The Development and Evaluation of Homogenously Weighted Moving Average Control Chart based on an Autoregressive Process

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.


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 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 firstorder 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.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.

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.

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 {  ,  = 1,2, . . . ., }be a sequence of AR(p) process given as in Equation 1; where  0 is a constant of model   is coefficients of autoregressive  = 1,2, . . ., .

The HWMA Control Chart
Under the assumption that {  ;  = 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 (  ) is known as an upper HWMA statistic.  can be represented as in Equation 2 using the recursive formula.
where   is a sequence of the AR(p) process with exponential white noise, the constant value  ̄0 =  is an initial value;  ∈ [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: ],  > 1 Lower control limit: ],  > 1 where  0 is the target mean, σ is the process standard deviation and  1 is the width of the control limits.
where  ℎ is the stopping time and h is UCL.

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: where  1 and  2 are exponential smoothing coefficient with (0 <  1 ≤ 1) and (0 ≤  2 ≤  1 ) and the initial value is a constant,  0 = .The upper control limit (UCL) and lower control limit (LCL) of the Extended EWMA control chart are given by: where  0 is the target mean,  is the process standard deviation, and  2 is suitable control limit width.
The stopping time of the Extended EWMA control chart (  ) is given by: where   is the stopping time and is UCL.

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: where  is a reference value,  0 =  is the initial value of CUSUM statistic;  ∈ [0, ] and the CUSUM chart's stopping time is defined as   = { > 0;   > } and  is UCL.

Analytical Explicit Formulas of the ARL for AR(p) Model
From the recursion of HWMA statistics in Equation 2, Consequently, the HWMA statistics can be displayed as: Set LCL=0, UCL=ℎfor in control process and given  ̄0 =  then: The zero state at  = 1 is considered, therefore () can be calculated as follows: After changing the variable in Equation 5, the expression can be reformulated as: Thus: Since  = ∫ () , we have; Substituting  in (6), we obtain: .
The in-control process ( =  0 ), the ARL of the HWMA control chart can be expressed in the following formula: Additionally, the out-of-control process ( =  1 ), the ARL of the HWMA control chart can be mathematically represented as follows:

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.
The evaluation of an integral approximation is accomplished using the quadrature rule in the following: where   is a point and   is a weight that is defined by the quadrature rules.
By use the quadrature formula, we derive  ̃( 1,2, . . ., ; The system of linear equations is as follows; This system can be shown as: where ,   = (1,1, . . .,1) and Let  × is a matrix and define the  to  ℎ as an element of the matrix  as follows; If ( − ) −1 exists, the numerical approximation for the integral equation corresponds to the term of the matrix, Eventually, we replace  ℎ by , the numerical approximation of the integral for the function () represented as:

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 be an operation on the class of all continuous functions denoted by: ) (11) According to Banach's Fixed-point Theorem, if an operator  is a contraction, and then the fixed-point equation (()) = () has a unique solution.The following Theorem can be used to show that the equation in Equation 9exists and has a unique solution.
Proof of Theorem 2. Let V defined in ( 9) is a contraction mapping for 1 ,

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 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.
where represents type I error.In this investigation, ARL0 was fixed at 370, and it can be calculated by dividing SDRL0 and MRL0 by Equation 13to yield values of approximately 370 and 256, respectively.SDRL1 and MRL1 are calculated differently for out-of-control situations by substituting with , where represents 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: ]  =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  / .Moreover, the average extra quadratic loss (AEQL) may refer to the average extra loss incurred due to an out-ofcontrol 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]: where  represents the particular change in the process, and  represents the sum of number of divisions from   to   In this study,  = 9 is determined from   to   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 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:

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,  =  0 is given to the exponential white noise parameter.And  1 = (1 + ) 0 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 ( 0 ) 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().
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  1 = (1 + ) 0 by using Equation 9and 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 9and NIE method in Equation 10.
Step 6: Comparison the performance of HWMA with EEWMA and CUSUM control charts.

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.

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  1 = (1 + ) 0 .Here, the shift sizes() 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  0 = 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  = 0.01,0.015,0.02,0.025, 0.03, 0.10, 0.15 in-control process parameter  0 = 1.For example, in the case of AR(2) process given  0 = 0.01,  = 0.01,and  0 = 370 the control limit is equal to 0.0073234.2. 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  with  0 = 0.01,  0 = 370 is shown in Tables 2 and 3.The ARL results were obtained after setting  = 0.01, 0.02, 0.03 in Table 2 and  = 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 value 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.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  with 01 0.01, 0.1,

 
2 = 0.2,  0 = 370are compared.For example, for  change to 0.01, for  = 0.01 ARL decreases from 370 to 14.15958, for  = 0.10 ARL decreases from 370 to 7.237717 and for  = 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.

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.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  ̂ = 0.999 −1 .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 .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

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.

Figure 1 .
Figure 1.The process of the methodology

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.

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

Figure 3. The performance comparison of real data among (A) HWMA control chart, (B) Extended EWMA control chart and (C) CUSUM control chart when 𝝀
=0.2

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