Improving Sensitivity of the DEWMA Chart with Exact ARL Solution under the Trend AR(p) Model and Its Applications

The double exponentially weighted moving average (DEWMA) chart is a control chart that is a vital analytical tool for keeping track of the quality of a process, and the sensitivity of the control chart to the process is evaluated using the average run length (ARL). Herein, the aim of this study is to derive the explicit formula of the ARL on the DEWMA chart with the autoregressive with trend model and its residual, which is exponential white noise. This study shows that this proposed method was compared to the ARL derived using the numerical integral equation (NIE) approach, and the explicit ARL formula decreased the computing time. By changing exponential parameters that were relevant to evaluating in various circumstances, the sensitivity of AR(p) with the trend model with the DEWMA chart was investigated. These were compared with the EWMA and CUSUM charts in terms of the ARL, standard deviation run length (SDRL), and median run length (MRL). The results indicate that the DEWMA chart has the highest performance, and when it was small, the DEWMA chart had high sensitivity for detecting processes. Digital currencies are utilized to demonstrate the efficacy of the proposed method; the results are consistent with the simulated data.

shifts in the process mean or covariance matrix.In 2017, it was revised by Khan et al. [8].Naveed et al. (2018) [9] suggested a new modified EWMA-type chart, which refers to the extended exponentially weighted moving average (EEWMA) chart and has sensitivity for monitoring small changes.Moreover, there is a control chart that outperforms in monitoring small changes rapidly.It is a double exponentially weighted moving average (DEWMA) control chart, which Shamma & Shamma first showed in 2010 [10], with modifications made by Mahmoud and Woodall [11].
Typically, a control chart must make the assumption that the data produced by the primary procedure will be independent and have a normal distribution.Nevertheless, in reality, this assumption is frequently violated because the observations or real-world data may show various patterns, and the data is mostly related to time series characteristics and forecasting.Time series and forecasting data display seasonal, trend, and autocorrelation traits.Autoregressive (AR) and moving average (MA) time series components are frequently observed when analyzing real-world data.How to assess the errors is a crucial factor to think about when creating a model.A majority of the data is normally distributed with white noise, which indicates errors in the time series model when using autocorrelated data.However, in certain situations, white noise may follow an exponential distribution [12,13].
The ARL, which consists of two characteristics, can be used to evaluate control chart effectiveness.The in-control ARL, also known as ARL0, represents the average amount of observations taken by a process under control before it signals an indication of being out-of-control.Out-of-control ARL, also known as ARL1, is the average number of observations needed to identify an alteration in a process variable that is out of control.The ARL0 values should ideally be high.ARL1 values, on the other hand, should ideally be as low as possible to demonstrate that the procedure is sensitive enough to rapidly identify any out-of-control situations.Calculating the ARL as a starting point is the goal when developing a control chart.In many literary works, different approaches to calculating the ARL have been proposed.For example, Champ & Rigdon [14] studied and compared the Markov Chain and the NIE method for calculating the ARL of quality control charts.Brook & Evans [15] presented the Markov Chain method for computing ARL.Karoon et al. [16] proposed the NIE method for evaluating ARL.The method cited above can be applied to a variety of data characteristics, especially real-world data, which contains many forms of autocorrelation and whose distribution does not meet the assumptions.
In addition to the methods mentioned above, there is one method of evaluating ARL, namely explicit formulas, and many researchers have studied them for various control charts.In their article, Petcharat et al. [17] explicitly established the ARL of random observations from an MA process with exponential white noise acting on the CUSUM chart.With a long-memory ARFIMA process, Sunthornwat et al. [18] assigned explicit formulations for the analytical ARL on the EWMA chart and compared them to the NIE method.An exact formula for the ARL using data from the MA(p) model was put forth by Supharakonsakun [19].When the data are the AR (1) and AR(p) models, Karoon et al.'s explicit formula for ARL on the EEWMA chart was suggested in 2022 [20,21].Moreover, in the same year, they proposed exact formulas of the ARL based on the data that are autoregressive with seasonality for the EEWMA chart [22].Areepong & Peerajit [23] used the ARL that they obtained from explicit formulas of the CUSUM chart to detect changes in the long-memory SARFIMAX model.
Phanthuna & Areepong [24] presented the explicit formula of ARL running on the MEWMA chart that shows the detection sensitivity of a modified EWMA chart under a time series model with fractionality and integration.The MEWMA chart had high performance when compared to the EWMA chart in all situations.Next, the explicit formula of the ARL underlying the data, which is seasonal autoregressive with explanatory variables on the CUSUM chart, was presented and expressed the performance of it by Phanyaem (2022) [25].Peerajit & Areepong [26] presented the ARL of an autoregressive fractionally integrated process with exponential white noise running on the modified EWMA control chart.And also, Silpakob et al. [27] presented the explicit ARL formulas on the new MEWMA chart running on the AR(p) process.In the same year, Silpakob et al. [28] presented the explicit formula of the ARL under the ARMA with explanatory variables that is running on the MEWMA chart, and the results show that it outperformed the EWMA and CUSUM charts.Peerajit [29] said that while the process is operating on long memory under SFIMAX for the CUSUM chart, the explicit formula of ARL outperforms the ARL that is obtained using the NIE approaches.Of special interest, several researchers have also adapted analytical integral equations based on explicit formulas and the NIE technique on a control chart for models with trend variables, which are basic characteristics of the presently available data.For instance, Phanthuna et al. [30] proposed explicit formulas for the ARL that can be used to detect shifts in the MEWMA chart.These formulas are based on the trend-stationary AR(1) process.
Petcharat [31] presented the exact formulas of the ARL on the CUSUM chart running on trend with a stationary SAR process.Supharakonsakun and Areepong [32] improved the performance of the MEWMA chart by using the explicit formula of the ARL, which is processed based on the observations and is a trend autoregressive with an explanatory model.And recently, Karoon et al. [33,34] presented the exact ARL formulas on the EEWMA chart based on the data: trend AR and quadratic trend AR models, and compared the efficiency with the EWMA chart, finding that it had more sensitivity than the EWMA chart.
All of the above-mentioned literature makes me realize that the exact formula deriving the ARL of the DEWMA chart based on the data requires an autoregressive with trend model, or trend AR(p), which has not been done before.Hence, the main objective of this paper is to use the DEWMA chart to generate specific ARL formulas for the data and compare them with the NIE method, which utilizes autoregressive with trend models.The DEWMA chart is then created using the precise ARL formula, which is enlarged to allow for a comparison of the control chart's sensitivity to the EWMA and CUSUM charts that underlie both simulated and real-world data.Then, the sensitivity of the DEWMA chart was calculated using the SDRL and MRL values, and control chart performance measures, namely the extra-square loss mean (AEQL) and comparative efficiency index (PCI), were used to confirm the results of the proposed ARL of the DEWMA chart.Moreover, the applications that were used to illustrate this research are related to digital currency, specifically referring to cryptocurrencies, namely Bitcoin and Ethereum, which are well-known among investors and are popular investments in the present.Finally, the significance of this proposed ARL is to improve the sensitivity of detecting changes in DEWMA charts using an exact ARL solution, which can be useful for the actual data generated in the autocorrelation with the trend autoregressive model to increase the efficiency of the control chart while the process changes are slight.

