Available online at www.HighTechJournal.org HighTech and Innovation Journal Vol. 5, No. 4, December, 2024 901 ISSN: 2723-9535 Performance Evaluation of Extended EWMA Chart for AR Model with Exogenous Variables Totsaporn Muangngam 1, Yupaporn Areepong 1* , Saowanit Sukparungsee 1 1 Department of Applied Statistics, Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok, Bangkok, 10800, Thailand. Received 25 August 2024; Revised 12 November 2024; Accepted 18 November 2024; Published 01 December 2024 Abstract The extended exponentially weighted moving average (Extended EWMA) control chart is an effective statistical process control method for monitoring and identifying shifts in process mean, particularly when dealing with autocorrelated data. One key performance measure used to evaluate the capability of control charts in detecting changes is the average run length (ARL). The primary goal of this study is to present the explicit formulas for calculating the ARL of the extended EWMA control chart for autoregressive models with exogenous variables (ARX) and exponential white noise. Another purpose is to compare the performance of the extended EWMA and the classical EWMA control charts under various conditions. The explicit formulas are derived from the ARL integral equation, which is expressed by the Fredholm integral equation. The accuracy of the exact solutions has been verified using the numerical integral equation (NIE) methods that employ four different composite quadrature rules. The result shows that the ARL values obtained from both methods are similar, and the computation time for the proposed explicit formulas is less than 0.001 second. In comparing the two control charts, it is evident that the extended EWMA control chart outperforms the traditional control chart in detecting shifts in the process mean, as confirmed by various overall performance criteria. Additionally, two real datasets, namely SCB stock price and GDP percentage expansions, are applied to demonstrate the effectiveness of the relevant control charts. Keywords: Average Run Length; Extended EWMA Chart; Explicit Formula; Autoregressive with Exogenous Variables. 1. Introduction Statistical Process Control (SPC) is a powerful set of problem-solving methods primarily used in the manufacturing industry to maintain and improve the quality of processes and products by reducing variability. A key visual tool in SPC is the control chart, which is extensively used to monitor process stability and detect special-cause variations or unnatural shifts in process parameters, such as mean and variance. These shifts can lead to the production process becoming out of control. The faster a control chart responds to changes, the quicker the process can be addressed and brought back into a controlled state. The concept of the traditional Shewhart chart, introduced by Shewhart [1], is classified as a memory-less control chart. One significant limitation of memory-less charts is their ineffectiveness in