Average Run Length Computations of Autoregressive and Moving

Average Process using the Extended EWMA Procedure

PHUNSA MONGKOLTAWAT1, YUPAPORN AREEPONG*, SAOWANIT SUKPARUNGSEE

1Department of Applied Statistics, Faculty of Applied Statistics,

King Mongkut’s University of Technology North Bangkok,

Bangkok 10800,

THAILAND

*Corresponding Author

Abstract: - In the past, the control chart served as a statistical tool for detecting process changes. The

Exponentially Weighted Moving Average (EWMA) control chart is highly effective for detecting small

changes. This research introduces the Extended Exponentially Weighted Moving Average (Extended EWMA)

control chart for the Autoregressive and Moving average process with order p = 1 and q = 1 (ARMA(1,1)) The

Extended EWMA control chart incorporates two smoothing parameters (

1



and

2



) derived from the EWMA

control chart. A comparative analysis of the performance of the EWMA control chart. The Average Run

Length (ARL) value as determined by the explicit formulas in this research, serves as a metric for evaluating

the performance of the Extended EWMA control chart and the EWMA control chart. The Numerical Integral

Equation (NIE) method is used to verify the accuracy of the ARL for the explicit formulas of the two control

charts which has not been before discovered. The effectiveness of control charts can also be evaluated by

analyzing SDRL, ARL, MRL, RMI, AEQL, and PCI values as metrics for various design parameter values.

After analyzing the results of the ARL and all five performance meters, it was determined that the Extended

EWMA control chart is better than the EWMA control chart at all shift sizes of process changes. Finally, the

assessment of the ARMA process is being conducted to evaluate the ARL using a dataset on PM2.5 dust levels

in Bangkok, Thailand during January and February of 2024.

Key-Words: - Average Run Length, Autoregressive and Moving Average Process, Extended Exponentially

Weighted Moving Average control chart, Explicit Formula, Numerical Integral Equation, ARL,

NIE method.

Received: March 14, 2024. Revised: April 19, 2024. Accepted: April 24, 2024. Published: May 20, 2024.

1 Introduction

The statistically important tool is the control chart.

[1], invented the control chart, which detects the

alteration of the control diagram, which is

commonly used in the manufacturing industry. The

Shewhart control chart is very efficient with small

change detection as well. [2], introduced an

EWMA control chart for better detection of small

changes. The Cumulative Sum (CUSUM) control

chart, developed by [3], is extensively utilized in

statistical control charting. In 2017, [4], developed

and presented a more efficient, Modified EWMA

control chart than the EWMA control chart in the

detection of minor changes. [5], presented The

Extended EWMA control chart as a powerful chart

designed to detect minor changes in the process

being examined. The effectiveness of control charts

can be assessed by utilizing the ARL, [6]. The ARL

is divided into two values, [7], The ARL0 is the

number of expected observations required before a

process is under control, and The ARL1 refers to

the amount of observations expected from an

uncontrollable process and should be reduced. In

1990, [8], are presented a quantitative analysis

comparing the EWMA control chart and the

CUSUM control chart. [9], examined the design of

the optimal EWMA control chart and compared it

to the CUSUM control chart. [10], are presented,

and the autocorrelation data will be linked to a

statistical model. The ARMA process is a

frequently employed model in real-world data

analysis. [11], have introduced datasets with

exponential white noise to the control chart. [12],

shows the performance of the CUSUM control

chart for autocorrelated seasonal consistency of

trends using the Midpoint Rule method with

exponential white noise. [13], presented an

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2024.23.40

Phunsa Mongkoltawat,

Yupaporn Areepong, Saowanit Sukparungsee

E-ISSN: 2224-2880

371

Volume 23, 2024

approximating of the ARL of changes in the mean

of a Seasonal Time Series Model with exponential

white noise running on a CUSUM control chart.

[14], presented the explicit formulas and NIE of

ARL when observations were seasonal

autoregressive models with an exogenous variable

SARX(P,r)L with exponential white noise based on

the CUSUM control chart. [15], proposed an

explicit formula for the ARL using the Fredholm

integral equation method in the EWMA control

chart on the MAX(q,r) process. [16], are improving

the CUSUM control chart for monitoring a change

in processes based on a seasonal autoregressive

model with one exogenous variable. [17],

introduced a Modified EWMA control chart that is

derived from its particular examples. [18],

developed the explicit formula for the ARL on a

Modified EWMA control chart for the AR(1)

process. Subsequently, [19], presents analytical

explicit formulas of the ARL of Homogenously

Weighted Moving Average control chart

(HEWMA) based on a MAX process.

However, the derivation of the explicit formula

for the ARL on the Extended EWMA control chart

for the Autoregressive and Moving Average

process with parameters p = 1 and q = 1

(ARMA(1,1)), when there are two smoothing

parameters (

1



and

2



) for the ARMA(1,1)

process has not been reported previously. The

objective of this research is to determine the

explicit formula for the ARL on the Extended

EWMA control chart for the ARMA(1,1). The

explicit formula for the ARL was compared with

the NIE method. This research differs from that of

other researchers, using five additional statistical

performance measurements, Standard of Deviation

Run Length (SDRL), Median Run Length (MRL),

[20], Related Mean Index (RMI), [21], Average

Extra Quadratic Loss (AEQL), [22] and

Performance Comparison Index (PCI). The

comparison of performance between the Extended

EWMA control chart and the EWMA control chart

using a dataset on PM2.5 dust levels in Bangkok,

Thailand during January and February of 2024.