2-Structures of the Control Charts with Trend AR Model
This part includes the DEWMA statistical structure, data from the autoregressive with trend model (trend AR(p)), followed by the obtained explicit formula, and the NIE method of the ARL.

2-1-The EWMA Chart
First, Robert [3] initially suggested the original idea for the EWMA chart.It is frequently used to monitor the process and identify slight deviations from the mean.The statistics of the EWMA chart can be described using the expression in Equation 1 below: where the EWMA chart parameter   is a sequence of autoregressive with trend or (trend AR(p)) model and a sequence data at  = 1,2,3, ... with exponential white noise,  is an exponential smoothing parameter (0,1],   at  = 0 is the initial value of the EWMA statistics.Its mean equals and variance of   equals  2 (2−) . The mean () and standard deviation () can be used to characterize both the upper and lower control limits (UCL and LCL) and had a control width limit with  ̃ in Equation 2 as follows: The stopping time of the EWMA chart can be specified as   * = { ≥ 0:   > }.

2-2-The DEWMA Chart
Second, after Shamma & Shamma first suggested the DEWMA control chart in 1992 [10], Mahmoud & Woodall [11] developed it in 2010.It was explained from the EWMA control chart after being smoothed twice exponentially.The expression in Equation 3 below can be used to explain the statistics of the DEWMA control chart.
where the DEWMA chart parameter   is a sequence of autoregressive with trend (trend AR(p)) model and sequence data at  = 1,2,3, ... with exponential white noise,  1 and  2 are exponential smoothing parameters equals(0,1],   at  = 0 is the initial value of the DEWMA statistics.Its mean equals and variance of ].The mean () and standard deviation () can be used to characterize both the upper and lower control limits (UCL and LCL), and had a control width limit with D in Equation 4as follows: ].
The stopping time of the EWMA chart can be specified as:   = { ≥ 0:   > }.Additionally, the DEWMA statistic becomes the EWMA statistic if  1 = 1.