detecting minor changes. To overcome this, memory-type control charts, such as the cumulative sum (CUSUM) control chart [2] and the exponentially weighted moving average (EWMA) control chart [3], were developed to rapidly identify small to moderate variations in processes. Subsequently, several researchers have proposed enhanced control charts. For instance, Patel & Divecha [4] presented the modified exponentially weighted moving average (MEWMA) control chart to detect small shifts in process mean, and Khan et al. [5] improved upon this with a generalized form of MEWMA. Abbas et al. [6] combined the CUSUM and EWMA control charts, demonstrating that the mixed CUSUM-EWMA control chart performs better than either individual chart. In 2018, Naveed et al. [7] developed a new design for an EWMA-based * Corresponding author: yupaporn.a@sci.kmutnb.ac.th http://dx.doi.org/10.28991/HIJ-2024-05-04-03 οƒ˜ This is an open access article under the CC-BY license (https://creativecommons.org/licenses/by/4.0/). Β© Authors retain all copyrights. https://creativecommons.org/licenses/by/4.0/ https://orcid.org/0000-0002-5103-9867 https://orcid.org/0000-0001-5248-8173 HighTech and Innovation Journal Vol. 5, No. 4, December, 2024 902 statistic called the extended EWMA control chart, which showed greater sensitivity in monitoring small changes compared to the classical EWMA control chart. The extended EWMA scheme has since been studied to assess its performance in various simulated and practical situations [8, 9]. The fundamental assumption underlying traditional control charts is that the observations are independent and identically distributed. However, in real applications, successive samples from many processes are often dependent on time intervals and exhibit serial correlation. This dependence can negatively impact the performance of standard control charts, leading to incorrect indications and conclusions [10, 11]. To address the issue of autocorrelated data, researchers have investigated alternative strategies. One such approach involves fitting an appropriate time series model and then applying the uncorrelated property of the residuals, or white noise process, in the statistical control chart procedure [12, 13]. The time series autoregressive (AR) and moving average (MA) models, as well as the models comprising AR and MA, are usually employed for modeling and predicting autocorrelated data that depend on themselves. In some cases, the independent factors can impact the behavior of the process, improving prediction accuracy. Consequently, the time series models incorporated with the exogenous variables, such as ARX, MAX, or ARMAX, have been studied across various fields [14]. Regarding the random error of time series models known as white noise, which usually follows normal distribution, however, the white noise can also exhibit an exponential distribution [15]. Generally, the performance and sensitivity of control charts for monitoring and detecting variation in the process