2 Materials and Methods

2.1 The Exponential Weighted Moving

Average Control Chart

This research describes the properties of the EWMA

control chart for ARMA(p,q) processes. The

EWMA control chart is defined by a recursive

equation, [2].

 

1

1 , 1,2,3,...

t t t

Z W tZ





   

(1)

where

t

W

is a sequence of an ARMA(p,q)

processes with exponential white noise and

01





,

0

W

is the initial value of the EWMA

statistics,

0

Zu

. The control limits of the EWMA

control chart consist of the upper control limit

(UCL) and the lower control limit (LCL) are:

0

2

UCL

LCL





 



 







(2)

where

0



is process mean of ARMA(p,q) process,



is the process standard deviation parameter,



is

a suitable control limit width. Let b be an UCL on

t

Z

. The stopping time

()

b



of the Extended

EWMA control chart is defined as

inf{ 0; }

bt

t Z b



  

3 The Exact Solutions of the ARL on

the Extended EWMA Control

Chart

3.1 The Explicit Formula of the ARL on the

Extended EWMA Control Chart for

ARMA(p,q) process

The ARMA(p,q) process are defined by the

following recursion:

0 1 1 2 2 ...

t t t p t p

W W W W

   

  

    

(3)

1 2 2 ...

t t t t q t q

      

  

    

where

t

W

is a sequence of the ARMA(p,q)

processes with exponential white noise,

t



is

autoregressive parameter,

0

Wu

is the initial

value, where

[0, ]ub

and b is UCL of the

Extended EWMA control chart

[6], presented The Extended EWMA control

chart. The performance control chart is highly

efficient in detecting small changes in the

monitored process. The Extended EWMA statistic

can be derived as:

 

11 2 1 2 1

1 , 1,2,3,...

t t t t

WtW

    



   

(4)

where

t

W

is a process with mean,

1



and

2



are

exponential smoothing parameters with

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2024.23.40

Phunsa Mongkoltawat,

Yupaporn Areepong, Saowanit Sukparungsee

E-ISSN: 2224-2880

372

Volume 23, 2024

21

01



  

and

0



is the initial value of the

Extended EWMA statistics,

0u





and

0v





.

The UCL and the LCL are:

22

1 2 1 2 1 2

02

1 2 1 2

22

1 2 1 2 1 2

02

1 2 1 2

2 (1 )

2( ) ( )

2 (1 ) ,

2( ) ( )

UCL

LCL

     

     

     

     

   

  

   

  





(5)

where

0



is the process mean of moving average

process,



is the process standard deviation

parameter and



is a suitable control limit width,

and

0 1 1 2 2 ...

t t t p t p

W W W W

   

  

    

(6)

1 2 2 ...

t t t t q t q

      

  

    

Hence, the formulation of the Extended EWMA

control chart for the ARMA(p,q) process is as.

 

11 2 1 0 1 1 2 2

1( ...

ttt t p t p

W W W

       

  

      

1 2 2 2 1, 1,2,3,...... )

t t t t q t q t tW

       

         

When t = 1

 

101 2 1 0 1 0 2 1 1

1( ... pp

W W W

       



      

1 0 2 1 1 2 0 1 1, 1,2,3,...... )

qq tW

        

      

Let

0 1 0 2 1 1

... pp

W W W

   



     

1 0 2 1 1

... qq

     



   

So

 

101 2 1 2 1 1

1 , 1, 2,3,...tu

      

      

(7)

Let's examine the in-control process, where the

UCL = b and the LCL = 0.

1

0b





 

01 2 1 2 1 1

10ub

     

     

 

01 1 1 2 1 2

10bu

     

      

 

01 2 1 2

1

0bu

  



    



 

1 2 1 2

1

0b u u





    



The function

()u



can be obtained by

Fredholm integral equation of the second kind as

follows;

1

0

( ) 1 ( ) ( )

b

u f d

    





(8)

 

01 2 1 2 1

0

1( ) 1 ( ) ( )

b

u u f d

        

     



Therefore, the function

()u



is obtained as

follows:

 

1 2 1 2 1

11

0

1

( ) 1 ( ) ( )

bb u u

u k f dk

  





     

 

Given that

()

tExp



is determined,

1

()

k

f k e







 

1 2 1 2 1

1

( ) 1 ( )

b u u k

b

a

e

u k e dk



    

 





     

 

Setting

 

1 2 1 2 1

1

()

b u u

ue

    



     



and

1

0

()

k

bk e dk









Thus

1

()

( ) 1 u

u





  

(9)

Consider



and take run

()k



 

1

1 2 1 2 12

1

1()

12

( 1)

1

1 ( 1)

()

b

b u u b

e

ee





    





  







     











(10)

By substituting the value of an equal to

1



into

()u



, we can get the explicit formula of the ARL1

on the Extended EWMA control chart as follows:

 

12

11

12

1

11

1 2 1 2 1

11

(1 )

12

()

12

11

( ) ( 1)

1

( ) ( 1)

ub

b

uu

ee

ARL



   





    

  



 



    





  

 

12

11

12

1

11

1 2 1 2 1

1 1 1 1

(1 )

12

()

12

11

( ) ( 1)

1

( ) ( 1)

ub

b

uu

ee

ARL



   





    

  



 



    





  

11ARL 

 

12

11

12

1

11

1 2 1 0 1 0 2 1 1 1 0 2 1 1 2 1

11

(1 )

12

()

12

1 ( ... ... )

( ) ( 1)

p p q q

ub

b

u W W W u

ee



   





              

  



   

 



            



  



(11)

From Equation (11), the ARL finished explicit

formula of the Extended EWMA control chart for

the ARMA(p,q) process is to be compared to the

EWMA control chart.

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2024.23.40

Phunsa Mongkoltawat,

Yupaporn Areepong, Saowanit Sukparungsee

E-ISSN: 2224-2880

373

Volume 23, 2024

3.2 The Numerical Integral Equation of the

ARL on the Extended EWMA Control

Chart of ARMA(p,q) process.

Let

()u



denote the estimated value of the ARL

determined from the m linear equation systems

using the composite midpoint quadrature rule.

The evaluation of the ARL approaching NIE on

the Extended EWMA control chart is carried out in

the following manner:

 

1

0

( ) ( ) ( )

bs

t j j

j

u f k dk c f x

  









(12)

The system of s linear equations is represented

as

1 1 1

1

s s s s s

L R L

   



or

 

1

11

1

s s s s s

L I R 

  



     

1 1 1

, ,..., ,

T

s NIE NIE NIE s

L L x L x L x





 

1,1,...,1

s

I diag

and

 

1

1 1,1,...,1 .

T

s

Let

ss

R

be a matrix. The s to sth element matrix

R is defined as follows:

1

ij j

Rc









1 2 1 0 1 0 2 1 1 1 0 2 1 1 2 1

1

(1 ) ( ... ... )

j p p q q

k u W W W u

f

              



   

             







The answer to the NIE can be succinctly

expressed as.

1

( ) 1

s

j

uc





 

1 2 1 0 1 0 2 1 1 1 0 2 1 1 2 1

1

(1 ) ( ... ... )

j p p q q

k u W W W u

f

              



   

             







(13)

where kj is a set of the division point on the interval

[0,b] as

1, 1,2,..., .

2

jj

k j c j s



  





j

c

is a weight of

composite midpoint formula

.

jb

cs



From Equation (13), the NIE method is a

comparative criterion to the explicit formulas that

the explicit formula is accurate. As a result, both

ARL values are similar.

4 Existence and Uniqueness of ARL

The answer is obtained from the explicit formula

using the ARL of the existence of the NIE, as

proven by Banach's fixed-point theorem, [23]. In

this study, let T denote an operation on the set of all

continuous functions that are defined.

1

( ( )) 1

s

j

T u c



 

(14)

1 2 1 0 1 0 2 1 1 1 0 2 1 1 2 1

1

(1 ) ( ... ... )

j p p q q

k u W W W u

f dk

              



   

             







According to Banach’s fixed-point theorem, if

an operator T is guaranteed to satisfy the criterion

of being a contraction,

( ( )) ( )T u u





has a one

solution, as previously stated. For Equation (14) to

have a solution that is both present and unique, the

Banach fixed-point theorem can be utilized. The

Banach fixed-point theorem, also referred to as the

contraction mapping theorem, was initially

introduced concretely in Banach's.

Typically, it is employed to determine the

existence of a solution to an integral problem.

Following this, [24], have extensively utilized this

tool to address numerous problems related to the

presence of solutions in diverse mathematical

domains, due to its straightforwardness and

practicality. The following information provides

the specific details.

Theorem 1 Banach’s Fixed-point Theorem :

Assume that

:D X X

is a contraction mapping

with contraction constant

0 1,S

such that

 

12

()DD





12

S





12

,X





, meets this

criterion. [25], have established the existence of a

single unique a.

(.) X





such that

( ( )) ( )D u u





has a unique fixed point in

.X

Proof: To demonstrate the value of T, as

determined by the equation

( ( ))Tu



is a

contraction mapping for

12

, [0, ].b





that

 

1 2 1 2

()D D S

   

  

,

12

, [0, ].b



 

with

01S

under the norm

[0, ]

sup ( )

ub u







From

()u



and

( ( ))Du



.

