Available online at www.HighTechJournal.org  

HighTech and Innovation  
Journal 

Vol. 5, No. 1, March, 2024 

 

 

16 

 

ISSN: 2723-9535 

 

The Development and Evaluation of Homogenously Weighted 

Moving Average Control Chart based on an Autoregressive Process 

Rapin Sunthornwat 1 , Saowanit Sukparungsee 2 , Yupaporn Areepong 2*  

1 Industrial Technology and Innovation Management Program, Faculty of Science and Technology, Pathumwan Institute of Technology 10330, Thailand. 

2 Department of Applied Statistics, Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok, Bangkok, 10800, Thailand. 

Received 02 November 2023; Revised 14 February 2024; Accepted 21 February 2024; Published 01 March 2024 

Abstract 

This research aims to investigate a Homogenously Weighted Moving Average (HWMA) control chart for detecting minor 

and moderate shifts in the process mean. A mathematical model for the explicit formulae of the average run length (ARL) 

of the HWMA control chart based on the autoregressive (AR) process is presented. The efficacy of the HWMA control 

chart is evaluated based on the average run length, the standard deviation of run length (SDRL), and the median run length 

(MRL). As illustrations of the design and implementation of the HWMA control chart, numerical examples are provided. 

In numerous instances, a comparative analysis of the HWMA control chart relative to the Extended Exponentially 

Weighted Moving Average (Extended EWMA) and cumulative sum (CUSUM) control charts with mean process shifts is 

performed in detail. Additionally, the relative mean index (RMI), the average extra quadratic loss (AEQL), and the 

performance comparison index (PCI) are utilized to evaluate the performance of control charts. For various shift sizes, the 

HWMA control chart is superior to the Extended EWMA and CUSUM control charts. This study applies empirical data 

from the area of economics to validate the explicit formula of ARL values for the HWMA control chart. 

Keywords: Integral Equation; Average Run Length; Autoregressive Process. 

1. Introduction 

Statistical process control (SPC) provides several benefits, including a reduction in defects, an increase in 

productivity, a reduction in waste, an increase in customer satisfaction, and an improvement in the overall performance 

of the process. It is extensively utilized across industries to maintain product quality and process consistency. A control 

chart is a statistical process control tool that continuously monitors and visually represents the performance of a process, 

product, or operation over time. Control charts are extensively utilized in manufacturing, healthcare, service industries, 

and virtually any setting where processes must be monitored and controlled. Various control charts are utilized to monitor 

and analyze different parts of a process. 

The Shewhart control chart, which Shewhart [1] first proposed, the cumulative sum (CUSUM) control chart, which 

Page [2] first proposed, and the exponentially weighted moving average (EWMA) control chart, which Roberts [3] first 

published, are the three process control charts that are most frequently used. The EWMA and CUSUM control charts 

are designed to gather data over time to identify subtle adjustments in process parameters. In contrast, Shewhart control 

charts are mainly utilized for promptly detecting significant process shifts. Previous studies have indicated that the 

EWMA and CUSUM control charts exhibit superior performance compared to the Shewhart control chart in detecting 

                                                           
* Corresponding author: yupaporn.a@sci.kmutnb.ac.th 

 
http://dx.doi.org/10.28991/HIJ-2024-05-01-02 

 This is an open access article under the CC-BY license (https://creativecommons.org/licenses/by/4.0/). 

© Authors retain all copyrights. 

https://creativecommons.org/licenses/by/4.0/
https://orcid.org/0000-0001-8981-5107
https://orcid.org/0000-0001-5248-8173
https://orcid.org/0000-0002-5103-9867


HighTech and Innovation Journal         Vol. 5, No. 1, March, 2024 

17 

 

minor variations in the process [4, 5]. The Extended Exponentially Weighted Moving Average (Extended EWMA) 

control chart was introduced by Neveed et al. [6] to expand the conventional EWMA control chart. The purpose of this 

design is to identify changes in both the mean and the standard deviation of a process. A homogeneously weighted 

moving average (HWMA) control chart was recently proposed by Abbas [7] as a control charting statistic that gives the 

present and past samples a specific weight. The impact of non-normality on the HWMA control chart's performance is 

examined, and adjustments to the control chart's parameters may improve its performance against non-normality. 

Furthermore, Abbas [7] showed that the HWMA control chart outperformed the CUSUM and EWMA control charts in 

terms of effectiveness. In order to compare how successfully the charts identified process changes, the authors therefore 

aimed to offer an exact formula for the average run time of HWMA control charts. Riaz et al. [8] utilized Monte Carlo 

simulation to examine the performance of the HWMA control chart in zero and steady states at various shifts. The 

HWMA control chart is compared to the EWMA control chart with time-varying limits to conduct the comparative 

analysis. It has been determined that, for several shift sizes under zero state, the HWMA control chart is superior to the 

EWMA chart. 

Control charts are often designed to be used in processes that have identically distributed (i.i.d.) data points for 

monitoring and analysis. Processes can, in fact, display autocorrelation, whereby prior observations impact the current 

data point. A type of autoregressive integrated moving average (ARIMA) model, AR models mix moving average and 

differencing components to address non-stationarity in the data. AR models are useful when there is a correlation 

between the values at various time points and the time series shows signs of autocorrelation. In this study, the criteria 

for choosing the ARIMA model with the lowest mean absolute percentage error (MAPE) and root mean square error 

(RMSE) were examined. Noise usually follows white noise; however, exponential white noise can also be followed by 

noise. As Fellag & Ibazizen [9] have done, a specific example of white noise with an exponential distribution will be 

considered. 

The Average Run Length (ARL) is a measure of the expected or average number of samples that need to be collected 

before a control chart signals an out-of-control condition. The ARL is used to evaluate the performance and efficiency 

of a control chart in maintaining the process in an in-control state. The ARL has two essential components: ARL for the 

in-control state (ARL0) refers to the average run length when the process is in the control condition. In an ideal scenario, 

the ARL for the in-control state should be relatively large, indicating that the chart does not frequently generate false 

alarms. ARL for out-of-control state (ARL1) refers to the average run length when the process is out of control. It 

measures the average time required for the control chart to identify and signal a real problem or deviation from the 

intended process conditions. A small ARL1 indicates that the control chart can rapidly detect a process change. Many 

approaches have been provided for evaluating the Average Run Length (ARL). For example, Champ & Rigdon [10] 

studied the Markov chain and integral equation approaches that are often used to evaluate the run length distribution of 

quality control charts to evaluate the cumulative sum (CUSUM) and the exponentially weighted moving average 

(EWMA) control charts. The product midpoint rule approximates the integral in the integral equation. Furthermore, 

Sukparungsee & Areepong [11] introduced an autoregressive model-based explicit analytical solution for the average 

run length of the EWMA control chart. They utilized the numerical integral equation method to compare the outcomes 

of the ARL. The use of explicit formulas for determining Average Run Length (ARL) values has yielded precise results 

and expedited computational processes. Consequently, many researchers have studied the derivation of Average Run 

Length (ARL) values using precise mathematical expressions. 

Sunthornwat et al. [12] estimate the fractional differencing parameter and the optimal smoothing value for the 

EWMA control chart to assess the Average Run Length (ARL) and compare the analytical EWMA and CUSUM control 

charts. Peerajit et al. [13] recently proposed explicit formulas for the average run length (ARL) of the CUSUM chart for 

non-seasonal and seasonal ARFIMA models. The accuracy of explicit ARL was compared with the numerical integral 

equation (NIE) method based on the Gauss-Legendre quadrature rule. On a modified EWMA control chart for a first-

order moving-average process with exponential white noise, Supharakonsakun [14] examined explicit formulations for 

both the one-sided and two-sided ARL. A comparison was made between the EWMA control chart's performance and 

the solution's accuracy as determined by the numerical integral equation method. Following this, Supharakonsakun [15] 

conducts an analysis on the efficacy of a modified EWMA chart in determining the precise average run length. The 

observations were obtained from a general-order moving average process accompanied by exponential white noise. 

Additionally, a comparison of the effectiveness of the EWMA and modified EWMA control charts is also presented. In 

the meanwhile, Phanyaem [16] introduced the explicit formula of the ARL for seasonal autoregressive with explanatory 

variables on the CUSUM chart. Subsequently, Petcharat [17] determined the average run length (ARL) for the 

cumulative sum (CUSUM) control chart by employing the Fredholm integral equation method and the SAR(P)L with 

trend process. The use of Banach's fixed point theorem guarantees the existence and uniqueness of the solution. 

Phanthuna et al. [18] have recently developed explicit analytical solutions for the ARL of a modified EWMA control 

chart with exponential white noise using a time series model with fractionality and integration. Peerajit and Areepong 

[19] presented the ARL of an autoregressive fractionally integrated process with exponential white noise using a 

modified EWMA control chart to detect variations in the mean process. 



HighTech and Innovation Journal         Vol. 5, No. 1, March, 2024 

18 

 

Furthermore, Silpakob et al. [20] created a modified exponentially weighted moving average (EWMA) control chart 

to determine a change in the mean process. They derived explicit formulas for both one-sided and two-sided ARL for 

autoregressive processes. The findings revealed that the performance of the newly modified EWMA control chart was 

superior to that of the conventional EWMA and the modified EWMA control charts. Peerajit [21] recently introduced 

an explicit formula for ARL for monitoring variations in the mean for the CUSUM control chart under the SFIMAX 

model. Karoon et al. [22] investigated the exact run length on a two-sided extended EWMA control chart that was 

autoregressive with a trend model to monitor the mean process. The effectiveness of the extended EWMA control chart 

for controlling process mean based on autocorrelated data was examined by Karoon et al. [23]. Peerajit [24] developed 

precise methods based on analytical integral equations to obtain the ARL. The proof of these formulas' existence and 

uniqueness relied on Banach's fixed-point theorem. This work examines the FIMAX model's CUSUM chart, which takes 

exogenous factors and a fractionally integrated moving average into account. The model assumes that there is an 

underlying exponential white noise. Using an analytical formula based on an integral equation, Peerajit [25] gave a 

precise estimation of the ARL for long-memory models in the same year. Examples of these models include fractionally 

integrated MAX processes (FIMAX) with exponential white noise operating on an EWMA control chart. Its efficacy 

was contrasted with the ARL determined by the widely recognized numerical integral equation (NIE) method. 