are typically assessed through the average run length, or ARL. This measure indicated the expected number of in-control observations before an out-of-control signal is detected. There are two types of ARL; ARL0 refers to the average run length for an in-control process with no changes. Ideally, this value should be large, indicating that the control chart is stable and effective. ARL1 denotes the average run length when the process is out of control and reflects the detection capabilities for various magnitudes of shifts. A smaller value ARL1 is desirable, as it shows that the control chart can quickly identify any out- of-control conditions. Evaluating the ARL values is a crucial aspect when studying and developing control charts, as it allows for comparison of their capability. Previous research has employed different methodologies to calculate ARL values. For example, Champ & Rigdon [16] studied and compared the Markov Chain and the numerical integral equation (NIE) method for calculating the ARL of quality control charts. Naveed et al. [7] utilized Monte Carlo simulation for assessing the ARL and the proposed extended EWMA control chart. Nevertheless, the mentioned approaches can be time-consuming in terms of calculations. Several researchers have investigated the derivation of the Average Run Length (ARL) integral equation under conditions of autocorrelation, particularly when the white noise process follows an exponential distribution, leading to the establishment of explicit formulas. Paichit [17] derived an exact solution for the ARL of the Cumulative Sum (CUSUM) control chart for an Autoregressive (AR) process with one exogenous variable (ARX(1)), where the white noise is characterized by an exponential distribution. The accuracy of the ARL was confirmed with Numerical Integration Evaluation (NIE) using the Gauss-Legendre rule, showing excellent agreement. Phanyaem [18] also presented an explicit formula for the ARL, comparing the accuracy of this formula against the NIE method using different quadrature rules for the CUSUM control chart when the observations belong to a seasonal ARX model with exponential white noise. The ARL from the explicit formula closely matched the NIE results, with an absolute percentage difference of less than 1%. Suriyaket & Phetcharat [19] developed an explicit formula for the ARL of the Maximum (MAX) process operating on an Exponentially Weighted Moving Average (EWMA) chart using techniques from Fredholm integral equations. They applied numerical integration methods, including Gaussian, midpoint, and trapezoidal rules, to verify the accuracy of the explicit formula. The results indicated that the ARL derived from their proposed method approximated the NIE results and outperformed the numerical methods in terms of computational time. Supharakonsakun [20] derived the ARL for a modified EWMA control chart applied to a Seasonal Moving Average (SMA) of order q (SMA(q)), where the white noise is exponentially distributed. The