2-3-The CUSUM Chart
Third, Page (1959) designed the CUSUM chart for quality control, which can be used to spot small differences in process mean.The statistics of the CUSUM chart can be expressed using the algorithm in Equation 5 as follows: where is non-zero constant,  0 =  is the initial value of CUSUM;  ∈ [0,  ′ ] and the CUSUM chart's stopping time is described as   ′ = {  > 0;   > }.

2-4-The Trend AR(p) Model of DEWMA Chart
The two types of time-series data are steady data and non-stationary data.Gathering time-series data using stationary data does not reveal any trends or periodic effects.Non-stationary time-series data refers to time-series datasets that exhibit patterns or periodic impacts, unlike stationary time-series data that only contain random errors as a source of variance.Data points collected over time may contain internal structures (such as autocorrelation, trend, or seasonal fluctuation).Other measures, such as the moving average (MA(q)), the autoregressive moving average (ARMA(p,q)), and others, can also be used to describe a trend model.The trend AR(p) model, also known as the autoregressive with trend model, was examined in this paper.The trend autoregressive model for lag p, called trend AR (p), is written in Equation 6 as: where  is the constant of the model,  is a slop,

3-Methods and Measurement of Efficiency for Control Chart
For the DEWMA chart on underlying autoregressive with trend model, the initial value of ARL denoted (), and the initial value of the monitoring DEWMA statistic  0 =  represented at  ∈ [, ].As the result, the function () is given as = () =  ∞ (  ).Thus, where  ∞ (⋅)represents the expectation with the density function as (, ).Next, it can be used in the following section about process detecting.The change-point in model is considered as follows: Herein,  = ∞ is the in-control ARL (ARL0) and there has been no change in the statistical control process.In contrast,  = 1 denotes the first time point in the statistical control process when a change occurs from  0 to, which is referred to the out-of-control ARL (ARL1).

3-1-Analytical Explicit Formulas of the ARL for Trend AR(p) Model
This section solves the mathematically explicit formula for the ARL on the DEWMA chart using a trend autoregressive model with an exponential noise distribution.The LCL and UCL are both assumed to be equivalent to and , respectively.The explicit formula of the ARL is derived on the DEWMA chart with the trend AR(p) model.Let's start by substituting Equation 6 into Equation 3 as follows: where the first time  = 1 such that  0 =  is determined, then the initial values  0 =  and  −1 ,  −2 , . . .,  − equals 1.The following is a description of the DEWMA data with trend AR(p): In control process, the interval of  1 between the lower and upper bound control limits are expressed to be and can be written as follows below.The interval  1 between the lower and upper bound control limits, can be represented as follows: On the variable  1 , it is possible to rewrite this interval as: Next, the Fredholm integral equation is used to describe the integral equation of the ARL on the DEWMA chart for the trend AR(p) model with an initial value  0 = .The equation rearranged is Let () denote the ARL on the DEWMA chart for the trend AR(p) model.We use the second kind of Fredholm integral equation to solve the ARL [35].The formula is displayed in Equation 7as follows: Eventually, the function () expresses the error terms, or the function  1 , as an exponential distribution function.Hence, the following is a description of the function () in Equation 8: The fixed-point theorem of Banach is used to confirm the ARL solution.In terms of its existence and uniqueness, this is characterized as an ARL solution [36].From Equation 8, suppose that Therefore, the ARL solution that is obtained by Equation 8 can be rewritten that showed in Equation 9 as follows: Later, the integral equation , which can be expressed as: Finally, Equation 10, which is replaced in Equation 9, is substituted into the solution of , and the following result is obtained in Equation 11 as: As the trend AR(p) model is applied to the DEWMA chart, Equation 11 provides the explicit ARL formula.Moreover,  0 is used to replace the in-control process, while 1 ;  1 = (1 + ) 0 is used to replace the out-of-control process.And also,  stands for the shift size in the monitoring process.