Sunthornwat et al.’s [26] recent study examined the HWMA control chart's explicit formula for the MAX model and 

contrasted its performance with that of the CUSUM control chart. When utilizing the EARL, ESDRL, and EMRL 

criteria, the HWMA control chart outperformed the CUSUM control chart. Therefore, the purpose of this study is to 

examine the HWMA control chart and evaluate its effectiveness in comparison to the CUSUM control chart and the 

Extended Exponentially Weighted Moving Average (Extended EWMA) control chart. The autoregressive model 

(AR(p)), a new model that will be used in numerous real-world applications, will be employed to apply this control chart. 

This study also identifies a new performance criterion that consists of PCI, AEQL, and RMI values. 

Thus, the primary objective of this research is to evaluate the ARL formulas derived from the HWMA control chart 

for an autoregressive model under zero state and compare them to the ones utilized by the NIE method. In addition, the 

HWMA control chart is enlarged to allow for a comparison of the control chart's efficiency to the Extended EWMA and 

CUSUM control charts that underlie both simulated and real-world data for various shift sizes in the process mean. Then, 

the efficiency of the HWMA chart was calculated using the SDRL and MRL values. The outcomes of the HWMA control 

chart were verified using the performance measures, which include the performance comparison by index (PCI), average 

extra quadratic loss (AEQL), and relative mean index (RMI). Furthermore, the application used to illustrate this research 

is related to natural gas prices. Changes in the price of oil play an equal role in predicting the fundamental future 

movement of the exchange rate (Brahmasrene et al. [27]). Moreover, the oil price is a significant factor directly related 

to the economy. In this situation, control charts aim to identify patterns or movements in price behavior that may signal 

a change in market conditions. The remaining article is organized as follows: Section 2 describes the structure of the 

HWMA, Extended EWMA, and CUSUM processes and control charts. The Average Run Length is evaluated in Section 

3 using explicit formulas and numerical integral equations. The fourth section of the report provides numerical results. 

Finally, concluding remarks are summarized in Section 5. 

2. Process and Control Charts 

This section includes the statistical scheme of the HWMA control chart, using data derived from the autoregressive 

model (AR(p)). Subsequently, the explicit formula derived from the analysis and the NIE method for calculating the 

Average Run Length (ARL) is shown. 

2.1. The Autoregressive Process 

There are stationary and non-stationary components in time-series data. The moving average (MA(q)), autoregressive 

(AR(p), and autoregressive and moving average (ARMA(p,q)) models are the methods available for creating stationary 

time series. In this work, the autoregressive model, or AR(p) model, was studied. Equation 1 expresses the AR(p) as 

follows: 

Definition 2.1 Let {𝑌𝑡 , 𝑡 = 1,2, . . . . , }be a sequence of AR(p) process given as in Equation 1; 

𝑌𝑡 = 𝜙0 + 𝜙1𝑌𝑡−1 + 𝜙2𝑌𝑡−2+. . . +𝜙𝑝𝑌𝑡−𝑝 + 휀𝑡                                                                                  (1) 

where 𝜙0 is a constant of model 

 𝜙𝑖 is coefficients of autoregressive 𝑖 = 1,2, . . . , 𝑝. 

휀𝑡 is a exponential white noise process (휀𝑡 ∼ 𝐸𝑥𝑝(𝛼)) 

The probability density function of 휀𝑡 is defined as𝑓(𝑦, 𝛼) =
1

𝛼
𝑒−

𝑦

𝛼; 𝛼 > 0, and then initial values of the AR(p) model 

are 𝑌0, 𝑌−1, . . . , 𝑌1−𝑝.  



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2.2. The HWMA Control Chart 

Under the assumption that {𝐻𝑡; 𝑡 = 1,2,3, . . . } is a sequence of i.i.d continuous random variables with a probability 

density function, the HWMA statistic is taken into consideration. Based on the AR(p) procedure, the HWMA statistic 

(𝐻𝑡) is known as an upper HWMA statistic. 𝐻𝑡  can be represented as in Equation 2 using the recursive formula. 

𝐻𝑡 = 𝜆𝑌𝑡 + (1 − 𝜆)�̄�𝑡−1, for 𝑡 = 1, 2, 3, . ..                                                                   (2) 

where 𝑌𝑡 is a sequence of the AR(p) process with exponential white noise, the constant value �̄�0 = 𝜐 is an 

initial value; 𝜐 ∈ [0,h] where h is a upper control limit of HWMA control chart. 

The control limits of HWMA control chart consist of; 

Upper control limit:  𝑈𝐶𝐿𝑡 =

{
 

 𝜇0 + 𝐿1√
𝜎2

𝑛
𝜆2, 𝑡 = 1                   

𝜇0 + 𝐿1√
𝜎2

𝑛
[𝜆2 +

(1−𝜆)2

(𝑡−1)
], 𝑡 > 1

   

Lower control limit:   𝐿𝐶𝐿𝑡 =

{
 

 𝜇0 − 𝐿1√
𝜎2

𝑛
𝜆2, 𝑡 = 1                   

𝜇0 − 𝐿1√
𝜎2

𝑛
[𝜆2 +

(1−𝜆)2

(𝑡−1)
], 𝑡 > 1

   

where 𝜇0 is the target mean, σ is the process standard deviation and 𝐿1is the width of the control limits. 

The HWMA stopping time (𝜏ℎ)is defined as; 

𝜏ℎ ={t>0;H_t≥h}, for h> 𝜐. 

where 𝜏ℎ is the stopping time and h is UCL. 

2.3. The Extended EWMA Control Chart 

The Extended EWMA control chart was presented by Neveed et al. [6]. By giving more weight to recent data points, 

it is allowed to rapidly monitor and detect small to moderate changes in the mean process. The Extended EWMA statistic 

is given by: 

𝐸𝑡 = 𝜆1𝑌𝑡 − 𝜆2𝑌𝑡−2 + (1 − 𝜆1 − 𝜆2)𝐸𝑡−1, 𝑡 = 1, 2, . ..                                                          (3) 

where 𝜆1 and 𝜆2 are exponential smoothing coefficient with (0 < 𝜆1 ≤ 1) and (0 ≤ 𝜆2 ≤ 𝜆1) and the initial value is a 

constant, 𝐸0 = 𝑢. The upper control limit (UCL) and lower control limit (LCL) of the Extended EWMA control chart 

are given by: 

𝑈𝐶𝐿 = 𝜇0 + 𝐿2𝜎√
𝜆1
2 + 𝜆2

2 − 2𝜆1𝜆2(1 − 𝜆1 + 𝜆2)

2(𝜆1 − 𝜆2) − (𝜆1 − 𝜆2)
2

, 

𝐿𝐶𝐿 = 𝜇0 − 𝐿2𝜎√
𝜆1
2 + 𝜆2

2 − 2𝜆1𝜆2(1 − 𝜆1 + 𝜆2)

2(𝜆1 − 𝜆2) − (𝜆1 − 𝜆2)
2

, 

where 𝜇0 is the target mean, 𝜎 is the process standard deviation, and 𝐿2 is suitable control limit width. 

The stopping time of the Extended EWMA control chart (𝜏𝑏) is given by:  

𝜏𝑏 = {𝑡 > 0; 𝐸𝑡 ≥ 𝑏}, 

where 𝜏𝑏 is the stopping time and 𝑏is UCL. 

2.4. The CUSUM Chart 

For quality control, Page [2] produced the CUSUM control chart, which can be used to detect small changes in the 

process mean. Using the procedure described in Equation 4, the statistics of the CUSUM control chart can be expressed 

as follows:                                                 

𝐶𝑡 = 𝑚𝑎𝑥( 0, 𝐶𝑡−1 + 𝑌𝑡 −𝜛), 𝑡 = 1,2,3, . ..                                                                          (4) 

where 𝜛 is a reference value, 𝐶0 = 𝜍 is the initial value of CUSUM statistic; 𝜍 ∈ [0, 𝑠] and the CUSUM chart's stopping 

time is defined as 𝜏𝑠 = {𝑡 > 0; 𝐶𝑡 > 𝑠} and 𝑠 is UCL. 



HighTech and Innovation Journal         Vol. 5, No. 1, March, 2024 

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3. Evaluation of Average Run Length 

3.1. Analytical Explicit Formulas of the ARL for AR(p) Model 

From the recursion of HWMA statistics in Equation 2, 

𝐻𝑡 = 𝜆𝑌𝑡 + (1 − 𝜆)�̄�𝑡−1 

and 𝑌𝑡 = 𝜙0 + 𝜙1𝑌𝑡−1 + 𝜙2𝑌𝑡−2+. . . +𝜙𝑝𝑌𝑡−𝑝 + 휀𝑡 

Consequently, the HWMA statistics can be displayed as: 

 𝐻𝑡 = 𝜆(𝜙0 + 𝜙1𝑌𝑡−1 + 𝜙2𝑌𝑡−2+. . . +𝜙𝑝𝑌𝑡−𝑝 + 휀𝑡) + (1 − 𝜆)�̄�𝑡−1 

For t=1,  

𝐻1 = 𝜆(𝜙0 + 𝜙1𝑌0 + 𝜙2𝑌−1+. . . +𝜙𝑝𝑌1−𝑝 + 휀1) + (1 − 𝜆)�̄�0 

𝐻1 = 𝜆(𝜙0 + 𝜙1𝑌0 + 𝜙2𝑌−1+. . . +𝜙𝑝𝑌1−𝑝) + 𝜆휀1 + (1 − 𝜆)�̄�0 

Let 𝐵 = 𝜆(𝜙0 + 𝜙1𝑌0 + 𝜙2𝑌−1+. . . +𝜙𝑝𝑌1−𝑝) 

Set LCL=0, UCL=ℎfor in control process and given �̄�0 = 𝜐 then: 

0 < 𝐻𝑡 < ℎ 

0 < 𝜆𝐵 + (1 − 𝜆)�̄�𝑡−1 < ℎ 

The zero state at 𝑡 = 1 is considered, therefore 𝐾(𝜐) can be calculated as follows: 

𝐾(𝜐) = 1 + ∫ 𝐾(𝐵 + 𝜆𝑦 + (1 − 𝜆)𝜐)𝑓(𝑦)𝑑𝑦
ℎ−(1−𝜆)𝜐−𝐵

𝜆
0

                                (5)  

Let 𝑤 = 𝐵 + 𝜆𝑦 + (1 − 𝜆)𝜐, then 𝑑𝑦 =
1

𝜆
𝑑𝑤. 