findings showed good agreement between the explicit formula and the numerical integral equation method. Karoon et al. [21] explored explicit formulas for the ARL of an extended EWMA control chart designed for a trend AR(p) model, comparing its accuracy to that of the NIE method. Zhang et al. [22] formulated explicit expressions for the ARL and Average Delay Time (ADT) for the CUSUM control chart associated with a seasonal SMA(Q)s model. The performance comparison between the results obtained from the explicit formulas and the numerical integration approach indicated that the explicit formulas considerably reduced computational time. Recently, Peerajit [23] introduced an analytical solution for calculating the ARL of a long-memory ARFIMA(1, d, 1)(1, D, 1)s process with exponential white noise operating on a CUSUM control chart. The NIE method was employed to verify the accuracy of this proposed approach. The results from both methods were in close agreement, but the time required for computing the ARL using the proposed method was significantly shorter. Sunthornwat et al. [24] suggested explicit formulas for the ARL of the Homogeneously Weighted Moving Average (HWMA) control chart based on an AR process. Phanthuna et al. [25] examined the explicit formula for the ARL of a double- modified exponentially weighted moving average (DMEWMA) control chart applied to an AR process. They compared the ARLs computed using the explicit formula and numerical integral equation method to validate the former. Finally, Phanyaem [26] developed an exact formula for computing the average run length in an EWMA control chart, specifically for a SARX(P,r)L model. The results showed that the average run length calculated using the proposed method is close with from the numerical integral equation method. HighTech and Innovation Journal Vol. 5, No. 4, December, 2024 903 According to the efficiency of the extended EWMA control chart and the approaches to calculate the ARL that are mentioned above, we are interested in deriving the ARL integral equation of the extended EWMA control chart when the observation is in the pattern of AR process with an exogenous variable that has not been proposed before. Therefore, the aim of this study is to derive the explicit formulas of the ARL on the extended EWMA chart for the ARX model when the white noise follows an exponential distribution and compare it to the numerical integral equation method in four different composite quadrature rules. Moreover, the comparison of the sensitivity of detecting changes of the extended EWMA and the EWMA control chart is conducted under various conditions. The two real datasets are studied to assess the proposed explicit formula for the control charts and presented in this article. 2. Preliminaries The definitions of the time series model and the control charts, including their characteristics, are given in this section. 