 

12

()DD







1 2 1 0 1 0 2 1 1 1 0 2 1 1 2 1

1

(1 ) ( ... ... )

()

[0, ]

12

0

1

sup

( ( ) ( ))

j p p q q

k u W W W u

b

ub

e

k k e dk

              







   

             









(1 ) ( ... ... )

1 2 1 0 1 0 2 1 1 1 0 2 1 1 2 1

()

1

[0, ]

12

0

1

sup

( ( ) ( ))

k u W W W u

j p p q q

b

ub

e

k k e dk

              







             

   











WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2024.23.40

Phunsa Mongkoltawat,

Yupaporn Areepong, Saowanit Sukparungsee

E-ISSN: 2224-2880

374

Volume 23, 2024

1 2 1 0 1 0 2 1 1 1 0 2 1 1 2 1

1

(1 ) ( ... ... )

()

12

[0, ]

12

sup

11

j p p q q

k u W W W u

ub

b

e

eS

              







   

             











   





where

1 2 1 0 1 0 2 1 1 1 0 2 1 1 2 1

1

(1 ) ( ... ... )

()

[0, ]

sup

11

j p p q q

k u W W W u

ub

b

Se

e

              





   

             















,

0 1,S

The uniqueness of the solution is

ensured by Banach's fixed-point theorem.

5 Numerical Results

In this research, we evaluate the ARL0 and ARL1

by employing explicit formulas and the NIE for an

ARMA(p,q) process on the Extended EWMA

control chart. In addition, performance indicators

such as the SDRL and MRL are used to assess the

effectiveness of control charts. The computation for

SDRL and MRL for the in-control process is as

follows.

0

0 0 0

2

0 0 0

1

log( )

11 2

,,

( ) log(1 )

ARL SDRL MRL



  



   

(15)

where

0



represents an error of type I. This

research analysis determined that ARL0 = 370. The

value of the ARL0 can be computed using Equation

(15) as SDRL0 and MRL0 with an approximate

value. Conversely, ARL1, SDRL1 and MRL1 are

calculated using Equation (16).

1

1 1 1

2

1 1 1

1

log( )

11 2

,,

( ) log(1 )

ARL SDRL MRL



  



   

(16)

where

1



represents an error of type II

The minimum values of the ARL1, SDRL1 and

MRL1 indicate a higher ability to promptly detect

variations in the process mean. To conduct a

comparison analysis, we will examine the Extended

EWMA control charts and the EWMA control

charts for the ARMA(p,q) process.

RMI is employed to assess the efficacy of the

Extended EWMA control chart. RMI can be

computed.

1

( ) ( )

1,

()

nii

ii

RMI ARL MAX ARL MIN

n ARL MIN













(17)

where

()

i

ARL MAX

is the ARLi of row i on the

control chart under examination.

()

i

ARL MIN

is

the minimum of the ARL1 for row i. A control chart

is deemed more effective when it has a lower RMI.

Furthermore, performance measurements can be

utilized to evaluate the effectiveness of control

charts across a range of modifications. In addition,

the AEQL may pertain to costs that have been

accrued as a result of an unmanageable situation.

This comparison may entail the utilization of

various control chart kinds to determine the most

efficient strategy for a specific procedure. These

include the study model, the research data set, the

appropriate parameter value, the control chart that

the research is introduced, as well as the application

of the actual data to make the chart of this research

result as desired.

The AEQL can be determined by using the

following formula.

max

min

2

1( ( ))

i

shift

ii

shift shift

AEQL shift ARL shift







(18)

where shift refers to a distinct change in the

process.



denotes the aggregate of number of

divisons from shiftmin(

min



) to shiftmax(

max



). In

this research,

min

10, 0.01



  

and

max 3.00





. The most effective control chart is the one with

the minimum AEQL value.

Additionally, the examination of control chart

performance can be carried out by utilizing the

performance evaluation criteria of the PCI. The

determination of the PCI value entails comparing

the AEQL of a certain control chart to the AEQL of

the control chart with the minimum value. This

helps identify the control chart that has the highest

level of efficiency. The PCI can be computed:

min

AEQL

PCI AEQL



(19)

The ARL was approximated by NIE using the

composite midpoint rule on the Extended EWMA

control chart for the ARMA(p,q) process with a

sample size of 1,000 nodes. When ARL0 = 370,

0



= 0.5,

1



= 0.1,

1



= -0.1 and 0.1,

2



= -0.2

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DOI: 10.37394/23206.2024.23.40

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E-ISSN: 2224-2880

375

Volume 23, 2024

and 0.2,

3



= 0.3,

1



= 0.05, 0.10,

2



=

0.01, and



= 0.01, 0.03, 0.05, 0.10, 0.30, 0.50,

1.00, 2.00 and 3.00, The in-control process is

01





. The results indicate that the ARL of the

explicit formulas and the NIE are very similar for

the ARMA(1,1), ARMA(1,2), and ARMA(1,3)

processes in Table 1, Table 2 and Table 3 in

Appendix. The results of the ARL of the Extended

EWMA control chart using an explicit formula are

shown in Table 4, Table 5 and Table 6 in

Appendix. These Tables compare the performance

of the Extended EWMA control chart against the

EWMA control chart for ARMA(1,1), ARMA(1,2),

and ARMA(1,3) processes. Based on the results,

the Extended EWMA control chart outperformed

the EWMA control chart in terms of the ARL,

SDRL, and MRL for

1



values of 0.05 and 0.10. In

addition, the results suggest that the Extended

EWMA control chart with

1



= 0.10 exhibits the

minimum values for RMI, AEQL, and PCI.

Therefore, it can be inferred that the Extended

EWMA control chart exhibits greater performance

when compared to the EWMA control chart.