After changing the variable in Equation 5, the expression can be reformulated as: 

𝐾(𝜐) = 1 +
1

𝜆
∫ 𝐾(𝜐)

1

𝛼
𝑒−

1
𝛼
[
𝑤−(1−𝜆)𝜐−𝐵

𝜆
]𝑑𝑤

ℎ

0

 

Since we determine 휀1 ∼ 𝐸𝑥𝑝(𝛼) then𝑓(𝑦) =
1

𝛼
𝑒−

𝑦

𝛼.  

Thus: 

𝐾(𝜐) = 1 +
𝑒
(1−𝜆)𝑢+𝐵

𝛼𝜆

𝛼𝜆
∫ 𝐾(𝑤)

1

𝛼
𝑒−

𝑤
𝛼𝜆𝑑𝑤

ℎ

0

 

We setting that 𝐶(𝜐) =
𝑒
(1−𝜆)𝜐+𝐵

𝛼𝜆

𝛼𝜆
 and 𝑃 = ∫ 𝐾(𝑤)

1

𝛼
𝑒−

𝑤

𝛼𝜆𝑑𝑤
ℎ

0
 

So that 𝐾(𝜐) = 1 + 𝐶(𝜐)𝑃.                                                                                   (6) 

Since 𝑃 = ∫ 𝐾(𝜐)
1

𝛼
𝑒−

𝑤

𝛼𝜆𝑑𝑤
ℎ

0
, we have; 

            = ∫ (1 + 𝐶(𝑤)𝑃)𝑒
−𝑤

𝛼𝜆𝑑𝑤
ℎ

0
 = ∫ 𝑒

−𝑤

𝛼𝜆𝑑𝑤 +
𝑃

𝛼𝜆
∫ 𝑒

𝑤−𝜆𝑤+𝐵−𝑤

𝛼𝜆 𝑑𝑤
ℎ

0

ℎ

0
 = ∫ 𝑒

−𝑤

𝛼𝜆𝑑𝑤 +
𝑃𝑒

𝐵
𝛼𝜆

𝛼𝜆
∫ 𝑒

−𝜆𝑤

𝛼𝜆 𝑑𝑤
ℎ

0

ℎ

0
 

           𝑃 = −𝛼𝜆(𝑒−
ℎ

𝛼𝜆 − 1) −
𝑃𝑒

𝐵
𝛼𝜆

𝜆
(𝑒−

ℎ

𝛼 − 1) 

          𝑃 =
−𝛼𝜆[𝑒

−ℎ
𝛼𝜆−1]

[1+
𝑒
𝐵
𝛼𝜆

𝜆
(𝑒
−ℎ
𝛼 −1)]

 

Substituting 𝑃 in (6), we obtain: 

𝐾(𝜐) = 1 −
[𝑒
−ℎ
𝛼𝜆−1]𝑒

(1−𝜆)𝜐+𝐵
𝛼𝜆

1+
𝑒
𝐵
𝛼𝜆

𝜆
(𝑒
−ℎ
𝛼 −1)

.   

𝐾(𝜐) = 1 −
[𝑒
−ℎ
𝛼𝜆−1]𝑒

(1−𝜆)𝜐+𝜆(𝜙0+𝜙1𝑌0+𝜙2𝑌−1+...+𝜙𝑝𝑌1−𝑝)

𝛼𝜆

1+
𝑒

𝜆(𝜙0+𝜙1𝑌0+𝜙2𝑌−1+...+𝜙𝑝𝑌1−𝑝)

𝛼𝜆

𝜆
(𝑒
−ℎ
𝛼 −1)

.                                                          (7)  



HighTech and Innovation Journal         Vol. 5, No. 1, March, 2024 

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The in-control process (𝛼 = 𝛼0), the ARL of the HWMA control chart can be expressed in the following formula:     

𝐴𝑅𝐿0 = 1 −
[𝑒

−ℎ
𝛼0𝜆−1]𝑒

(1−𝜆)𝜐+𝜆(𝜙0+𝜙1𝑌0+𝜙2𝑌−1+...+𝜙𝑝𝑌1−𝑝)

𝛼0𝜆

1+
𝑒

𝜆(𝜙0+𝜙1𝑌0+𝜙2𝑌−1+...+𝜙𝑝𝑌1−𝑝)

𝛼0𝜆

𝜆
(𝑒

−ℎ
𝛼0−1)

.                                                                         (8) 

Additionally, the out-of-control process (𝛼 = 𝛼1), the ARL of the HWMA control chart can be mathematically 

represented as follows:  

𝐴𝑅𝐿1 = 1 −
[𝑒

−ℎ
𝛼1𝜆−1]𝑒

(1−𝜆)𝜐+𝜆(𝜙0+𝜙1𝑌0+𝜙2𝑌−1+...+𝜙𝑝𝑌1−𝑝)

𝛼1𝜆

1+
𝑒

𝜆(𝜙0+𝜙1𝑌0+𝜙2𝑌−1+...+𝜙𝑝𝑌1−𝑝)

𝛼1𝜆

𝜆
(𝑒

−ℎ
𝛼1−1)

.                                                                   (9) 

3.2. The Numerical Integral Equation Method 

For an autoregressive model with exponential white noise, the analytical NIE technique for the ARL on the HWMA 

control chart is solved in this section. The ARL in this study is assessed using the Gauss-Legendre rule.   

𝐾(𝜐) = 1 +
1

𝜆
∫ 𝐾(𝑤)𝑓(

𝑤−(1−𝜆)−𝜆(𝜙0+𝜙1𝑌0+𝜙2𝑌−1+...+𝜙𝑝𝑌1−𝑝)

𝜆

ℎ

0
)𝑑𝑤  

The evaluation of an integral approximation is accomplished using the quadrature rule in the following: 

approach:∫ 𝑓(𝑥)𝑑𝑥
ℎ

0
≈ ∑ 𝑤𝑘𝑓(𝑎𝑘)

𝑛
𝑘=1  

where 𝑎𝑘 is a point and 𝑤𝑘 is a weight that is defined by the quadrature rules. 

By use the quadrature formula, we derive 𝐾(𝑎ℎ) = 1 +
1

𝜆
∑ 𝑤𝑘𝐾(𝑎𝑘)𝑓(

𝑤−(1−𝜆)𝜐−𝜆(𝜙0+𝜙1𝑌0+𝜙2𝑌−1+...+𝜙𝑝𝑌1−𝑝)

𝜆
),  ℎ =𝑛

𝑘=1

1,2, . . . , 𝑛; 

The system of 𝑛linear equations is as follows; 

𝐾(𝑎ℎ) = 1 +
1

𝜆
∑ 𝑤𝑘𝐾(𝑎𝑘)𝑓(

𝑤−(1−𝜆)𝜐−𝜆(𝜙0+𝜙1𝑌0+𝜙2𝑌−1+...+𝜙𝑝𝑌1−𝑝)

𝜆
),  ℎ = 1,2, . . . , 𝑛𝑛

𝑘=1   

𝐾(𝑎1) = 1 +
1

𝜆
∑ 𝑤𝑘𝐾(𝑎𝑘)𝑓(

𝑎𝑘−(1−𝜆)𝜐−𝜆(𝜙0+𝜙1𝑌0+𝜙2𝑌−1+...+𝜙𝑝𝑌1−𝑝)

𝜆
)𝑛

𝑘=1    

𝐾(𝑎2) = 1 +
1

𝜆
∑ 𝑤𝑘𝐾(𝑎𝑘)𝑓(

𝑎𝑘−(1−𝜆)𝜐−𝜆(𝜙0+𝜙1𝑌0+𝜙2𝑌−1+...+𝜙𝑝𝑌1−𝑝)

𝜆
)𝑛

𝑘=1   

   ⋮ 

𝐾(𝑎𝑛) = 1 +
1

𝜆
∑ 𝑤𝑘𝐾(𝑎𝑘)𝑓(

𝑎𝑘−(1−𝜆)𝜐−𝜆(𝜙0+𝜙1𝑌0+𝜙2𝑌−1+...+𝜙𝑝𝑌1−𝑝)

𝜆
)𝑛

𝑘=1   

This system can be shown as: 

𝐾𝑛×1 = (𝐼𝑛 − 𝑅𝑛×𝑛)
−11𝑛×1, 

where 𝐾𝑛×1 =

[
 
 
 
𝐾(𝑎1)

𝐾(𝑎2)
⋮

𝐾(𝑎𝑛)]
 
 
 

, 𝐼𝑛 = 𝑑𝑖𝑎𝑔(1,1, . . . ,1) and 1𝑛×1 = [

1
1
⋮
1

]. 

Let 𝑹𝑛×𝑛 is a matrix and define the 𝑛 to 𝑛𝑡ℎ as an element of the matrix 𝑹 as follows; 

[𝑹ℎ𝑘] ≈
1

𝜆
𝑤𝑘𝑓(

𝑤−(1−𝜆)𝜐−𝜆(𝜙0+𝜙1𝑌0+𝜙2𝑌−1+...+𝜙𝑝𝑌1−𝑝)

𝜆
)  

If (𝑰 − 𝑹)−1exists, the numerical approximation for the integral equation corresponds to the term of the matrix, 

𝐾𝑛×1 = (𝐼𝑛×1 − 𝑅𝑛×𝑛)
−11𝑛×1 

Eventually, we replace 𝑎ℎ by 𝜐, the numerical approximation of the integral for the function 𝐾(𝜐) represented as:  

𝐾(𝜐) = 1 +
1

𝜆
∑ 𝑤𝑘𝐾(𝑎𝑘)𝑓(

𝑎𝑘−(1−𝜆)𝜐−𝜆(𝜙0+𝜙1𝑌0+𝜙2𝑌−1+...+𝜙𝑝𝑌1−𝑝)

𝜆
)𝑛

𝑘=1                                                  (10)  

3.3. The Existence and Uniqueness of Exact ARL Solution 

The ARL formula's accuracy is theoretically verified by the Banach’s Fixed-point Theorem, which guarantees that 

explicit formulations have a unique solution to the integral equation. Let 𝑉be an operation on the class of all continuous 

functions denoted by: 



HighTech and Innovation Journal         Vol. 5, No. 1, March, 2024 

22 

 

𝑉(𝐾(𝜐)) = 1 +
1

𝜆
∫ 𝐾(𝑤)𝑓(

𝑤−(1−𝜆)−𝜆(𝜙0+𝜙1𝑌0+𝜙2𝑌−1+...+𝜙𝑝𝑌1−𝑝)

𝜆

ℎ

0
)𝑑𝑤                              (11) 

According to Banach’s Fixed-point Theorem, if an operator 𝑉  is a contraction, and then the fixed-point equation 
𝑉(𝐾(𝜐)) = 𝐾(𝜐) has a unique solution. The following Theorem can be used to show that the equation in Equation 9 
exists and has a unique solution. 