2.1. Time Series ARX Model Let π‘Œπ‘‘ be a sequence observation from the ARX(p,r) model defined as: π‘Œπ‘‘ = πœ‡ + βˆ‘ πœ™π‘–π‘Œπ‘‘βˆ’π‘– 𝑝 𝑖=1 + νœ€π‘‘ + βˆ‘ 𝛽𝑗𝑋𝑗𝑑 π‘Ÿ 𝑗=1 , 𝑑 = 1,2,3, . .. (1) where 𝝁 is a constant, π“π’Š ∈ (βˆ’πŸ, 𝟏) is an autoregressive coefficient, 𝑿𝒋𝒕 is an exogenous variable, πœ·π’‹ is a coefficient of 𝑿𝒋𝒕 and πœΊπ’• is an error term or a white noise process assumed to follow the exponentially distributed, therefore, πœΊπ’• ∼ 𝑬𝒙𝒑(𝜢). 2.2. Extended EWMA Control Chart The extended EWMA statistic improved by Naveed et al. [7] can be defined by the recursive equation: 𝐸𝑑 = πœ†1π‘Œπ‘‘ βˆ’ πœ†2π‘Œπ‘‘βˆ’1 + (1 βˆ’ πœ†1 + πœ†2)πΈπ‘‘βˆ’1, 𝑑 = 1, 2, 3, …. (2) where π‘Œπ‘‘ is a sequence observation from the ARX process, πœ†1 ∈ (0,1] and πœ†2 ∈ [0, πœ†1) are smoothing constants. The upper control limit (UCL) and the lower control limit (LCL) are: UCL=πœ‡0 + πœ”πœŽβˆš πœ†1 2 + πœ†2 2 βˆ’ 2πœ†1πœ†2(1 βˆ’ πœ†1 + πœ†2) 2(πœ†1 βˆ’ πœ†2) βˆ’ (πœ†1 βˆ’ πœ†2)2 LCL=πœ‡0 βˆ’ πœ”πœŽβˆš πœ†1 2 + πœ†2 2 βˆ’ 2πœ†1πœ†2(1 βˆ’ πœ†1 + πœ†2) 2(πœ†1 βˆ’ πœ†2) βˆ’ (πœ†1 βˆ’ πœ†2)2 (3) where πœ‡0 is a target mean, 𝜎 is a process standard deviation and πœ” is an appropriate control limit width. The stopping time for the extended EWMA control chart is πœπ‘Ž,𝑏 = 𝑖𝑛𝑓{ 𝑑 > 0: 𝐸𝑑 < π‘Ž βˆͺ 𝐸𝑑 > 𝑏} where π‘Ž and 𝑏 represent the LCL and UCL, respectively. The extended EWMA control chart converts to the classical EWMA scheme, 𝑍𝑑 = πœ†1π‘Œπ‘‘ + (1 βˆ’ πœ†1)π‘π‘‘βˆ’1 when πœ†2 = 0. Similarly, the LCL (π‘Žβ€²) and UCL (𝑏′) of the EWMA control chart can be determined by equation (3) whenπœ†2 = 0 with constant width πœ” = πœ”π‘, therefore, the stopping time for the EWMA control chart is πœπ‘Žβ€²,𝑏′ = 𝑖𝑛𝑓{ 𝑑 > 0: 𝑍𝑑 < π‘Žβ€² βˆͺ 𝑍𝑑 > 𝑏′}. 2.3. Average Run Length Let νœ€π‘‘ , 𝑑 = 1,2, . .. be a sequence of independent random variables with a probability density function 𝑓(𝑀, 𝛼) where 𝛼is the parameter. The in-control state is normally with the parameter 𝛼 = 𝛼0 and assumed that there is no change in the process. On the contrary, the parameter 𝛼 = 𝛼1 when the process has changed to out-of-control state at the change-point time, πœ‘. Average run length or ARL is the common characteristic of control charts to measure and compare their performance in detecting changes in parameters. Ideally, the ARL for in-control process denoted as 𝐴𝑅𝐿0 are required to be sufficiently large in order to reduce the number of false out-of-control signals, whereas, the ARL for out-of-control state or 𝐴𝑅𝐿1 must be small to quickly detect a correct out-of-control signal. In this study, the stopping time (πœπ‘Ž,𝑏) are used as the alarm signals, therefore, the ARL is defined as: 𝐴𝑅𝐿 = { 𝐴𝑅𝐿0 = οΏ½Μ‚οΏ½πœ‘(πœπ‘Ž,𝑏), πœ‘ = ∞ 𝐴𝑅𝐿1 = οΏ½Μ‚οΏ½πœ‘(πœπ‘Ž,𝑏|πœπ‘Ž,𝑏 β‰₯ 1), πœ‘ = 1 (4) HighTech and Innovation Journal Vol. 5, No. 4, December, 2024 904 where οΏ½Μ‚οΏ½πœ‘ is the expectation of the stopping time under the assumption that the change-point time occur at time, πœ‘. ARL0 represents the in-control ARL which implies that the change-point time does not exist, whereas ARL1 denotes the out- of-control ARL when