3-2-Analytical NIE of the ARL for Trend AR(p) Model
This section solves the analytical NIE approach for the ARL on the DEWMA chart using a trend autoregressive model with an exponential noise distribution.Let  ̂() represent the midpoint quadrature rule-computed ARL of the DEWMA chart for the trend AR(p) model with an exponential white noise.In particular, we computed in terms of the m linear equation systems with the midpoint rule on the interval [, ], and this method was split into  ≤  1 ≤  2 ≤. . .≤   ≤  using a set of constant weights   = ( − )/ after using a quadrature rule.The approximation for an integral can be determined by applying the quadrature rule, which is represented in Equation 12below (12) The NIE approach;  ̂(  ), which is evaluated by a linear equation, had shown in Eq. ( 13), as follows Finally,   is instead of  into  ̂(  ), the NIE approximating of the ARL is rewritten in Equation 14as follows where   is the division point within the interval  ≤  1 ≤  2 ≤. . .≤   ≤  as well as  = ( − 0.5)  +  for  = 1,2, . . ., .And then, a weight of the composite midpoint formula is given as   = ( − )/

3-3-The Existence and Uniqueness of Exact ARL Solution
In this part, this study also uses Banach's fixed-point theorem to prove the ARL solution's existence and uniqueness because the explicit ARL formula must prove its existence and uniqueness.Let represent the operation in the class of all continuous functions, which is expressed as:

𝑑𝜐
Theorem 1 Banach's Fixed-point Theorem: Let (, ) and :  → represent a complete metric space and the contraction mapping, respectively.And then, T is referred to unique on fixed point.There exists a unique solution to the fixed point when (()) = () ∈ .

3-4-The Measurement of Efficiency for the Control Chart
The average run length (ARL) is a popular measurement used to assess the performance of control charts.The ARL, produced by explicit equations, and the NIE method, which uses an autoregressive with trend process, or trend AR process, to detect changes, were employed to compare their findings on the DEWMA chart.The percentage accuracy (%Acc) indicates the relative efficiency of two methods of the ARL, which is given in Equation 15 as follows: Next, the efficiency of the ARL is calculated with different parameter values based on the DEWMA chart.It is then used for comparison with the EWMA and CUSUM charts.In addition, some other characteristics of the run length (RL) exist, namely the standard deviation run length (SDRL) and the median run length (MRL).Those are additional metrics for the evaluation of control charts.The equations in Equation 16 are used to determine SDRL and MRL for in-control [26]: where type I error represents  = 1 − ( <   < | 0 ).In this study, ARL0 was fixed at 370.Form an ARL0 value that can be calculated as SDRL0 and MRL0 by Equation 16at approximately 370 and 256, respectively.On the other hand, SDRL and MRL are calculated by the formulas in Equation 17for out-of-control situations.The values obtained by those formulas are the lowest; the result indicates that control charts provide the best performance [26] where type II error represents  = 1 − ( <   < | 1 ).Moreover, when shift sizes differ, using the ARL measurement to assess how well control charts affect the process is reasonable.Many studies recommend using overall performance metrics to evaluate a control chart's success during different changes (  ).Some of them feature performance measurements, including the average extra quadratic loss (AEQL) and the performance comparison index (PCI), that are used to evaluate their effectiveness [37].
The mathematical formula for the AEQL is: )   ∑ =  (18) where  is the particular change in the process, and ∆ is the sum of number of divisions from   to   .In this study,  = 7 is determined from   to   .The control charts with the lowest AEQL values perform the best.
The PCI measurement is the ratio between the AEQLs 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 PCI value of the most efficient control chart is 1, while the less efficient control chart will give a PCI value greater than 1.

3-5-The Procedure of Analytical Results of the ARL
he ARL for spotting changes in the process is a standard metric for evaluating a control chart's effectiveness.The effectiveness of the explicit formula and the NIE approach for calculating the ARL for monitoring shifts were evaluated in this study.The DEWMA chart running on the trend AR models, specifically the trend AR(1), trend AR(2), and trend AR(3), was used to evaluate the performance with exponential white noise.The residual of an exponential distribution with uncorrelated data is what is known as "exponential white noise," as was previously mentioned.As a consequence,  = 0 represents an in-control process, while  > 0 represents an out-of-control process.The NIE approach is used to determine the number of division points, m = 500, using the ARL approximation.The NIE approach is used to determine the number of division points, m = 500, using the ARL approximation.The explicit ARL and the NIE approach to the ARL were computed using the Mathematica program.Through research, the Intel(R) Xeon(R) CPU X5680 @ 3.33 GHz (3 processors) RAM 32.0 GB specification for Mathematica was evaluated.The following is a brief description of the procedure: Step 1: Give the input parameters, such as the coefficients of autoregressive (  ), the initial values of the autoregressive;  −1 ,  −2 , . . .,  − , and the control chart parameters set as  1 = 0.05,0.10, 2 = 0.05.
Step 2: Determine the initial value of known parameter (), slope value of the trend AR(p) model () and the initial value of the DEWMA statistic ( 0 =  and 0 = ).
Step 3: Impose the parameter of exponential white noise;  =  0 for the in-control process.
Step 4: Specify the lower control limit; = 0 and fixed the ARL value of in-control equals 370 to compute the upper control limit ().
Step 5: Define the upper control limit () from Step 4. Compute the ARL values of out-of-control by the explicit formula and the NIE method, determine parameter of exponential white noise ( =  1 ) where  1 =  0 (1 + ), and change shift sizes () equals 0.001, 0.002, 0.01, 0.02, 0.1, 0.5, and 1, respectively.Also, solutions can be found using the approach depicted in Figure 1.