Moreover, the RMI, AEQL, and PCI derived from

each control chart are utilized to assess the

effectiveness of the aforementioned control charts.

The Extended EWM control chart exhibited

superior performance. Based on the minimal values

for RMI, AEQL, and PCI, all of them were equal to

one.

6 Application to Real-world Data

In this study, the explicit formulas of the ARL on

the Extended EWMA control chart for the

ARMA(1,1) prcess were applied to the dataset on

PM2.5 dust levels in Bangkok, Thailand during

January and February of 2024 and generated a

forecasting process. The ARL was calculated using

explicit formulas on the Extended EWMA control

chart with ARL0 = 370 for

1



= 0.05, 0.10 and

2



= 0.01, shift

()



equal to 0.01, 0.03, 0.05,

0.10, 0.30, 0.50, 1.00, 2.00, 3.00 and sample size =

1,000 nodes. The performance of the control chart

was evaluated by comparing it to the EWMA

control chart using a dataset on PM2.5 dust levels

in Bangkok, Thailand during January and February

of 2024. The coefficient parameters estimated for

the ARMA(1,1) process are determined using

maximum likelihood estimation:

0



= 27.3401,

1



= 0.947,

1



= 0.616. The ARMA(1,1) process can

be defined by utilizing the parameter of this

forecasting process.

11

ˆ0.947 0.616

t t t

WW







Using the explicit formula, we compare the

ARL values of the ARMA(1,1) process on the

Extended EWMA control chart with the ARL,

SDRL, and MRL of the EWMA control charts.

This comparison evaluates their efficiency. The

results are presented in Table 7 (Appendix) and

Figure 1 (Appendix), demonstrating a clear

agreement with the findings seen in Table 4, Table

5 and Table 6 in Appendix. Figure 2 (Appendix)

displays a comparison of the RMI, AEQL, and PCI

derived from each control chart. The purpose is to

assess the effectiveness of the control charts.

In this research, the performance of the ARL of

the Extended EWMA control chart is assessed and

contrasted with that of the EWMA control chart.

The findings suggest that the Extended EWMA

control charts are superior to the EWMA control

chart for the ARMA(1,1) process. Additionally, the

Extended EWMA control chart, with

1



= 0.10,

better than all three control charts.

7 Conclusions

In this study, the formula was successful in finding

the ARL value and the accuracy of the Extended

EWMA control chart for the ARMA(1,1) compared

to the EWMA control chart. the efficacy of control

charts was evaluated for the ARL by utilizing the

NIE, the explicit formula is subjected to

comparison, And use all five measurements as an

additional criterion to compare the performance of

the two control charts. Both methods demonstrate

that the ARL values are similar. The Extended

EWMA control chart for the ARMA(1,1) process

has superior performance compared to the EWMA

control chart. When assessing the comparative

efficacy of the ARL under different smoothing

factors, it is recommended to utilize a smoothing

parameter of

1



= 0.10. The simulation research

and the real-world dataset on PM2.5 dust levels in

Bangkok, Thailand during January and February of

2024, ultimately, the outcomes were the same.

Further research, the extended EWMA control

chart can be applied to other aspects, such as health

or economics, as well as using an NIE comparison

method with the explicit formulas. Several other

methods of comparison will generate new control

charts.

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2024.23.40

Phunsa Mongkoltawat,

Yupaporn Areepong, Saowanit Sukparungsee

E-ISSN: 2224-2880

376

Volume 23, 2024

Acknowledgement:

The authors gratefully acknowledge the editor and

referees for their valuable comments and

suggestions which greatly improve this paper. The

research was funding by King Mongkut’s

University of Technology North Bangkok Contract

no. KMUTNB-67-BASIC-02

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WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2024.23.40

Phunsa Mongkoltawat,

Yupaporn Areepong, Saowanit Sukparungsee

E-ISSN: 2224-2880

377

Volume 23, 2024

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Contribution of Individual Authors to the

Creation of a Scientific Article (Ghostwriting

Policy)

- Phunsa Mongkoltawat carried out the writing-

original draft preparation and simulation.

- Yupaporn Areepong has organized the

conceptualization, writing-review and editing,

and validation

- Saowanit Sukparungsee has implemented the

methodology and solfware.

Sources of Funding for Research Presented in a

Scientific Article or Scientific Article Itself

The research was funding by King Mongkut’s

University of Technology North Bangkok Contract

no. KMUTNB-67-BASIC-02.

Conflicts of Interest

The authors declare no conflict of interest.

Creative Commons Attribution License 4.0

(Attribution 4.0 International, CC BY 4.0)

This article is published under the terms of the

Creative Commons Attribution License 4.0

https://creativecommons.org/licenses/by/4.0/deed.e

n_US

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Volume 23, 2024

APPENDIX

Table 1. ARL comparison of the Extended EWMA control chart for ARMA(p,q) using explicit formulas

against NIE method when

0



= 0.5.

1



= 0.1,

1



= 0.1,

0



= 1 for ARL0 = 370

Table 2. ARL comparison of the Extended EWMA control chart for ARMA(p,q) using explicit formulas

against NIE method when

0



= 0.5.