the change point appears at the first time. 3. ARL Evaluation Methods In this section, the explicit formulas of ARL for the extended EWMA control chart of the ARX(p,r) model with exponential white noise derived from a Fredholm integral equation of the second kind are presented. Moreover, an approximated ARL from the numerical integral equation (NIE) method is used to confirm the accuracy of the exact solutions. Here, the in-control state of extended EWMA scheme for ARX(p,r) model can be rewritten in the form of white noise process νœ€π‘‘ as: [ π‘Žβˆ’(πœ†1πœ™1βˆ’πœ†2)π‘Œ0βˆ’(1βˆ’πœ†1+πœ†2)𝜈 πœ†1 βˆ’ πœ‡ βˆ’ βˆ‘ πœ™π‘–π‘Œπ‘‘βˆ’π‘– 𝑝 𝑖=2 βˆ’ βˆ‘ 𝛽𝑗𝑋𝑗𝑑 π‘Ÿ 𝑗=1 ] < νœ€π‘‘ < [ π‘βˆ’(πœ†1πœ™1βˆ’πœ†2)π‘Œ0βˆ’(1βˆ’πœ†1+πœ†2)𝜈 πœ†1 βˆ’πœ‡ βˆ’ βˆ‘ πœ™π‘–π‘Œπ‘‘βˆ’π‘– 𝑝 𝑖=2 βˆ’ βˆ‘ 𝛽𝑗𝑋𝑗𝑑 π‘Ÿ 𝑗=1 ] (5) where 𝜈 is an initial value of the extended EWMA control chart and π‘Œ0 is an initial value of ARX(p,r) model. Let πœ•(𝜈) be an ARL of an initial value 𝜈 which can be described by applying the Fredholm integral equation second kind according to the method of Champ & Rigdon [16] of as follows: πœ•(𝜈) = 1 + ∫ πœ•(𝐸𝑑) π‘βˆ’(πœ†1πœ™1βˆ’πœ†2)π‘Œ0βˆ’(1βˆ’πœ†1+πœ†2)𝜈 πœ†1 βˆ’πœ‡βˆ’βˆ‘ πœ™π‘–π‘Œπ‘‘βˆ’π‘– 𝑝 𝑖=2 βˆ’βˆ‘ 𝛽𝑗𝑋𝑗𝑑 π‘Ÿ 𝑗=1 π‘Žβˆ’(πœ†1πœ™1βˆ’πœ†2)π‘Œ0βˆ’(1βˆ’πœ†1+πœ†2)𝜈 πœ†1 βˆ’πœ‡βˆ’βˆ‘ πœ™π‘–π‘Œπ‘‘βˆ’π‘– 𝑝 𝑖=2 βˆ’βˆ‘ 𝛽𝑗𝑋𝑗𝑑 π‘Ÿ 𝑗=1 𝑓(νœ€π‘‘)π‘‘νœ€π‘‘ (6) After setting new variables, then, the πœ•(𝜈) can be defined as πœ•(𝜈) = 1 + 1 πœ†1 ∫ πœ•(𝑀)𝑓 ( π‘€βˆ’(πœ†1πœ™1βˆ’πœ†2)π‘Œ0βˆ’(1βˆ’πœ†1+πœ†2)𝜈 πœ†1 βˆ’ πœ‡ βˆ’ βˆ‘ πœ™π‘–π‘Œπ‘‘βˆ’π‘– 𝑝 𝑖=2 βˆ’ βˆ‘ 𝛽𝑗𝑋𝑗𝑑 π‘Ÿ 𝑗=1 ) 𝑑𝑀 𝑏 π‘Ž (7) White noise process is assumed to be random variables exponentially distributed so that the pdf is 𝑓(𝑀) = 1 𝛼 𝑒 βˆ’π‘€ 𝛼 , therefore, the ARL can be rearranged as the following equation: πœ•(𝜈) = 1 + 1 πœ†1 ∫ πœ•(𝑀)𝑒 βˆ’π‘€+(πœ†1πœ™1βˆ’πœ†2)π‘Œ0+(1βˆ’πœ†1+πœ†2)𝜈 π›Όπœ†1 + πœ‡+βˆ‘ πœ™π‘–π‘Œπ‘‘βˆ’π‘– 𝑝 𝑖=2 +βˆ‘ 𝛽𝑗𝑋𝑗𝑑 π‘Ÿ 𝑗=1 𝛼 𝑑𝑀 𝑏 π‘Ž (8) The exact and the approximated solutions of the integral Equation 8 are revealed in the next subsections. 3.1. The Proposed Explicit Formula From the integral Equation 8, we can rewrite in Equation 9: πœ•(𝜈) = 1 + 𝐢(𝜈) π›Όπœ†1 𝐷 (9) where 𝐢(𝜈) = 𝑒 (πœ†1πœ™1βˆ’πœ†2)π‘Œ0+(1βˆ’πœ†1+πœ†2)𝜈 π›Όπœ†1 + πœ‡+βˆ‘ πœ™π‘–π‘Œπ‘‘βˆ’π‘– 𝑝 𝑖=2 +βˆ‘ 𝛽𝑗𝑋𝑗𝑑 π‘Ÿ 𝑗=1 𝛼 and 𝐷 = ∫ πœ•(𝑀)𝑒 βˆ’π‘€ π›Όπœ†1𝑑𝑀 𝑏 π‘Ž . Then, consider the variable D in this form; 𝐷 = ∫ (1 + 𝐢(𝑀) π›Όπœ†1 𝐷) 𝑒 βˆ’π‘€ π›Όπœ†1𝑑𝑀 𝑏 π‘Ž = ∫ 𝑒 βˆ’π‘€ π›Όπœ†1𝑑𝑀 𝑏 π‘Ž + 𝐷 π›Όπœ†1 ∫ 𝑒 (πœ†1πœ™1βˆ’πœ†2)π‘Œ0+(1βˆ’πœ†1+πœ†2)𝑀 π›Όπœ†1 + πœ‡+βˆ‘ πœ™π‘–π‘Œπ‘‘βˆ’π‘– 𝑝 𝑖=2 +βˆ‘ 𝛽𝑗𝑋𝑗𝑑 π‘Ÿ 𝑗=1 𝛼 βˆ’ 𝑀 π›Όπœ†1 𝑏 π‘Ž 𝑑𝑀 = βˆ’π›Όπœ†1(𝑒 βˆ’π‘ π›Όπœ†1βˆ’π‘’ βˆ’π‘Ž π›Όπœ†1) 1+ 1 πœ†1βˆ’πœ†2 𝑒 (πœ†1πœ™1βˆ’πœ†2)π‘Œ0 π›Όπœ†1 + πœ‡+βˆ‘ πœ™π‘–π‘Œπ‘‘βˆ’π‘– 𝑝 𝑖=2 +βˆ‘ 𝛽𝑗𝑋𝑗𝑑 π‘Ÿ 𝑗=1 𝛼 (𝑒 βˆ’(πœ†1βˆ’πœ†2)𝑏 π›Όπœ†1 βˆ’π‘’ βˆ’(πœ†1βˆ’πœ†2)π‘Ž π›Όπœ†1 ) (10) After substituting D into Equation 9, we finally have the explicit formula for in-control state as, πœ•(𝜈) = 1 βˆ’ (πœ†1βˆ’πœ†2)(𝑒 βˆ’π‘ 𝛼0πœ†1βˆ’π‘’ βˆ’π‘Ž 𝛼0πœ†1)𝑒 (1βˆ’πœ†1+πœ†2)𝜈 𝛼0πœ†1 (πœ†1βˆ’πœ†2)𝑒 βˆ’(πœ†1πœ™1βˆ’πœ†2)π‘Œ0 𝛼0πœ†1 𝑒 βˆ’(πœ‡+βˆ‘ πœ™π‘–π‘Œπ‘‘βˆ’π‘– 𝑝 𝑖=2 +βˆ‘ 𝛽𝑗𝑋𝑗𝑑 π‘Ÿ 𝑗=1 ) 𝛼0 +(𝑒 βˆ’(πœ†1βˆ’πœ†2)𝑏 𝛼0πœ†1 βˆ’π‘’ βˆ’(πœ†1βˆ’πœ†2)π‘Ž 𝛼0πœ†1 ) (11) and the out-of-control exact solution for ARL as; HighTech and Innovation Journal Vol. 5, No. 4, December, 2024 905 πœ•(𝜈) = 1 βˆ’ (πœ†1βˆ’πœ†2)(𝑒 βˆ’π‘ 𝛼1πœ†1βˆ’π‘’ βˆ’π‘Ž π›Όπœ†1)𝑒 (1βˆ’πœ†1+πœ†2)𝜈 𝛼1πœ†1 (πœ†1βˆ’πœ†2)𝑒 βˆ’(πœ†1πœ™1βˆ’πœ†2)π‘Œ0 𝛼1πœ†1 𝑒 βˆ’(πœ‡+βˆ‘ πœ™π‘–π‘Œπ‘‘βˆ’π‘– 𝑝 𝑖=2 +βˆ‘ 𝛽𝑗𝑋𝑗𝑑 π‘Ÿ 𝑗=1 ) 𝛼1 +(𝑒 βˆ’(πœ†1βˆ’πœ†2)𝑏 𝛼1πœ†1 βˆ’π‘’ βˆ’(πœ†1βˆ’πœ†2)π‘Ž 𝛼1πœ†1 ) (12) Furthermore, Banach’s fixed point theorem from mathematical analysis is utilized to confirm the existence and uniqueness of the proposed explicit formula which is the solution to the ARL integral equation. Definition 1. Let (𝑆, π›₯)be a metric space. An operator 𝑀: 𝑆 β†’ 𝑆is a contractive mapping or a contraction if there exists a constant πœ… ∈ (0,1)such that π›₯(𝑀(𝑠1), 𝑀(𝑠2)) ≀ πœ…π›₯(𝑠1, 𝑠2)for all 𝑠1,𝑠2in 𝑆. Theorem 1. Banach’s fixed point theorem: Let (𝑆, π›₯)be a complete metric space and 𝑀: 𝑆 β†’ 𝑆be a contraction on 𝑆. Then 𝑀 has a unique fixed point such that 𝑀(𝑠) = 𝑠, 𝑠 ∈ 𝑆. In this present work, we consider the ARL Equation 8 in the set of all continuous function denoted by 𝐢[π‘Ž, 𝑏]. Consequently, the space (𝐢[π‘Ž, 𝑏], β€–. β€–βˆž) is complete with a norm given by β€–πœ•(𝑣)β€–βˆž = 𝑠𝑒𝑝 π‘£βˆˆ[π‘Ž,𝑏] |πœ•(𝑣)|. The following theorem and its proof provide the second condition of Theorem 1 which can imply that the ARL integral equation has an only one solution. Theorem 2. Let 𝑀: 𝐢[π‘Ž, 𝑏] β†’ 𝐢[π‘Ž, 𝑏] be an operator defined as, 𝑀(πœ•(𝜈)) = πœ•(𝜈) = 1 + 1 π›Όπœ†1 ∫ πœ•(𝑀)π‘˜(𝜈, 𝑀)𝑑𝑀 𝑏 π‘Ž (14) where π‘˜(𝑣, 𝑀) = 𝑒 βˆ’π‘€+(πœ†1πœ™1βˆ’πœ†2)π‘Œ0+(1βˆ’πœ†1+πœ†2)𝑣 π›Όπœ†1 + πœ‡+βˆ‘ πœ™π‘–π‘Œπ‘‘βˆ’π‘– 𝑝 𝑖=2 +βˆ‘ 𝛽𝑗𝑋𝑗𝑑 π‘Ÿ 𝑗=1 𝛼 is a kernel function. Then, 𝑀 is a contraction. Proof. Let πœ•1(𝜈) and πœ•2(𝜈) are two arbitrary functions in 𝐢[π‘Ž, 𝑏], then, consider; ‖𝑀(πœ•1(𝑣)) βˆ’ 𝑀(πœ•2(𝑣))β€–βˆž = 𝑠𝑒𝑝 π‘£βˆˆ[π‘Ž,𝑏] |∫ |πœ•1(𝑣) βˆ’ πœ•2(𝑣)|𝑑𝑀 𝑏 π‘Ž | ≀ 𝑠𝑒𝑝 π‘£βˆˆ[π‘Ž,𝑏] ∫ |π‘˜(𝑣, 𝑀)| 𝑏 π‘Ž |πœ•1(𝑣) βˆ’ πœ•2(𝑣)|𝑑𝑦 ≀ 𝑠𝑒𝑝 π‘£βˆˆ[π‘Ž,𝑏] ∫ |π‘˜(𝑣, 𝑀)| 𝑏 π‘Ž 𝑑𝑀‖𝑀(πœ•1(𝑣)) βˆ’ 𝑀(πœ•2(𝑣))β€–βˆž = πœ…β€–π‘€(πœ•1(𝑣)) βˆ’ 𝑀(πœ•2(𝑣))β€–βˆž where πœ… < 1and πœ… = 𝑠𝑒𝑝 π‘£βˆˆ[π‘Ž,𝑏] ∫ |π‘˜(𝑣, 𝑀)| 𝑏 π‘Ž 𝑑𝑀is a positive constant. This implies that 𝑀is a contraction. 3.2. Numerical Integral Equation Method The NIE method for approximating a solution of an integral equation is the use of a quadrature rule which determined by the set of nodes or points, {π‘₯𝑗 , 𝑗 = 0,1, . . . , π‘š} obtained from the partition of an integral limit [π‘Ž, 𝑏] intoπ‘šsubintervals and the set of weights, {𝑀𝑗 , 𝑗 = 0,1, . . . , π‘š}, generally, the approximation of an integral can be expressed as ∫ π‘Š(π‘₯)𝑓(π‘₯)𝑑π‘₯ 𝑏 π‘Ž β‰ˆ βˆ‘ 𝑀𝑗𝑓(π‘₯𝑗)π‘š 𝑗=1 . The integral equation to evaluate the ARL in (8) can be estimated by the solution of π‘š linear equation systems, πœ•(π‘₯𝑖) = 1 + 1 πœ†1 βˆ‘ π‘€π‘—πœ•(π‘₯𝑗)π‘š 𝑗=1 𝑓 ( π‘₯π‘—βˆ’(πœ†1πœ™1βˆ’πœ†2)π‘Œπ‘‘βˆ’1βˆ’(1βˆ’πœ†1+πœ†2)π‘₯𝑖 