Theorem 2. Banach’s Fixed-point Theorem 

Let (𝑋, 𝑑) defined on a complete metric space and 𝑉: 𝑋 → 𝑋 satisfies the conditions of a contraction mapping with 

contraction constant 0 ≤ 𝜂 < 1  such that‖𝑉(𝐾1) − 𝑉(𝐾2)‖ ≤ 𝜂‖𝐾1 − 𝐾2‖, ∀𝐾1, 𝐾2 ∈ 𝑋. Then there exists a unique 

𝐾(. ) ∈ 𝑋 such that𝑉(𝐾(𝜐)) = 𝐾(𝜐), i.e., a unique fixed-point in 𝑋. 

Proof of Theorem 2. Let V defined in (9) is a contraction mapping for 1,K 𝐾2 ∈ 𝐹[0, ℎ], such that‖𝑉(𝐾1) − 𝑉(𝐾2)‖ ≤

𝜂‖𝐾1 − 𝐾2‖, ∀𝐾1, 𝐾2 ∈ 𝐹[0, ℎ] with 0 ≤ 𝜂 < 1 under the norm  

                          ‖𝐾‖∞ = 𝑠𝑢𝑝
𝜐∈[0,ℎ]

|𝐾(𝜐)|, so 

      ‖𝑉(𝐾1) − 𝑉(𝐾2)‖∞ = 𝑠𝑢𝑝
𝜐∈[0,ℎ]

|
1

𝛼𝜆
𝑒
(1−𝜆)𝜐+𝜆(𝜙0+𝜙1𝑌0+𝜙2𝑌−1+...+𝜙𝑝𝑌1−𝑝)

𝛼𝜆 ∫ (𝐾1(𝑤) − 𝐾2(𝑤))𝑒
−
𝑤

𝛼𝜆𝑑𝑤
ℎ

0
| 

                                   = ‖𝐾1 − 𝐾2‖∞ 𝑠𝑢𝑝
𝜐∈[0,ℎ]

|𝑒
(1−𝜆)𝜐+𝜆(𝜙0+𝜙1𝑌0+𝜙2𝑌−1+...+𝜙𝑝𝑌1−𝑝)

𝛼𝜆 | |1 − 𝑒−
ℎ

𝛼𝜆| ≤ 𝜂‖𝐾1 − 𝐾2‖∞ 

where 𝜂 = 𝑠𝑢𝑝
𝜐∈[0,ℎ]

|𝑒
(1−𝜆)𝜐+𝜆(𝜙0+𝜙1𝑌0+𝜙2𝑌−1+...+𝜙𝑝𝑌1−𝑝)

𝛼𝜆 | |1 − 𝑒−
ℎ

𝛼𝜆|; 𝜂 ∈ [0,1).                        

3.4. The Performance Measurement 

This section presents a simulation analysis comparing the NIE approach and explicit formulae' accuracy for the ARL 

of the AR(p) process on the HWMA control chart. The accuracy of the ARL values is compared with the percentage 

accuracy which can be obtained from 

%𝐴𝑐𝑐𝑢𝑟𝑎𝑐𝑦 = 100 - |
𝐾(𝜐)−𝐾(𝜐)

𝐾(𝜐)
| × 100%                                                     (12) 

Further, the efficacy of control charts in identifying out-of-control conditions is investigated using the Standard 

Deviation Run Length (SDRL) and Median Run Length (MRL) (Fonseca et al. [28]). The SDRL and MRL for the 

process under control are calculated as follows. 

𝐴𝑅𝐿0 =
1

𝛼
, 𝑆𝐷𝑅𝐿0 = √

1−𝛼

𝛼2
, 𝑀𝑅𝐿0 =

𝑙𝑜𝑔(0.5)

𝑙𝑜𝑔(1−𝛼)
,                                    (13) 

where 𝛼represents type I error. In this investigation, ARL0 was fixed at 370, and it can be calculated by dividing SDRL0 

and MRL0 by Equation 13 to yield values of approximately 370 and 256, respectively. SDRL1 and MRL1 are calculated 

differently for out-of-control situations by substituting 𝛼with 𝛽, where 𝛽represents type II error. 

The efficacy of the control chart in detecting various types of process variations can be evaluated, and informed 

decisions about its performance. A control chart with the lowest ARL1, SDRL1, and MRL1 values is considered to have 

the best performance for rapidly identifying shifts in the process mean. Additionally, the performance efficiency of the 

HWMA control chart is compared with the Extended EWMA and CUSUM control charts by using the relative mean 

index (RMI) [29]. If the RMI is a small value, this control chart will have a quick and robust performance for detecting 

shifts. RMI is defined as: 

𝑅𝑀𝐼(𝑐) =  
1

𝑛
∑ [

𝐴𝑅𝐿𝑖(𝑐)−𝐴𝑅𝐿𝑖(𝑠)

𝐴𝑅𝐿𝑖(𝑠)
]𝑛

𝑖=1                                                 (14) 

where ARLi(c) is denoted the ARL of the control chart for the shift size of row i, while the smallest ARL among all 

control charts for the same shift size is represented by ARLi(s). Furthermore, the performance metrics can be employed 

to evaluate the effectiveness of a control chart under different changes 𝛿𝑚𝑎𝑥/𝑚𝑖𝑛. 

Moreover, the average extra quadratic loss (AEQL) may refer to the average extra loss incurred due to an out-of-

control condition. During out-of-control periods, it could be calculated as the average difference between the observed 

values and the target or desired values. The calculation for AEQL is as follows [30]: 

𝐴𝐸𝑄𝐿 =
1

𝛥
∑ (𝛿𝑖

2 × 𝐴𝑅𝐿(𝛿𝑖))
𝛿𝑚𝑎𝑥∑
𝛿𝑖=𝛿𝑚𝑖𝑛

                                                 (15) 

where 𝛿 represents the particular change in the process, and 𝛥 represents the sum of number of divisions from 𝛿𝑚𝑖𝑛 to 

𝛿𝑚𝑎𝑥 In this study, 𝛥 = 9 is determined from 𝛿𝑚𝑖𝑛 to 𝛿𝑚𝑎𝑥 The control chart with the lowest AEQL values perform the 

best. Additionally, the performance comparison index (PCI) is a measurement used to compare the performance of 



HighTech and Innovation Journal         Vol. 5, No. 1, March, 2024 

23 

 

different control charts. The PCI measurement is the ratio between the AEQL of the control chart and the most efficient 

control chart, which is shown as the lowest AEQL. The mathematical formula for the PCI is: 

𝑃𝐶𝐼 =
𝐴𝐸𝑄𝐿

𝐴𝐸𝑄𝐿𝑙𝑜𝑤𝑒𝑠𝑡
                                                 (16) 

3.5. The ARL Procedure for Analytical Results 

This section will outline the procedures for calculating the ARL value using the explicit formula and the NIE method. 

When the process is in-control, 𝛼 = 𝛼0 is given to the exponential white noise parameter. And 𝛼1 = (1 + 𝛿)𝛼0 is set 

when the process goes out of control. The computation of ARL involved in comparing ARL values from both methods 

are as follows and also shown in Figure 1: 

Step 1: Determining the parameters of control chart and AR(p) process: 

a) Set the exponential white noise (𝛼0) and smoothing parameters for the in-control process. 

b) Set the initial values for the AR(p) process and the HWMA statistic. 

c) Determine suitable values for ARL0 and the shift sizes(𝛿). 

Step 2: Calculating the upper control limit (h) that yields the desired ARL for the control process by using Equation 8. 

Step 3: Calculating ARL for the in-control process: 

a) Calculate ARL0 by using Equation 8 when given the upper control limit (h) from Step 1. 

b) Calculate the value of ARL0 via the explicit formula by using Equation 8. 

c) Determine the value of ARL0 using the NIE approach by using Equation 10. 

d)  Adjust the value of h to correspond with the targeted ARL0 value. 

Step 4: Calculating ARL for the out-of-control process: 

a) Calculate ARL1 for various shift sizes and 𝛼1 = (1 + 𝛿)𝛼0 by using Equation 9 and the value of h from Step.1  

b) Approximate ARL1 via the NIE method by using Equation 10. 

Step 5: Comparison ARL: 

a) Compare the ARL values obtained using the explicit formula in Equation 9 and NIE method in Equation 10. 

Step 6: Comparison the performance of HWMA with EEWMA and CUSUM control charts. 

 

Figure 1. The process of the methodology 

Input parameters and set 

ARL0=370 ARL0=370 

Start 

Calculate the UCL 

Compare the ARL of the explicit 

formula with NIE method 

Compute ARL values 

End 

Compare the performance of the 

HWMA with the Extended EWMA and 

CUSUM control charts 

Print results 



HighTech and Innovation Journal         Vol. 5, No. 1, March, 2024 

24 

 

4. Results and Discussions 

This section will present two main points: firstly, a comparison of the accuracy between the ARL explicit formula 

and the approximate ARL by NIE method for the AR(p) process on the HWMA control chart with various change levels; 

and secondly, a comparison of the performance of the HWMA control chart with the EWMA and CUSUM control charts 

in detecting changes in process means. ARL should be sufficiently large to support the in-control process when the 

under-study process is operating in an in-control process to avoid false alarms from occurring regularly. ARL1 should 

be small for the out-of-control process to allow quick shift detection. A minimized ARL1 value indicates a more effective 

control chart. 

4.1. The Simulated Results 

We consider the change in the process mean and process standard deviation subject to the changes in the 

exponential white noise parameter 𝛼1 = (1 + 𝛿)𝛼0. Here, the shift sizes(𝛿) take the values 0.004, 0.008, 0.01, 0.04, 

0.08, 0.10, and 0.40. Furthermore, the evaluation of the control charts' capacity to identify unusual shifts is conducted 

by varying the smoothing parameter values for  𝐴𝑅𝐿0 = 370 The main insights regarding the outcomes are expanded 

in the following: 

1. In Table 1, the control limits of HWMA control chart with AR(1), AR(2), and AR(3) processes are provided. The 

control limits were obtained after setting 𝜆 = 0.01,0.015,0.02, 0.025, 0.03, 0.10, 0.15 in-control process parameter 

𝛼0 = 1. For example, in the case of AR(2) process given 𝜙0 = 0.01, 𝜆 = 0.01,and 𝐴𝑅𝐿0 = 370 the control limit is equal 

to 0.0073234.  