4-1-The Simulated Results
For the in-control scenario, the simulated data is frequently given with ARL0 = 370, allowing the beginning parameters to be explored at  0 = 1.On the other hand,  1 =  0 (1 + ) is researched in the out-of-control scenario and computed to determine shift sizes.At the interval [0, b], the lower and higher control limits are investigated.First, the ARL was evaluated using the explicit formula and the NIE method, and the capability of the methods was compared with the %Acc.The results were computed based on specified known parameters, such as  1 = 0.1,  2 = 0.2,  3 = 0.3,  = 0, and  = 0.5 .There are three models, namely the trend AR(1), trend AR(2), and trend AR(3) models, that were determined with exponentially smoothing parameters as follows in Table 1.The explicit formula technique's ARL values, represented as (), are calculated using Equation 11.The  ̂() is then indicated using the NIE approach and is computed using Equation 14.All circumstances have a very high percentage of acceptance, about 100.However, the explicit formula appears very instantly in all circumstances, but the ARL values obtained using the NIE approach take roughly 3-4 seconds to compute.As a result, it makes sense to proceed with these precise formulations.The comparative results of accuracy and computational speed on the DEWMA chart are consistent with the research of Areepong & Peerajit (2022) [23], which obtained an explicit formula for ARL in the CUSUM chart.Then it's consistent with the research of Karoon et al. [33], which offers an explicit ARL solution for detecting changes on EEWMA with Trend's AR(p) process and validating the proposal with ARL's NIE method.Obviously, it takes almost no computation time compared to the NIE method.Next, the capability of the explicit ARL based on the DEWMA chart running on the trend AR(p) model is compared with the EWMA and CUSUM charts and then investigated using different .For out-of-control, the results for contrasting capability between the DEWMA, EWMA, and CUSUM charts based on different situations are shown in Tables 2 and 3.The results showed that the DEWMA chart obtained lower ARL1, SDRL1, and MRL1 values than the EWMA and CUSUM charts in all situations for both the trend AR(1) and trend AR(2) models, whereas the RL results based on shift sizes that are defined as  ≥ 0.1show that the results of the EWMA chart are very close to the results of the DEWMA chart.In addition, the AEQL and PCI values are also utilized to validate their efficacy.The ARL1 values of all charts were used for calculating the AEQL and PCI values, which were computed from Equations 18 and 19, respectively.The results show that the AEQL values of the DEWMA chart are lower than the AEQL values of the EWMA and CUSUM charts, and the PCI values of the DEWMA chart with = 0 .05 are equal to 1, just as they are for the trend AR(1) and trend AR(2) models.From the research results mentioned above, the performance of DEWMA shows superior performance in detecting transition changes compared to EWMA and CUSUM charts, and the smaller the exponential smoothing parameter, the greater the capabilities of the DEWMA chart.The findings are consistent with previously presented studies showing that the explicit formula of the ARL generalized modified EWMA-type was more effective than the original EWMA; see [21,24].