1



= 0.1,

1



= -0.1,

0



= 1 for ARL0 = 370

Table 3. ARL comparison of the Extended EWMA control chart for ARMA(p,q) using explicit formulas

against NIE method when

0



= 0.5.

1



= 0.1,

1



= -0.1,

2



= -0.2,

0



= 1 for ARL0 = 370



2



=0.01



1



= 0.1

1



= 0.1,

2



= 0.2

1



= 0.1,

2



= 0.2,

3



= 0.3

Process

ARMA(1,1)

ARMA(1,2)

ARMA(1,3)

1



0.05

0.10

0.05

0.10

0.05

0.10

b

2.96646

3.63105

2.96808

3.63040

2.96850

3.63300

0.00

Explicit

NIE

370.00200

370.55058

370.50773

370.04983

370.54021

370.59006

0.01

Explicit

NIE

358.74338

357.14330

359.22690

356.66069

359.25546

357.17068

0.03

Explicit

NIE

337.80512

332.44859

338.24730

331.99912

338.26862

332.45412

0.05

Explicit

NIE

318.76091

310.26908

319.16611

309.84910

319.18091

310.25553

0.10

Explicit

NIE

278.06230

263.82544

278.39071

263.46626

278.39183

263.77384

0.30

Explicit

NIE

177.21156

155.21847

177.36603

154.99343

177.33527

155.09284

0.50

Explicit

NIE

125.76868

104.21405

125.84709

104.05419

125.80204

104.06690

1.00

Explicit

NIE

69.34975

52.92752

69.36142

52.83141

69.30529

52.78121

2.00

Explicit

NIE

34.94470

24.95056

34.93253

24.89444

34.87852

24.83357

3.00

Explicit

NIE

3.16420

1.17294

3.14954

1.13249

3.10190

1.07843



2



=0.01



1



= -0.1

1



= -0.1,

2



= 0.2

1



= -0.1,

2



= 0.2,

3



= 0.3

Process

ARMA(1,1)

ARMA(1,2)

ARMA(1,3)

1



0.05

0.10

0.05

0.10

0.05

0.10

b

2.96750

2.96808

2.96900

3.63200

2.96850

3.63300

0.00

Explicit

NIE

370.50578

370.50773

370.91904

370.77768

370.56128

370.67882

0.01

Explicit

NIE

359.23026

359.22690

359.62363

357.35786

359.27603

357.25643

0.03

Explicit

NIE

338.26024

338.24730

338.61718

332.64039

338.28826

332.53439

0.05

Explicit

NIE

319.18744

319.16611

319.51189

310.44085

319.19975

310.33096

0.10

Explicit

NIE

278.42876

263.80566

278.68608

263.95661

278.40903

263.83937

0.30

Explicit

NIE

177.43595

177.21156

155.23175

177.54395

155.26439

154.26439

177.34909

155.13671

0.50

Explicit

NIE

125.92494

104.23758

125.97011

104.22719

125.81467

104.10135

1.00

Explicit

NIE

69.43484

52.95400

69.42895

52.91705

69.31705

52.80609

2.00

Explicit

NIE

34.98837

24.97123

34.96830

24.93601

34.88929

24.85146

3.00

Explicit

NIE

3.19345

1.18909

3.17426

1.16019

3.11154

1.09275

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2024.23.40

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Yupaporn Areepong, Saowanit Sukparungsee

E-ISSN: 2224-2880

379

Volume 23, 2024

Table 4. ARL comparison of the Extended EWMA control chart for ARMA(1,1) against EWMA control charts

when

0



= 0.5.