Table 1. Control limits of HWMA control chart with AR processes 

Models 
Coefficients 

𝝓𝟎 𝝓𝟏 𝝓𝟐 𝝓𝟑 𝝀 =0.01 𝝀 =0.015 𝝀 =0.02 𝝀 =0.025 𝝀 =0.03 

AR(1) 0.01 0.1   0.0089552 0.0134816 0.0180257 0.0225896 0.0271740 

AR(2) 0.01 0.1 0.2  0.0073234 0.0110215 0.0147309 0.0184533 0.0221891 

AR(3) 0.01 0.1 0.2 0.3 0.0054177 0.0081506 0.0108890 0.0136343 0.0163869 

AR(1) 0.01 -0.1   0.0109530 0.0164957 0.0220661 0.0276667 0.0332983 

AR(2) 0.01 -0.1 -0.2  0.0133999 0.0201910 0.0270250 0.0339048 0.0408317 

AR(3) 0.01 -0.1 -0.2 -0.3 0.0181425 0.0273652 0.0366698 0.0460603 0.0555391 

 𝝓𝟎 𝝓𝟏 𝝓𝟐 𝝓𝟑 𝝀 =0.10 𝝀 =0.15 𝝀 =0.20 𝝀 =0.25 𝝀 =0.30 

AR(1) 0.01 0.1   0.0936550 0.1439970 0.1970060 0.2529770 0.3122650 

AR(2) 0.01 0.1 0.2  0.0760033 0.1162925 0.1582710 0.2020860 0.2479060 

AR(3) 0.01 0.1 0.2 0.3 0.0557332 0.0848140 0.1147650 0.1456390 0.1774947 

AR(1) 0.01 -0.1   0.1156515 0.1789200 0.2464580 0.3188850 0.3969650 

AR(2) 0.01 -0.1 -0.2  0.1431980 0.2233075 0.3103940 0.4057850 0.5112400 

AR(3) 0.01 -0.1 -0.2 -0.3 0.1986260 0.3150140 0.4467470 0.5984990 0.7774610 

2. The comparison of the ARL1 values produced by the numerical ARL methods and the explicit formula on the 

HWMA control chart for the AR(2) model with differing choice of 𝜆 with 𝜙0 = 0.01, 𝐴𝑅𝐿0 = 370 is shown in Tables 

2 and 3. The ARL results were obtained after setting 𝜆 = 0.01, 0.02, 0.03 in Table 2 and 𝜆 = 0.1, 0.2, 0.3 in Table 3. 

The results indicate that the ARL are extremely similar and that the percentage accuracy is equal to 100 when both 

approaches are computed based on accuracy percentage. Nonetheless, the explicit formula's CPU time of about 0.001 is 

less than that of the NIE technique, which is about 1.6 seconds. In addition, it is found that when the 𝜆value increases, the 

ARL value decreases at the same level of change. Furthermore, the SDRL and MRL values were the same direction as 

the ARL values.  

 

 



HighTech and Innovation Journal         Vol. 5, No. 1, March, 2024 

25 

 

Table 2. ARL results of explicit formulas and NIE method with AR(2) process for different choices of 𝝀 with 𝝓𝟎 = 𝟎.𝟎𝟏, 𝑨𝑹𝑳𝟎 = 𝟑𝟕𝟎 

𝝀 
Coefficients of process 

Methods 
Shift size (𝜹) 

𝝓𝟏 𝝓𝟐 b 0.004 0.008 0.01 0.04 0.08 0.1 0.4 

0.01 

0.1 

  Explicit 184.0047 122.3969 104.8452 33.29889 17.48924 14.15958 3.954241 

  CPUExp (<0.001) (<0.001) (<0.001) (<0.001) (<0.001) (<0.001) (<0.001) 

0.2 0.0073234 NIE 184.0047 122.3969 104.8452 33.29890 17.48924 14.15958 3.954240 

  CPUNIE (1.609) (1.625) (1.641) (1.609) (1.641) (1.609) (1.625) 

  %Acc 100.00 100.00 100.00 100.00 100.00 100.00 100.00 

  Explicit 203.8752 140.6707 121.7941 40.46458 21.46985 17.41639 4.833761 

 0.010953 CPUExp (<0.001) (<0.001) (<0.001) (<0.001) (<0.001) (<0.001) (<0.001) 

-0.2  NIE 203.8753 140.6708 121.7941 40.46460 21.46985 17.41639 4.833760 

  CPUNIE (1.625) (1.625) (1.609) (1.625) (1.625) (1.594) (1.610) 

  %Acc 100.00 100.00 100.00 100.00 100.00 100.00 100.00 

0.2 

  Explicit 179.9503 118.8366 101.5852 31.98378 16.76081 13.56267 3.788305 

 0.0066231 CPUExp (<0.001) (<0.001) (<0.001) (<0.001) (<0.001) (<0.001) (<0.001) 

0.2  NIE 179.9504 118.8366 101.5852 31.98378 16.76081 13.56267 3.788310 

  CPUNIE (1.641) (1.594) (1.609) (1.609) (1.625) (1.610) (1.593) 

  %Acc 100.00 100.00 100.00 100.00 100.00 100.00 100.00 

  Explicit 198.2067 135.2958 116.7679 38.27715 20.25324 16.42226 4.571254 

  CPUExp (<0.001) (<0.001) (<0.001) (<0.001) (<0.001) (<0.001) (<0.001) 

-0.2 0.0099036 NIE 198.2068 135.2958 116.7679 38.27716 20.25325 16.42227 4.571250 

  CPUNIE (1.610) (1.594) (1.594) (1.594) (1.625) (1.625) (1.625) 

  %Acc 100.00 100.00 100.00 100.00 100.00 100.00 100.00 

0.02 

0.1 

  Explicit 139.8667 86.42654 72.62748 21.83289 11.64208 9.534070 3.088052 

  CPUExp (<0.001) (<0.001) (<0.001) (<0.001) (<0.001) (<0.001) (<0.001) 

0.2 0.0147309 NIE 139.8667 86.42655 72.62749 21.83289 11.64208 9.534070 3.088050 

  CPUNIE (1.594) (1.578) (1.610) (1.640) (1.656) (1.641) (1.594) 

  %Acc 100.00 100.00 100.00 100.00 100.00 100.00 100.00 

  Explicit 159.4424 101.7907 86.27203 26.73764 14.31027 11.71729 3.719865 

  CPUExp (<0.001) (<0.001) (<0.001) (<0.001) (<0.001) (<0.001) (<0.001) 

-0.2 0.0220662 NIE 159.4425 101.7907 86.27207 26.73765 14.31028 11.71729 3.719866 

  CPUNIE (1.609) (1.609) (1.625) (1.641) (1.609) (1.640) (1.625) 

  %Acc 100.00 100.00 100.00 100.00 100.00 100.00 100.00 

0.2 

  Explicit 136.0627 83.54507 70.09196 20.94726 11.15933 9.138001 2.969554 

  CPUExp (<0.001) (<0.001) (<0.001) (<0.001) (<0.001) (<0.001) (<0.001) 

0.2 0.0133184 NIE 136.0627 83.54509 70.09197 20.94726 11.15933 9.138000 2.969554 

  CPUNIE (1.593) (1.625) (1.609) (1.625) (1.625) (1.609) (1.640) 

  %Acc 100.00 100.00 100.00 100.00 100.00 100.00 100.00 

  Explicit 153.6162 97.13764 82.12037 25.22385 13.48867 11.04640 3.530567 

  CPUExp (<0.001) (<0.001) (<0.001) (<0.001) (<0.001) (<0.001) (<0.001) 

-0.2 0.0199428 NIE 153.6163 97.13768 82.12040 25.22386 13.48868 11.04640 3.530570 

  CPUNIE (1.641) (1.609) (1.640) (1.641) (1.625) (1.609) (1.625) 

  %Acc 100.00 100.00 100.00 100.00 100.00 100.00 100.00 



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0.03 

0.1 

  Explicit 126.2992 76.41419 63.89156 18.99983 10.21095 8.400650 2.865010 

  CPUExp (<0.001) (<0.001) (<0.001) (<0.001) (<0.001) (<0.001) (<0.001) 

0.2 0.0221892 NIE 126.2993 76.41420 63.89157 18.99984 10.21095 8.400651 2.865008 

  CPUNIE (1.625) (1.594) (1.625) (1.625) (1.641) (1.610) (1.609) 

  %Acc 100.00 100.00 100.00 100.00 100.00 100.00 100.00 

  Explicit 145.4102 90.77216 76.49872 23.35693 12.57243 10.33406 3.437505 

  CPUExp (<0.001) (<0.001) (<0.001) (<0.001) (<0.001) (<0.001) (<0.001) 

-0.2 0.0332983 NIE 145.4103 90.77220 76.49875 23.35694 12.57244 10.33406 3.437510 

  CPUNIE (1.610) (1.625) (1.625) (1.640) (1.609) (1.625) (1.625) 

  %Acc 100.00 100.00 100.00 100.00 100.00 100.00 100.00 

0.2 

  Explicit 122.5142 73.71040 61.54534 18.21682 9.785970 8.051790 2.758150 

  CPUExp (<0.001) (<0.001) (<0.001) (<0.001) (<0.001) (<0.001) (<0.001) 

0.2 0.0200545 NIE 122.5143 73.71041 61.54535 18.21682 9.785970 8.051790 2.758150 

  CPUNIE (1.625) (1.609) (1.609) (1.625) (1.625) (1.594) (1.578) 

  %Acc 100.00 100.00 100.00 100.00 100.00 100.00 100.00 

  Explicit 139.7719 86.42875 72.662080 22.00802 11.84285 9.737945 3.265459 

  CPUExp (<0.001) (<0.001) (<0.001) (<0.001) (<0.001) (<0.001) (<0.001) 

-0.2 0.0300784 NIE 139.7719 86.42878 72.66210 22.00803 11.84286 9.737950 3.265459 

  CPUNIE (1.625) (1.609) (1.641) (1.625) (1.610) (1.594) (1.625) 

  %Acc 100.00 100.00 100.00 100.00 100.00 100.00 100.00 

Note: The numerical results in parentheses are computational times in second. 