4-2-The Real-World Datasets
Two applications, the prices of Bitcoin and Ethereum, are brought up for analysis in this section, utilizing daily datasets and the trend AR(1) and trend AR(2) models, respectively.Those were fitted as the models by SPSS.The results for two datasets that are suitable inputs for the trend AR(p) model and display suitable parameters are shown in Table 4.The significance of white noise's fit to the exponential mean was then determined using the one-sample Kolmogorov-Smirnov test, as shown in Table 5.For application 1, daily data of the prices of Bitcoin (unit: 1,000 USD) from December 16, 2022, to March 5, 2023.It was fitted to the trend AR (1), which expressed itself as   = 16.946+ 0.085 + 0.948 −1 +   , where   ∼ ( 0 = 0.4357).
The ARL1 values for the DEWMA (different ; 0.05, 0.10), EWMA ( 1 = 1), and CUSUM charts are displayed in Table 6.According to the findings of the control chart comparison, the DEWMA chart with the lower  1 had a lower ARL1 and performed better than the EWMA chart in every scenario.In addition, to verify performance, the AEQL and PCI values were also used in the same way as the simulated results above.The results show that the AEQL value of the DEWMA chart is lower than the AEQL values of the EWMA and CUSUM charts, and the PCI value of the DEWMA chart has  1 equal to 1, as in the simulated data above.As a result, the results indicate that the outcomes of two applications with underlying trend AR(1) and AR(2) models are similar to simulated data, as illustrated in Figure 2.And then, the AEQL and PCI values supported the control chart's effectiveness by using ARL1 values in the formulas mentioned above.The outcomes demonstrate that, as shown in Figure 3, the DEWMA chart with equal to 0.05 outperformed the DEWMA chart with greater  1 , the EWMA and CUSUM charts, all of which had higher AEQL and PCI > 1, by having the lowest AEQL and PCI equal to 1.After that, Figure 4 further demonstrates how well the control chart functions to identify shift changes during the monitoring process.And also, the performance of the CUSUM chart shown above indicates that CUSUM charts are significantly less effective than EWMA types such as DEWMA and EWMA charts.The CUSUM chart is therefore not shown in the detection diagram section of Figure 4.In this part, the DEWMA with  and  equals 0.10 and 0.05, that was compared to EWMA chart.While the DEWMA chart for Application 1 acknowledges shifts as being out of control at the 8 th observation, the EWMA chart does so at the 14 th observation.While, the DEWMA chart for Application 2 acknowledges shifts as being out of control at the 12 th observation, but the EWMA chart does so at the 27 th observation.According to the findings, the double EWMA control chart may be able to identify shift changes more quickly than the EWMA control chart throughout the monitoring process.Therefore, an excellent option for spotting change is the DEWMA chart, which performs the best under the situations in this study.However, this finding was shown in cases where the data were only underlying a trend autoregressive process.If the data are from another process that is not a trend autoregressive model, it may not be appropriate to use this proposed ARL on the DEWMA chart for monitoring the process mean changes and may need to be studied further.

5-Conclusion
The exact ARL solution, compared in performance to the NIE approach's ARL, was found to help reduce processing time and was used to assess the sensitivity of DEWMA charts based on the running trend AR(p) model with exponential white noise.After that, the explicit ARL on running the DEWMA chart was compared to the EWMA and CUSUM charts under the out-of-control process with different shift sizes (compared by using ARL1, SDRL1, and MRL1 values).Next, the sensitivity of control charts is verified by two measures, such as AEQL and PCI.According to the results, the DEWMA chart has the highest performance, and when  was small, the DEWMA chart had high sensitivity for detecting processes.Additionally, actual data can be used to apply the exact ARL solutions, producing results that are identical to those of simulated data.Real-world data following the AR(p) trend model with exponential distribution white noise might be analyzed using these formulas, as might the prices of digital currencies such as Bitcoin and Ethereum, both of which were used.As a result, the explicit formula was a good method for determining the ARL for shift changes based on observations in the DEWMA chart to use a precise ARL solution, which improved the sensitivity of the DEWMA chart for parameter shift detection.In this study, the proposed explicit formula may have some limitations.It works very well only for data that is characterized as being autocorrelated with an autoregressive model.However, this research is a good starting point to further improve the sensitivity of detecting small process changes under various data formats in future scenarios.Last, future studies will be conducted on the explicit ARL formulas on the DEWMA chart to appropriate them with the other model of real data.Also, we will derive the explicit formula on modern control charts using this approach to improve their efficacy for detecting change in different situations.

6-2-Data Availability Statement
Publicly available datasets were analyzed in this study.These datasets to be Bitcoin and Ethereum prices can be found here: https://coinmarketcap.com.

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-66-04.

6-4-Acknowledgements
The authors are grateful to the referees for their constructive comments and suggestions which helped to improve this research.DEWMA Chart

Table 1 . The ARL values of the explicit formula against the NIE method for trend AR(p) models on the DEWMA chart with known parameters,
= .,   = .,   = .,   = .,  = , and  = . under different conditions

Table 3 . The ARL1 values of the explicit formula for trend AR(2) model on the DEWMA, EWMA, and CUSUM charts with known
parameters,   = .,  = , and  = . under different conditions.