1



= 0.1,

1



= 0.1,

2



=0.01,

0



= 1 for ARL0 = 370



Control

Chart

Extended EWMA

EWMA

1



= 0.05

1



= 0.10

1



= 0.05

1



= 0.10

UCL

2.96646

3.63105

2.96920

3.63200

0.00

ARL0

SDRL0

MRL0

370.00200

370.55058

370.96100

370.68423

0.01

ARL1

SDRL1

MRL1

358.74338

358.64338

247.30530

357.14330

357.04330

246.20230

359.66153

359.56153

247.93820

359.26077

359.16770

247.66200

0.03

ARL1

SDRL1

MRL1

337.80512

337.66369

232.87120

332.44859

332.30716

229.17860

339.64755

339.50612

234.14130

334.53658

334.39515

230.61800

0.05

ARL1

SDRL1

MRL1

318.76091

318.53730

219.74280

310.26908

310.04547

213.88880

319.53546

319.31185

220.27670

312.33100

312.10739

215.31020

0.10

ARL1

SDRL1

MRL1

278.06230

277.74607

191.68660

263.82544

263.50921

181.87220

279.69520

279.37897

192.81220

265.83410

265.51787

183.25690

0.30

ARL1

SDRL1

MRL1

177.21156

176.66383

122.16350

155.21847

154.67074

107.00230

179.51763

178.96990

123.75330

157.11150

156.56377

108.3072

0.50

ARL1

SDRL1

MRL1

125.76868

125.06157

86.70059

104.21405

103.50694

71.84157

127.92548

127.21837

88.18741

106.05878

105.35167

73.11326

1.00

ARL1

SDRL1

MRL1

69.34975

68.34975

47.80732

52.92752

51.92752

36.48641

69.36277

68.36277

47.81630

54.73000

53.73000

37.72897

2.00

ARL1

SDRL1

MRL1

34.94470

33.53048

24.08967

24.95056

23.53634

17.20005

36.88625

35.47203

25.42811

26.73442

25.32020

18.42979

3.00

ARL1

SDRL1

MRL1

3.16420

1.43214

1.18129

1.17294

0.55911

0.50858

23.08540

21.35334

15.42811

15.95210

14.22004

10.99683

RMI

AEQL

PCI

0.31766

28.89150

1.39533

0.0000

20.70582

1.00000

2.21926

47.67518

2.30250

1.41800

34.89650

1.68534

Table 5. ARL comparison of the Extended EWMA control chart for ARMA(1,2) against EWMA control charts

when

0



= 0.5.

1



= 0.1,

1



= 0.1,

2



= 0.2,

2



=0.01,

0



= 1 for ARL0 = 370

Control

Extended EWMA

EWMA



2



=0.01



1



= -0.1

1



= -0.1,

2



= -0.2

1



= -0.1,

2



= -0.2,

3



= 0.3

Process

ARMA(1,1)

ARMA(1,2)

ARMA(1,3)

1



0.05

0.10

0.05

0.10

0.05

0.10

b

2.96750

2.96808

2.96655

3.62855

2.96801

3.63012

0.00

Explicit

NIE

370.50578

370.50773

370.48056

370.10566

370.50499

370.00294

0.01

Explicit

NIE

359.23026

359.22690

359.21279

356.72779

359.22487

356.61687

357.61687

0.03

Explicit

NIE

338.26024

338.24730

338.25548

332.08623

338.24655

331.96088

0.05

Explicit

NIE

319.18744

319.16611

319.19400

309.95325

319.16650

309.81575

0.10

Explicit

NIE

278.42876

263.80566

278.45731

263.60281

278..39342

278.39342

263.44280

0.30

Explicit

NIE

177.43595

177.21156

155.23175

177.50268

155.18361

177.37365

154.99338

0.50

Explicit

NIE

125.92494

104.23758

125.99841

104.24435

125.85641

104.05877

1.00

Explicit

NIE

69.43484

52.95400

69.49913

52.99346

69.37115

52.84059

2.00

Explicit

NIE

34.98837

24.97123

35.03238

25.00843

34.94060

24.90295

3.00

Explicit

NIE

3.19345

1.18909

2.22606

1.21908

3.15615

1.13953

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DOI: 10.37394/23206.2024.23.40

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E-ISSN: 2224-2880

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Volume 23, 2024



Chart

1



= 0.05

1



= 0.10

1



= 0.05

1



= 0.10

UCL

2.96808

3.63040

2.96880

3.63190

0.00

ARL0

SDRL0

MRL0

370.50773

370.04983

370.73988

370.46651

0.01

ARL1

SDRL1

MRL1

359.22690

359.12690

247.63860

356.66069

356.56069

245.86960

359.44785

359.34785

247.79090

359.05123

358.95123

247.51750

0.03

ARL1

SDRL1

MRL1

338.24730

338.07409

233.17600

331.99912

331.82591

228.86870

339.44759

339.27438

234.00350

334.34198

334.16877

230.48380

0.05

ARL1

SDRL1

MRL1

319.16611

318.94250

220.02210

309.84910

309.62549

213.59930

319.34781

319.12420

220.14740

312.14968

311.92607

215.18520

0.10

ARL1

SDRL1

MRL1

278.39071

278.07448

191.91290

263.46626

263.15003

181.62450

279.53339

279.21716

192.70070

265.68011

265.36388

183.15070

0.30

ARL1

SDRL1

MRL1

177.36603

176.81830

122.27000

154.99343

154.44570

106.84710

179.41662

178.86889

123.68360

157.01868

156.47095

108.24330

0.50

ARL1

SDRL1

MRL1

125.84709

125.13998

86.75464

104.05419

103.34708

71.73137

127.85347

127.14636

88.13777

106.99335

106.28624

73.75752

1.00

ARL1

SDRL1

MRL1

69.36142

68.36142

47.81537

52.83141

51.83141

36.42015

69.42107

68.42107

47.85649

54.69192

53.69192

37.70272

2.00

ARL1

SDRL1

MRL1

34.93253

33.51831

24.08128

24.89444

23.48022

17.16137

36.86304

35.44882

25.41211

26.71276

25.29854

18.41485

3.00

ARL1

SDRL1

MRL1

3.14954

1.14174

1.14118

1.13249

0.59556

0.58700

23.06902

21.33696

15.90299

15.93662

14.20456

10.98616

RMI

AEQL

PCI

0.32930

28.87843

1.39977

0.00000

20.63083

1.00000

2.29702

28.88843

1.40025

1.47237

34.96265

1.69467

Table 6. ARL comparison of the Extended EWMA control chart for ARMA(1,3) against EWMA control charts

when

0



= 0.5.