Table 3. ARL results of explicit formulas and NIE method with AR(2) process for different choices of 𝝀 with 𝝓𝟎 = 𝟎.𝟎𝟏, 𝑨𝑹𝑳𝟎 = 𝟑𝟕𝟎 

𝝀 
Coefficients of process 

Methods 
Shift size (𝜹) 

𝝓𝟏 𝝓𝟐 b 0.004 0.008 0.01 0.04 0.08 0.1 0.4 

0.1 

0.1 

  Explicit 110.7426 65.49227 54.47828 16.07262 8.741300 7.237580 2.636631 

  CPUExp (<0.001) (<0.001) (<0.001) (<0.001) (<0.001) (<0.001) (<0.001) 

0.2 0.0760033 NIE 110.7426 65.49229 54.47830 16.07262 8.741300 7.237585 2.636630 

  CPUNIE (1.594) (1.625) (1.625) (1.610) (1.625) (1.625) (1.641) 

  %Acc 100.00 100.00 100.00 100.00 100.00 100.00 100.00 

  Explicit 131.2544 80.14457 67.18359 20.27702 11.00570 9.090510 3.192092 

  CPUExp (<0.001) (<0.001) (<0.001) (<0.001) (<0.001) (<0.001) (<0.001) 

-0.2 0.1156515 NIE 131.2546 80.14464 67.18364 20.27703 11.00570 9.090510 3.192090 

  CPUNIE (1.562) (1.578) (1.640) (1.609) (1.625) (1.610) (1.656) 

  %Acc 100.00 100.00 100.00 100.00 100.00 100.00 100.00 

0.2 

  Explicit 107.0277 62.91628 52.26109 15.35365 8.352197 6.918040 2.536803 

  CPUExp (<0.001) (<0.001) (<0.001) (<0.001) (<0.001) (<0.001) (<0.001) 

0.2 0.0685125 NIE 107.0277 62.91630 52.26110 15.35366 8.352200 6.918044 2.536803 

  CPUNIE (1.609) (1.625) (1.625) (1.609) (1.594) (1.641) (1.625) 

  %Acc 100.00 100.00 100.00 100.00 100.00 100.00 100.00 

  Explicit 125.0088 75.57754 63.20157 18.93942 10.28736 8.504056 3.021285 

  CPUExp (<0.001) (<0.001) (<0.001) (<0.001) (<0.001) (<0.001) (<0.001) 

-0.2 0.104043 NIE 125.0089 75.57759 63.20161 18.93943 10.28736 8.504058 3.021290 

  CPUNIE (1.593) (1.656) (1.625) (1.610) (1.625) (1.594) (1.609) 

  %Acc 100.00 100.00 100.00 100.00 100.00 100.00 100.00 



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0.2 

0.1 

  Explicit 111.4606 65.97207 54.88944 16.21152 8.822032 7.306020 2.664663 

  CPUExp (<0.001) (<0.001) (<0.001) (<0.001) (<0.001) (<0.001) (<0.001) 

0.2 0.158271 NIE 111.4608 65.97213 65.97213 16.21153 8.822034 7.306020 2.664660 

  CPUNIE (1.625) (1.594) (1.610) (1.578) (1.641) (1.610) (1.609) 

  %Acc 100.00 100.00 100.00 100.00 100.00 100.00 100.00 

  Explicit 136.1859 83.80362 70.38534 21.36064 11.58393 9.560768 3.322547 

  CPUExp (<0.001) (<0.001) (<0.001) (<0.001) (<0.001) (<0.001) (<0.001) 

-0.2 0.246458 NIE 136.1865 83.80387 70.38552 21.36066 11.58393 9.560774 3.322547 

  CPUNIE (1.641) (1.609) (1.593) (1.625) (1.640) (1.594) (1.609) 

  %Acc 100.00 100.00 100.00 100.00 100.00 100.00 100.00 

0.2 

  Explicit 107.1431 62.99641 52.33168 15.38787 8.378500 6.942550 2.553330 

  CPUExp (<0.001) (<0.001) (<0.001) (<0.001) (<0.001) (<0.001) (<0.001) 

0.2 0.142075 NIE 107.1432 62.99645 52.33171 15.38787 8.378500 6.942554 2.553330 

  CPUNIE (1.625) (1.625) (1.609) (1.610) (1.594) (1.594) (1.625) 

  %Acc 100.00 100.00 100.00 100.00 100.00 100.00 100.00 

  Explicit 128.3554 78.02164 65.33161 19.65383 10.67113 8.817430 3.112634 

  CPUExp (<0.001) (<0.001) (<0.001) (<0.001) (<0.001) (<0.001) (<0.001) 

-0.2 0.22019 NIE 128.3558 78.02180 65.33173 19.65384 10.67113 8.817434 3.112635 

  CPUNIE (1.578) (1.610) (1.610) (1.640) (1.610) (1.625) (1.625) 

  %Acc 100.00 100.00 100.00 100.00 100.00 100.00 100.00 

0.3 

0.1 

  Explicit 115.0221 68.45685 57.02951 16.89262 9.178866 7.594535 2.740620 

  CPUExp (<0.001) (<0.001) (<0.001) (<0.001) (<0.001) (<0.001) (<0.001) 

0.2 0.247906 NIE 115.0225 68.45698 57.02960 16.89263 9.178870 7.594540 2.740618 

  CPUNIE (1.610) (1.609) (1.593) (1.609) (1.610) (1.625) (1.625) 

  %Acc 100.00 100.00 100.00 100.00 100.00 100.00 100.00 

  Explicit 145.4781 90.86855 76.60295 23.48986 12.70819 10.46934 3.553795 

  CPUExp (<0.001) (<0.001) (<0.001) (<0.001) (<0.001) (<0.001) (<0.001) 

-0.2 0.396965 NIE 145.4797 90.86921 76.60342 23.48991 12.70821 10.46935 3.553800 

  CPUNIE (1.625) (1.594) (1.610) (1.594) (1.610) (1.625) (1.593) 

  %Acc 100.00 100.00 100.00 100.00 100.00 100.00 100.00 

0.2 

  Explicit 109.9660 64.94908 54.00954 15.91745 8.655680 7.166671 2.612544 

  CPUExp (<0.001) (<0.001) (<0.001) (<0.001) (<0.001) (<0.001) (<0.001) 

0.2 0.221471 NIE 109.9662 64.94917 54.00960 15.91746 15.91746 7.166670 2.612540 

  CPUNIE (1.610) (1.609) (1.579) (1.625) (1.609) (1.610) (1.609) 

  %Acc 100.00 100.00 100.00 100.00 100.00 100.00 100.00 

  Explicit 135.5077 83.31025 69.95301 21.19887 11.48539 9.475818 3.282781 

  CPUExp (<0.001) (<0.001) (<0.001) (<0.001) (<0.001) (<0.001) (<0.001) 

-0.2 0.351603 NIE 135.5088 83.31067 69.95331 21.1989 11.48540 9.475825 3.282782 

  CPUNIE (1.594) (1.593) (1.579) (1.594) (1.593) (1.625) (1.609) 

  %Acc 100.00 100.00 100.00 100.00 100.00 100.00 100.00 

Note: The numerical results in parentheses are computational times in seconds. 

3. In Tables 4, the ARL of HWMA control chart for AR(2) model using explicit formula against Extended EWMA 

and CUSUM control charts for different choices of 𝜆 with 
0 10.01, 0.1,   𝜙2 = 0.2, 𝐴𝑅𝐿0 = 370are compared. For 

example, for 𝛿 change to 0.01, for 𝜆 = 0.01 ARL decreases from 370 to 14.15958, for 𝜆 = 0.10 ARL decreases from 

370 to 7.237717 and for 𝜆 = 0.2 ARL decreases from 370 to 7.306020. Furthermore, it was discovered that HWMA 

control charts were the fastest at detecting changes at all change levels when compared to Extended EWMA and CUSUM 

control charts. The outcomes of Table 5's comparison of the efficacy of control charts with the AR(3) model are in the same 

direction as those of Table 4. Moreover, when the RMI and AEQL values from Tables 4 and 5 are considered, the HWMA 

control chart has the lowest RMI and AEQL values. Additionally, the PCI value of the HWMA control chart also equals 1, 

verifying that the HWMA control chart has the highest performance. 



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Table 4. The ARL of HWMA control chart for AR(2) using explicit formula against Extended EWMA and CUSUM control 

charts given 𝝓𝟎 = 𝟎. 𝟎𝟏,𝝓𝟏 = 𝟎. 𝟏,𝝓𝟐 = 𝟎.𝟐 and 𝜶𝟎 = 𝟏. 

