseville and I. V. Nikiforov, “Detection
of Abrupt Changes: Theory and Application”,
Englewood Cliffs: Prentice Hall (1993).
[7] M. A. Al.Osh and A. A. Alzaid, “An integer-
valued pth-order autoregressive structure
(INAR(p)) process,” Journal of Applied
Probability 27, 314-324 (1990).
[8] F. W. Steuel and K. Van Harn, “Discrete
analogues of self-decomposability and
stability,” Annals of Probability 7, 893-899
(1979).
[9] E. McKenzie, “Some simple models for
discrete variate time series,” Water Resources
Bulletin 21, 645-650 (1985).
-5
0
5
10
15
110 19 28 37 46 55 64 73 82 91 100
shewhart CL LCL UCL
-5
0
5
10
15
110 19 28 37 46 55 64 73 82 91 100
MA CL_MA LCL_MA UCL_MA
-5
0
5
10
15
1 9 17 25 33 41 49 57 65 73 81 89 97 105
DMA CL_DMA LCL_DMA UCL_DMA
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Volume 21, 2024
[10] E. McKenzie, “Discrete Variate Time Series in
Handbook of Statistics,” Elsevier, Amsterdam,
2003.
[11] M. A. Al.Osh and A. A. Alzaid, “An integer-
valued pth-order autoregressive structure
(INAR(p)) process,” Journal of Applied
Probability 27, 314-324 (1990).
[12] F. W. Steuel and K. Van Harn, “Discrete
analogues of self-decomposability and
stability,” Annals of Probability 7, 893-899
(1979).
[13] D. Jin-Guan and L. Yuan, “The integer-valued
autoregressive (INAR(p)) model,” Journal of
Time Series Analysis 12, 129-142 (1991).
[14] J. Franke and T. H. Seligmann, “Conditional
maximum likelihood estimates for INAR (1)
processes and their application to modeling
epileptic seizure counts,” Developments in
time series analysis, 310-330 (1993).
[15] M. Cardinal, R. Roy, and J. Lambert, “On the
application of integer‐valued time series
models for the analysis of disease incidence,”
Statistics in Medicine 18, 2025-2039 (1999).
[16] C. Gourieroux, and J. Jasiak, “Heterogeneous
INAR (1) model with application to car
insurance,” Insurance: Mathematics and
Economics 34, 177-192 (2004).
[17] K. K. Jose and B. Abraham, “Analysis of DAR
(1)/D/s queue with Quasi-negative binomial-II
as marginal distribution,” Applied Mathematics
2, 1159 (2011).
[18] C. H. Weiß and M. C. Testik, “CUSUM
monitoring of first-order integer-valued
autoregressive processes of Poisson counts,”
Journal of Quality Technology 41, 389-400
(2009).
[19] C. H. Weiß and S. Schweer, “Detecting
overdispersion in INARCH(1) processes,”
Statistica Neerlandica 69, 281-297 (2015).
[20] C. H. Weiß, “Thinning operations for modeling
time series of counts – a survey,” AStA
Advances in Statistical Analysis 92, 319-341
(2008).
[21] S. Phantu, S. Sukparungsee, and Y. Areepong,
“DMA chart monitoring of the first integer-
valued autoregressive process of Poisson
counts,” Advances and Applications in
Statistics 52, 97-119 (2017).
[22] C. H. Weiß and M. C. Testik, “The Poisson
INAR(1) CUSUM chart under overdispersion
and estimation error,” IIE Transactions 43,
805-818 (2011).
[23] P. A. Yontay, “Two-Sided CUSUM for First-
order integer-valued autoregressive process of
Poisson counts, Master thesis,” Middle East
Technical University, Ankara (2011).
[24] C. H. Weiß, “Controlling correlated processes
of Poisson counts,” Quality and Reliability
Engineering International 23, 741-754 (2007).
[25] D. Brook and D. A. Evans, “An approach to
the probability distribution of CUSUM run
length,” Biometrika 59, 539-548 (1972).
[26] C. W. Champ and S. E. Rigdon, “A
comparison of the Markov chain and the
integral equation approaches for evaluating the
run length distribution of quality control
charts,” Communications in Statistics:
Simulation and Computation 20, 191-204
(1991).
[27] M. S. Srivastava and Y. Wu, “Evaluation of
optimum weights and average run lengths in
EWMA control schemes,” Communications in
Statistics: Theory and Methods 26, 1253 -1267
(1997).
[28] Y. Areepong and S. Sukparungsee, “Explicit
expression for the average run length of double
moving average scheme for zero-inflated
binomial process,” International Journal of
Applied Mathematics and Statistics 53, 34-43
(2015).
[29] S. Sukparungsee and Y. Areepong, “A study of
the performance of EWMA chart with
transformed Weibull observation,” Thailand
Statistician 7, 179-191 (2009).
[30] C. Petru, “Statistical Tool to Estimate and
Optimize the Intensity of the Dependence
Between the Parameters of a Dynamic
System,” WSEAS Transactions on Computers
21, 165-170 (2022).
[31] F. W. Steuel and K. Van Harn, Discrete
analogues of self-decomposability and
stability. Annals of Probability, 7, 893-899
(1979).
[32] M. B. C. Khoo, “A moving average control
chart for monitoring the fraction non-
conforming,” Journal of Quality and Reliability
Engineering International, 20, 617-635 (2004).
[33] M. B. C. Khoo and V. H. Wong, “A double
moving average control chart,”
Communications in Statistics-Simulation and
Computation, 37, 1696-1708 (2008).
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[34] D. C. Montgomery, “Introduction to Statistical
Quality Control,” John Wiley & Sons (2009).
[35] U.S. Bureau of Labor Statistics, (data of in
Jan/1996 to Dec/2004), [Online].
http://www.bls.gov/wsp/ (Accessed Date:
January 18, 2024).
Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
- Suganya Phantu: writing an original draft,
software, data analysis, data curation, prove and
validate.
- Yupaporn Areepong: investigation, methodology,
validate.
- Saowanit Sukparungsee: conceptualization,
investigation, writing-review and editing, funding
acquisition, project administration, reviewing and
editing.
All authors have read and agreed to the published
version of the manuscript.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
The authors would like to express their sincere
gratitude to the Department of Applied Statistics for
allowance and financial support to the Faculty of
Applied Science, King Mongkut's University of
Technology North Bangkok, Thailand for supporting
research grants with contract no. 672158.
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
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