1



= 0.1,

1



= 0.1,

2



= 0.2,

3



= 0.3,

2



=0.01,

0



= 1 for ARL0 = 370

Control

Extended EWMA

EWMA

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2024.23.40

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381

Volume 23, 2024



Chart

1



= 0.05

1



= 0.10

1



= 0.05

1



= 0.10

UCL

2.96850

3.63300

2.96950

3.63400

0.00

ARL0

SDRL0

MRL0

370.54021

370.59006

370.92196

370.90797

0.01

ARL1

SDRL1

MRL1

359.25546

359.15546

247.65830

357.17068

357.07068

246.22110

359.62196

359.52196

247.91100

359.47073

359.37073

247.80670

0.03

ARL1

SDRL1

MRL1

338.26862

338.09541

233.19070

332.45412

332.28091

229.18240

339.60703

339.43382

234.11340

334.72153

334.54832

230.74550

0.05

ARL1

SDRL1

MRL1

319.18091

318.95730

220.03230

310.25553

310.03192

213.87950

319.49410

319.27049

220.24820

312.49391

312.27070

215.42250

0.10

ARL1

SDRL1

MRL1

278.39183

278.07560

191.91370

263.77384

263.45761

181.83660

279.65219

279.33596

192.78260

265.95230

265.63607

183.33830

0.30

ARL1

SDRL1

MRL1

177.33527

176.80754

122.24880

155.09284

154.54511

106.91570

179.47178

178.92405

123.72170

157.13560

156.58787

108.32390

0.50

ARL1

SDRL1

MRL1

125.80204

125.09493

86.72358

104.06690

103.35979

71.74013

127.87959

127.17248

88.15578

106.04689

105.33978

73.10507

1.00

ARL1

SDRL1

MRL1

69.30529

68.30529

47.77667

52.78121

51.78121

36.38554

69.42012

68.42012

47.85583

54.69363

53.69363

37.70390

2.00

ARL1

SDRL1

MRL1

34.87852

33.46430

24.04405

24.83357

23.41935

17.11941

36.85123

35.43701

25.40397

26.69774

25.28352

18.40450

3.00

ARL1

SDRL1

MRL1

3.10190

1.36984

1.13834

1.07843

0.65362

0.64343

23.05620

21.32414

15.89415

15.92080

14.18874

10.97525

RMI

AEQL

PCI

0.33929

28.80694

1.40149

0.00000

20.55447

1.00000

2.40922

47.63901

2.31769

1.54835

34.92037

1.69891

Table 7. ARL comparison of the Extended EWMA control chart for ARMA(1,1) using NIE against EWMA

control chart when

0



= 27.3401,

1



= 0.947,

1



= 0.616 for ARL0 = 370



Control

Chart

Extended EWMA

EWMA

1



= 0.05

1



= 0.10

1



= 0.05

1



= 0.10

UCL

81.14050

99.36650

81.20000

99.42950

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Volume 23, 2024

0.00

ARL0

SDRL0

MRL0

370.09745

370.04424

370.87888

370.78295

0.01

ARL1

SDRL1

MRL1

358.83352

358.73352

247.36740

346.65490

346.55490

245.86560

359.58227

354.48227

247.88360

357.35438

357.25438

246.34780

0.03

ARL1

SDRL1

MRL1

337.88490

337.71169

232.92620

321.99279

321.81958

228.86440

338.57353

338.40032

233.40090

332.62084

332.44763

229.29730

0.05

ARL1

SDRL1

MRL1

318.83135

318.60774

219.79130

299.84205

299.61844

213.59440

319.46609

319.24248

220.22890

310.40691

310.18330

213.98380

0.10

ARL1

SDRL1

MRL1

278.11315

277.79692

191.72160

253.45687

253.14064

181.61810

288.63544

288.31921

198.97530

273.89268

273.57645

188.81220

0.30

ARL1

SDRL1

MRL1

177.21618

176.66845

122.16670

144.97452

144.42679

106.83410

179.47908

178.93135

123.72670

155.13019

154.58246

106.94140

0.50

ARL1

SDRL1

MRL1

125.75122

125.04411

86.68855

94.01994

93.31283

71.70776

127.89571

127.18860

88.16689

106.05868

105.35157

73.11319

1.00

ARL1

SDRL1

MRL1

69.30894

68.30894

47.77919

42.77308

41.77308

36.37994

79.33972

78.33972

54.69406

62.70964

61.70964

43.22986

2.00

ARL1

SDRL1

MRL1

34.89093

33.47671

24.05260

14.80931

13.39509

17.10268

36.86497

35.45075

25.41344

26.70099

25.28677

18.40674

3.00

ARL1

SDRL1

MRL1

3.10585

1.37379

1.14106

1.03310

0.69895

0.61218

23.06385

21.33179

15.89942

15.91344

14.18138

10.97018

RMI

AEQL

PCI

0.35411

28.81308

1.40548

0.00000

20.50045

1.0000

2.53962

48.65270

2.37325

1.63711

35.70610

1.74172

Fig. 1: The ARL1 values on the control chart using a real-world dataset

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2024.23.40

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Volume 23, 2024

Fig. 2: Comparison the RMI, AEQL, and PCI values with the Extended EWMA control chart and the EWMA

control chart for

1



= 0.05, 0.10

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2024.23.40

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E-ISSN: 2224-2880

384

Volume 23, 2024