𝝀 𝝀𝟏 =0.01 𝝀𝟏 =0.1 𝝀𝟏 =0.2 

𝜹 

Control 

Chart 
HWMA 

EWMA 

𝝀𝟐=0.005 

CUSUM 

a=4 
HWMA 

EEWMA 

𝝀𝟐=0.05 

CUSUM 

a=4 
HWMA 

EEWMA 

𝝀𝟐=0.1 

CUSUM 

a=4 

UCL 0.0073234 0.0120775 1.432 0.0760035 0.1245097 1.432 0.158271 0.257305 1.432 

0.000 

ARL0 370.467 370.0205 370.348 370.4595 370.8249 370.348 370.7615 370.9559 370.348 

SDRL0 369.9666 369.5201 369.8477 369.9591 370.3246 369.8477 370.2612 370.4556 369.8477 

MRL0 256.4414 256.1319 256.3589 256.4362 256.6895 256.3589 256.6456 256.7803 256.3589 

0.002 

ARL1 245.8854 268.1707 365.981 170.2343 196.8122 365.981 171.0674 197.6334 365.981 

SDRL1 245.3849 267.6702 365.4807 169.7336 196.3115 365.4807 170.5667 197.1328 365.4807 

MRL1 170.0879 185.5349 253.332 117.6505 136.0729 253.332 118.228 136.6422 253.332 

0.004 

ARL1 184.0047 210.2855 361.681 110.7811 134.2153 361.681 111.4606 134.9623 361.681 

SDRL1 183.504 209.7849 361.1807 110.2799 133.7144 361.1807 110.9595 134.4614 361.1807 

MRL1 127.1954 145.412 250.3514 76.44048 92.68395 250.3514 76.91153 93.20173 250.3514 

0.008 

ARL1 122.3969 146.8759 353.283 65.50561 82.32452 353.283 65.97207 82.87974 353.283 

SDRL1 121.8959 146.375 352.7826 65.00368 81.82299 352.7826 65.47016 82.37822 352.7826 

MRL1 84.492 101.4596 244.5304 45.05757 56.71573 244.5304 45.3809 57.10058 244.5304 

0.01 

ARL1 104.8452 127.6324 349.182 54.48747 69.08969 349.182 54.88944 69.57705 349.182 

SDRL1 104.344 127.1315 348.6816 53.98515 68.58787 348.6816 54.38714 69.07524 348.6816 

MRL1 72.32603 88.12104 241.6878 37.42019 47.54191 241.6878 37.69882 47.87973 241.6878 

0.02 

ARL1 61.06833 77.11843 329.604 29.87616 38.60833 329.604 30.11616 38.91001 329.604 

SDRL1 60.56627 76.6168 329.1036 29.3719 38.10505 329.1036 29.61194 38.40675 329.1036 

MRL1 41.98182 53.1071 228.1173 20.36004 26.41317 228.1173 20.52641 26.62229 228.1173 

0.04 

ARL1 33.29889 43.06284 294.655 16.07336 20.93302 294.655 16.21152 21.10816 294.655 

SDRL1 32.79508 42.5599 294.1546 15.56533 20.4269 294.1546 15.70357 20.6021 294.1546 

MRL1 22.7327 29.50096 203.8925 10.79092 14.16026 203.8925 10.88672 14.28169 203.8925 

0.08 

ARL1 17.48924 22.91338 238.42 8.7415 11.36602 238.42 8.822032 11.46673 238.42 

SDRL1 16.98188 22.40781 237.9195 8.226319 10.85451 237.9195 8.306998 10.95533 237.9195 

MRL1 11.77264 15.5332 164.9133 5.705557 7.526435 164.9133 5.761445 7.59629 164.9133 

0.10 

ARL1 14.15958 18.59339 215.703 7.237717 9.387661 215.703 7.30602 9.472397 215.703 

SDRL1 13.65043 18.08648 215.2024 6.719139 8.873585 215.2024 6.787629 8.958455 215.2024 

MRL1 9.463871 12.53819 149.1671 4.661644 6.153952 149.1671 4.709074 6.212749 149.1671 

0.20 

ARL1 7.359294 9.67329 137.436 4.188452 5.356299 137.436 4.231024 5.407585 137.436 

SDRL1 6.841047 9.159653 136.9351 3.654406 4.830491 136.9351 3.697369 4.882048 136.9351 

MRL1 4.746068 6.352138 94.91638 2.540903 3.354202 94.91638 2.570592 3.389876 94.91638 

RMI 0 0 0.2383 6.9398 6.9398 0.2626 14.1634 0 0.262 

AEQL 0.0720 0.072 0.094 1.0943 1.0943 0.0494 1.0943 0.0388 0.0499 

PCI 1 1 1.3056 15.2017 15.2017 1.2873 28.4947 1 1.2867 

 

 



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Table 5. The ARL of HWMA control chart for AR(3) using explicit formula against Extended EWMA and CUSUM control 
charts given 𝝓𝟎 = 𝟎. 𝟎𝟏,𝝓𝟏 = 𝟎.𝟏,𝝓𝟐 = 𝟎.𝟐,𝝓𝟑 = 𝟎.𝟑 and 𝜶𝟎 = 𝟏. 

𝝀 𝝀𝟏 =0.01 𝝀𝟏 =0.1 𝝀𝟏 =0.2 

𝜹 

Control 

Chart 
HWMA 

EEWMA 

𝝀𝟐=0.005 

CUSUM 

a=4 
HWMA 

EEWMA 

𝝀𝟐=0.05 

CUSUM 

a=4 
HWMA 

EEWMA 

𝝀𝟐=0.1 

CUSUM 

a=4 

UCL 0.0054177 0.008935 1.129 0.0557333 0.091465 1.129 0.114765 0.187328 1.129 

0.000 

ARL0 370.59 370.2827 370.641 370.7671 370.5314 370.641 370.7544 370.617 370.641 

SDRL0 370.0897 369.7824 370.1407 370.2668 370.0311 370.1407 370.2541 370.1167 370.1407 

MRL0 256.5267 256.3137 256.562 256.6495 256.4861 256.562 256.6407 256.5454 256.562 

0.002 

ARL1 235.5205 253.8489 366.279 157.6979 177.7128 366.279 156.8015 176.4717 366.279 

SDRL1 235.02 253.3484 365.7787 157.1971 177.2121 365.7787 156.3007 175.971 365.7787 

MRL1 162.9036 175.6079 253.5385 108.9609 122.8342 253.5385 108.3396 121.974 253.5385 

0.004 

ARL1 172.5957 193.124 361.986 100.4174 117.1638 361.986 99.7031 116.0915 361.986 

SDRL1 172.095 192.6233 361.4857 99.91614 116.6627 361.4857 99.20184 115.5905 361.4857 

MRL1 119.2873 133.5165 250.5628 69.25688 80.8647 250.5628 68.76177 80.12145 250.5628 

0.008 

ARL1 112.4729 130.6312 353.599 58.43439 69.97283 353.599 57.96549 69.2239 353.599 

SDRL1 111.9718 130.1302 353.0986 57.93223 69.47103 353.0986 57.46332 68.72208 353.0986 

MRL1 77.61318 90.19961 244.7494 40.15606 48.15406 244.7494 39.83104 47.63494 244.7494 

0.01 

ARL1 95.7844 112.4411 349.503 48.42291 58.3434 349.503 48.02647 57.69983 349.503 

SDRL1 95.28309 111.94 349.0026 47.9203 57.84124 349.0026 47.52384 57.19765 349.0026 

MRL1 66.04551 77.59117 241.9103 33.21642 40.09299 241.9103 32.94162 39.64689 241.9103 

0.02 

ARL1 54.97847 66.29976 329.949 26.34255 32.1576 329.949 26.12474 31.78825 329.949 

SDRL1 54.47618 65.79786 329.4486 25.83771 31.65365 329.4486 25.61986 31.28426 329.4486 

MRL1 37.76054 45.60804 228.3565 17.91046 21.94155 228.3565 17.75946 21.68552 228.3565 

0.04 

ARL1 29.68477 36.44274 295.039 14.12756 17.34352 295.039 14.01952 17.15178 295.039 

SDRL1 29.18049 35.93926 294.5386 13.61838 16.8361 294.5386 13.51027 16.64427 294.5386 

MRL1 20.22736 24.912 204.1587 9.441665 11.67161 204.1587 9.366743 11.53866 204.1587 

0.08 

ARL1 15.49015 19.23176 238.851 7.68889 9.435694 238.851 7.64145 9.345016 238.851 

SDRL1 14.98181 18.72508 238.3505 7.171481 8.921694 238.3505 7.123925 8.830872 238.3505 

MRL1 10.38653 12.98078 165.2121 4.974914 6.187281 165.2121 4.941977 6.124362 165.2121 

0.10 

ARL1 12.5214 15.58568 216.146 6.373 7.810113 216.146 6.338045 7.740622 216.146 

SDRL1 12.011 15.07739 215.6454 5.851677 7.292993 215.6454 5.816594 7.223338 215.6454 

MRL1 8.327792 10.45276 149.4741 4.060999 5.059073 149.4741 4.036711 5.010829 149.4741 

0.20 

ARL1 6.48884 8.110357 137.877 3.711732 4.508561 137.877 3.70154 4.481941 137.877 

SDRL1 5.967931 7.593914 137.3761 3.172574 3.977256 137.3761 3.162255 3.950425 137.3761 

MRL1 4.141485 5.267499 95.22206 2.208101 2.764053 95.22206 2.200978 2.745504 95.22206 

RMI 0 0 0.1891 7.924 0 0.2013 16.1925 0 0.1979 

AEQL 0.0638 0.0638 0.0791 1.0971 0.0339 0.0414 1.0971 0.0338 0.041 

PCL 1 1 1.2405 17.206 1 1.2193 32.3349 1 1.2153 

4.2. The Real-World Datasets 

In this particular section, the explicit formulas for the average run length (ARL) of an autoregressive (AR) process 

on the EWMA control chart are applied and compared with the performance of the extended EWMA and cumulative 

sum (CUSUM) control charts. Following the subsequent steps, the ARL formula has been implemented using actual 

data. 

1. To estimate parameters from a dataset, it is necessary to include an autoregressive model of order p (AR(p)). 

2. To estimate the parameter of residuals that follow an exponential distribution. 

3. By utilizing the parameter values obtained from the previous two steps, we can calculate the Average Run Length 

(ARL) values in Equations 8 and 9. 



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4. To perform a performance comparison, the ARL value obtained from 3. was compared with the Extended EWMA 

and CUSUM control charts. 

5. To identify variations in the mean of a process, it is necessary to calculate the upper control limit (UCL) using the 

formula provided in Equation 4. Subsequently, the control chart statistics should be computed using actual data, 

and these statistics should be plotted on a graph to visualize any deviations. 

In the context of practical application, this study is carried out utilizing daily data of natural gas prices from January 

2, 2023 to April 4, 2023. The models were fitted using the SPSS program. The suitable model for dataset that correspond 

to AR(1) and AR(2) models is identified, and the relevant parameters are displayed in Table 6. As a result, the AR(1) 

model shows the lowest RMSE and MAPE values, implying that the AR(1) is the best model. The coefficient parameters 

for AR(1) are derived as shown in Table 6: �̂�1 = 0.999. As shown in Table 7, the mean parameter of exponential white 

noise was then determined using the one-sample Kolmogorov-Smirnov test. The in-control parameter is equal to 0.1223. 

The parameter of this prediction model, can be assigned as �̂�𝑡 = 0.999𝑌𝑡−1. 

Table 6. The coefficients for the trend AR(p) models using the real-world datasets 

Model AR(1) model AR(2) model 

Parameters Coefficient Std. Error t-Statistic p-value Coefficient Std. Error t-Statistic p-value 

AR(1) 0.999 0.002 660.400 0.000 0.742 0.106 7.000 0.000 

AR(2)  0.258 0.106 2.432 0.000 

RMSE 0.472 0.473 

MAPE 5.389 5.461 

Table 7. One-sample Kolmogorov test for the real-world datasets 

Residual of Application Residual AR(1) model 

Exponential parameter 0.1223 

One-sample Kolmogorov-Smirnov test 0.635 

p-value 0.814 

The explicit formula method was used to compare the ARL values for AR(1) on the HWMA, Extended EWMA, and 

CUSUM control charts; the results are shown in Table 8; it is evident that the results are consistent with those in Tables 

4 and 5. The findings indicate that the HWMA control chart exhibits the minimum RMI, AEQL, across all levels of 𝜆. 
Additionally, the PCI value on the HWMA control chart is 1. The outcomes presented in Table 8 are visually enhanced 

in Figure 2. In light of this, it can be concluded that the explicit formula for detecting mean process changes on the 

HWMA control chart is an acceptable alternative for practical applications. Figure 3 also displays the HWMA (Ht), 

Extended EWMA (Et), and CUSUM (Ct) statistics for the natural gas price corresponding to the AR(1) model. These 

results indicate that the HWMA control chart can detect a shift for the first time in the first observation, the Extended 

EWMA control chart can detect a shift for the first time in the third observation, and the CUSUM scheme is identified 

for the first time in the twelfth observation. The results therefore indicate that the HWMA control chart is preferable to 

the Extended EWMA and CUSUM control charts for the natural gas price dataset. 

 
(a)  

16.1868

16.735

10.6685

0.0162

0.0148

0.0379

0

0

0

0 2 4 6 8 10 12 14 16 18

RMI

HWMA

EEWMA

CUSUM



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31 

 

 
(b) 

  
(c) 

Figure 2. Comparison the RMI, AEQL and PCL values among HWMA, Extended EWMA and CUSUM control charts for 

AR(1) when (a) 𝝀𝟏 = 0.01, (b) 𝝀𝟏 =0.1 and (c) 𝝀𝟏 =0.2 

 

(a) 

0.1204

0.1204

0.1204

0.012

0.0117

0.014

0.0118

0.0116

0.0136

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

AEQL

HWMA EEWMA CUSUM

10.2031

10.408

8.8683

1.0152

1.0144

1.0305

1

1

1

0 2 4 6 8 10 12

PCI

HWMA EEWMA CUSUM

0.001

0.004

0.016

0.064

0.256

1.024
1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64 67 70 73 76 79 82 85

Sample No.

HWMA Control Chart  Sample No. is in control

 Sample No. is out of control

UCL=0.024862 

H
t
 



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(b) 

 
(c) 

Figure 3. The performance comparison of real data among (A) HWMA control chart, (B) Extended EWMA control chart 

and (C) CUSUM control chart when 𝝀𝟏 =0.2 

Table 8. The ARL of HWMA control chart for AR(1) using explicit formula against Extended EWMA and CUSUM control 

charts given 𝝓𝟏 = 𝟎. 𝟗𝟗𝟗 and 𝜶𝟎 = 𝟎. 𝟏𝟐𝟐𝟑 

𝝀 𝝀𝟏 =0.01 𝝀𝟏 =0.1 𝝀𝟏 =0.2 

𝜹 

Control 

Chart 
HWMA 

EEWMA 

𝝀𝟐=0.005 

CUSUM 

a=4.5 
HWMA 

EEWMA 

𝝀𝟐=0.05 

CUSUM 

a=4.5 
HWMA 

EEWMA 

𝝀𝟐=0.05 

CUSUM 

a=4.5 

UCL 0.001128 0.001216 0.2212 0.0118 0.01246988 0.2212 0.024862 0.0262728 0.2212 

0.000 

ARL0 370.0148 370.9366 370.0920 370.0900 370.7941 370.0920 370.8314 370.5368 370.0920 

SDRL0 369.5145 370.4363 369.5917 369.5896 370.2938 369.5917 370.3310 370.0364 369.5917 

MRL0 256.1280 256.767 256.1815 256.1801 256.6682 256.1815 256.6940 256.4898 256.1815 

0.002 

ARL1 68.54312 71.07629 336.1870 39.48426 39.99501 336.1870 40.89710 41.49404 336.1870 

SDRL1 68.04128 70.57452 335.6866 38.98105 39.49184 335.6866 40.39400 40.99099 335.6866 

MRL1 47.16305 48.91894 232.6803 27.02035 27.37439 232.6803 27.99970 28.41349 232.6803 

0.004 

ARL1 37.95438 39.49063 306.3130 21.38617 21.68080 306.3130 22.17772 22.52451 306.3130 

SDRL1 37.45104 38.98743 305.8126 20.88018 21.17489 305.8126 21.67195 22.01884 305.8126 

MRL1 25.95986 27.02477 211.9732 14.47442 14.67868 211.9732 15.02318 15.26361 211.9732 

0

0.02

0.04

0.06

0.08

0.1

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64 67 70 73 76 79 82 85

Sample No.

Extended EWMA Control Chart  Sample No. is in control

 Sample No. is out of control

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64 67 70 73 76 79 82 85

Sample No.

CUSUM Control Chart  Sample No. is in control

 Sample No. is out of control

E
t
 

UCL=0.0262728 

UCL=0.2212 

C
t
 



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0.008 

ARL1 20.22209 21.07776 256.4690 11.57006 11.73811 256.4690 11.99492 12.19157 256.4690 

SDRL1 19.71576 20.57168 255.9685 11.05876 11.22699 255.9685 11.48404 11.68087 255.9685 

MRL1 13.66738 14.26061 177.4240 7.667960 7.784525 177.4240 7.962644 8.099035 177.4240 

0.01 

ARL1 16.45130 17.15221 235.6180 9.538947 9.679884 235.6180 9.885292 10.04950 235.6180 

SDRL1 15.94346 16.6447 235.1175 9.025107 9.166257 235.1175 9.371964 9.536401 235.1175 

MRL1 11.05298 11.53896 162.9711 6.258925 6.356714 162.9711 6.499230 6.613156 162.9711 

0.02 

ARL1 8.684650 9.054566 159.7810 5.401260 5.485491 159.7810 5.584750 5.680963 159.7810 

SDRL1 8.169363 8.539941 159.2802 4.875690 4.960354 159.2802 5.060107 5.156780 159.2802 

MRL1 5.666103 5.922815 110.4048 3.385477 3.444061 110.4048 3.513091 3.579993 110.4048 

0.04 

ARL1 4.719190 4.911566 84.78130 3.284530 3.337927 84.78130 3.382941 3.442405 84.78130 

SDRL1 4.189459 4.383139 84.27982 2.739271 2.793534 84.27982 2.839252 2.899611 84.27982 

MRL1 2.910778 3.044726 58.41866 1.909164 1.946576 58.41866 1.978104 2.019735 58.41866 

0.08 

ARL1 2.760890 2.860518 34.76150 2.199470 2.234827 34.76150 2.253360 2.291816 34.76150 

SDRL1 2.204909 2.306956 34.25785 1.624253 1.661212 34.25785 1.680557 1.720640 34.25785 

MRL1 1.541239 1.611416 23.74658 1.143172 1.168423 23.74658 1.181646 1.209057 23.74658 

0.10 

ARL1 2.380200 2.460779 25.11730 1.976790 2.007815 25.11730 2.021350 2.054885 25.11730 

SDRL1 1.812499 1.895958 24.61222 1.389571 1.422500 24.61222 1.436839 1.472300 24.61222 

MRL1 1.271932 1.329121 17.06107 0.983246 1.005636 17.06107 1.015392 1.039531 17.06107 

0.20 

ARL1 1.647260 1.68883 9.108460 1.516840 1.537068 9.108460 1.541582 1.563148 9.108460 

SDRL1 1.032572 1.078571 8.593927 0.885417 0.908575 8.593927 0.913725 0.938235 8.593927 

MRL1 0.742032 0.772914 5.960214 0.643799 0.659194 5.960214 0.662620 0.678948 5.960214 

RMI 0 0.0379 10.6685 0 0.0148 16.7350 0 0.0162 16.1868 

AEQL 0.0136 0.0140 0.1204 0.0116 0.0117 0.1204 0.0118 00120 0.1204 

PCI 1 1.0305 8.8683 1 1.0144 10.4080 1 1.0152 10.2031 

5. Conclusion 

In this research, for an AR process with exponential white noise on an HWMA control chart, the ARL is proven and 

compared with the NIE technique. The results of the comparison showed that the ARL values obtained using the explicit 

formula and the NIE method were similar. Moreover, the existence and uniqueness of ARL derivatives according to 

clear formulas have been proven. In addition, the SDRL and MRL values were studied, which found that the results 

were in the same direction as the ARL values. 

Taking into account the variation of the parameters at different levels, the performance of the HWMA, Extended 

EWMA, and CUSUM control charts is studied by comparing the ARL values when the process is out of control. The 

RMI, AEQL, and PCI values were used to compare their performances on HWMA, Extended EWMA, and CUSUM 

control charts. The results indicated that the HWMA control chart exhibited lower RMI and AEQL values in 

comparison to the Extended EWMA and CUSUM control charts. Additionally, the HWMA control chart maintained a 

PCI value of 1. In conclusion, the HWMA control chart exhibits the most significant efficiency. Additionally, the price 

of natural gas is utilized as actual data to evaluate the HWMA control chart's performance. The benefit of this 

application is that it provides these results and conceptions for developing strategies to identify price-level changes. 

Natural gas is generally traded on commodity exchanges based on supply and demand forces in these markets. 

Consequently, if we use a control chart to monitor price fluctuations, this can be used as a guide by traders and 

investors to make trading decisions based on forecasts of future price movements by combining fundamental research, 

which looks at supply and demand factors, with technical analysis, which looks at graph patterns and historical price 

data. In conclusion, the findings show that the HWMA control chart outperformed the Extended EWMA and CUSUM 

control charts for all change magnitudes. Furthermore, the outcomes from the simulation study and a real -world 

situation concerning the price of natural gas agreed. While there is potential for the explicit formula derivation of the 

ARL to be implemented in other situations, it is limited to the AR model and exponential white noise. Alternative 

methods for calculating the ARL value, such as the NIE or Markov chain approach, may be required if the analyzed 

data contains additional white noise patterns. Finally, further research will be undertaken to apply the explicit ARL 

formulas displayed on the HWMA chart to other real-world data models, such as ARIMA and ARMA. In addition, we 

will apply this technique to derive the explicit formula for new control charts in order to enhance their ability to detect 

change in various situations. 



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6. Declarations  

6.1. Author Contributions 

Conceptualization, Y.A. and S.S.; methodology, R.S.; software, Y.A.; validation, R.S., S.S., and Y.A.; formal 

analysis, R.S.; investigation, S.S.; resources, Y.A.; data curation, Y.A.; writing—original draft preparation, Y.A.; 

writing—review and editing, Y.A.; visualization, R.S.; supervision, Y.A.; project administration, S.S.; funding 

acquisition, Y.A. All authors have read and agreed to the published version of the manuscript. 

6.2. Data Availability Statement 

The data presented in this study are available in the article. 

6.3. Funding 

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-67-B-11. 

6.4. Institutional Review Board Statement 

Not applicable.  

6.5. Informed Consent Statement 

Not applicable.  

6.6. Declaration of Competing Interest 

The authors declare that they have no known competing financial interests or personal relationships that could have 

appeared to influence the work reported in this paper. 

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