πœ†1 βˆ’ πœ‡ βˆ’ βˆ‘ πœ™π‘–π‘Œπ‘‘βˆ’π‘– 𝑝 𝑖=2 βˆ’ βˆ‘ 𝛽𝑗𝑋𝑗𝑑 π‘Ÿ 𝑗=1 ) , 𝑖 = 1, . . . , π‘š (15) The system of the π‘š linear equations is πΏπ‘šΓ—1 = (πΌπ‘š βˆ’ π‘…π‘šΓ—π‘š)βˆ’11π‘šΓ—1 where πΏπ‘šΓ—1 = [οΏ½ΜƒοΏ½(π‘Ž1) οΏ½ΜƒοΏ½(π‘Ž2) . . . οΏ½ΜƒοΏ½(π‘Žπ‘š)]𝑇. Let π‘…π‘šΓ—π‘š be a matrix and define the π‘š to π‘šπ‘‘β„Ž as elements of matrix 𝑅 as follows, [𝑅𝑖𝑗] β‰ˆ 1 πœ†1 𝑀𝑗𝑓 ( π‘₯π‘—βˆ’(πœ†1πœ™1βˆ’πœ†2)π‘Œπ‘‘βˆ’1βˆ’(1βˆ’πœ†1+πœ†2)π‘₯𝑖 πœ†1 βˆ’ πœ‡ βˆ’ βˆ‘ πœ™π‘–π‘Œπ‘‘βˆ’π‘– 𝑝 𝑖=2 βˆ’ βˆ‘ 𝛽𝑗𝑋𝑗𝑑 π‘Ÿ 𝑗=1 ) (16) Finally, the general numerical approximation of πœ•(𝜈) is expressed as: πœ•(𝜈) = 1 + 1 πœ†1 βˆ‘ π‘€π‘—πœ•(π‘₯𝑗)π‘š 𝑗=1 𝑓 ( π‘₯π‘—βˆ’(πœ†1πœ™1βˆ’πœ†2)π‘Œπ‘‘βˆ’1βˆ’(1βˆ’πœ†1+πœ†2)𝜈 πœ†1 βˆ’ πœ‡ βˆ’ βˆ‘ πœ™π‘–π‘Œπ‘‘βˆ’π‘– 𝑝 𝑖=2 βˆ’ βˆ‘ 𝛽𝑗𝑋𝑗𝑑 π‘Ÿ 𝑗=1 ) (17) The details of different composite quadrature rules including the location of nodes and their weights when setting the equal width β„Ž = (𝑏 βˆ’ π‘Ž)/π‘š and 𝐾𝑗 = π‘₯π‘—βˆ’(πœ†1πœ™1βˆ’πœ†2)π‘Œπ‘‘βˆ’1βˆ’(1βˆ’πœ†1+πœ†2)𝜈 πœ†1 βˆ’ πœ‡ βˆ’ βˆ‘ πœ™π‘–π‘Œπ‘‘βˆ’π‘– 𝑝 𝑖=2 βˆ’ βˆ‘ 𝛽𝑗𝑋𝑗𝑑 π‘Ÿ 𝑗=1 are presented in Table 1. HighTech and Innovation Journal Vol. 5, No. 4, December, 2024 906 Table 1. The composite quadrature rules Composite rules Equation Node(𝒙𝒋) Weight (π’˜π’‹) Midpoint �̃�𝑀(𝜈) = 1 + 1 πœ†1 βˆ‘ 𝑀𝑗𝐿(π‘₯𝑗) π‘š 𝑗=1 𝑓(𝐾𝑗) π‘Ž + (𝑗 βˆ’ 1 2 ) β„Ž β„Ž Trapezoidal �̃�𝑇(𝜈) = 1 + 1 πœ†1 βˆ‘ 𝑀𝑗𝐿(𝑀𝑗) π‘š 𝑗=0 𝑓(𝐾𝑗) π‘Ž + π‘—β„Ž β„Ž 2 ; 𝑗 = 0, π‘š, β„Ž; 𝑗 = 1, . . . , π‘š βˆ’ 1 Simpson’s �̃�𝑆(𝜈) = 1 + 1 πœ†1 βˆ‘ 𝑀𝑗𝐿(π‘₯𝑗) 2𝑛 𝑗=0 𝑓(𝐾𝑗) where m=2n π‘Ž + π‘—β„Ž β„Ž 3 ; 𝑗 = 0,2𝑛, 4β„Ž 3 ; 𝑗 = 1, . . ,2𝑛 βˆ’ 1, 2β„Ž 3 ; 𝑗 = 2, . . ,2𝑛 βˆ’ 2 Bool’s �̃�𝐡(𝜈) = 1 + 1 πœ†1 βˆ‘ 𝑀𝑗𝐿(π‘₯𝑗) 4𝑛 𝑗=0 𝑓(𝐾𝑗) where m=4n π‘Ž + π‘—β„Ž 14β„Ž 45 ; 𝑗 = 0,4𝑛, 64β„Ž 45 ; 𝑗 = 1, . . . ,4𝑛 βˆ’ 3,4𝑛 βˆ’ 1 24β„Ž 45 ; 𝑗 = 2, . . . ,4𝑛 βˆ’ 2, 28β„Ž 45 ; 𝑗 = 4, . . . ,4𝑛 βˆ’ 4 4. Simulation Result The details of simulation study, performance criteria and results for verifying the accuracy of the proposed explicit formula to assess the ARL of ARX(p,r) process running on the extended EWMA control chart are provided in subsection 4.1. The explicit formula and NIE method to evaluate the ARL were computed by the Mathematica program in the 64- bit operating system, AMD Ryzen 7 4700U with Radeon Graphics 2.00 GHz processor. In addition, the performance comparisons of the extended EWMA control chart and the classical EWMA control chart in detecting process mean change under different conditions are presented in subsection 4.2. The real datasets in finance and economics fields are studied and revealed in 4.3. 4.1. The Accuracy of the Proposed Explicit Formula The numerical algorithm for calculating the ARL can be concluded as the following steps. Step 1: Set the values of  The autoregressive coefficients (πœ™π‘–), the coefficient exogenous variables (𝛽𝑗), constant (πœ‡), the initial value of autoregressive: π‘Œπ‘‘βˆ’1, π‘Œπ‘‘βˆ’2, . . . , π‘Œπ‘‘βˆ’π‘ and the exogenous variables (𝑋𝑗𝑑) in the ARX(p,r) model.  The smoothing constants (πœ†1, πœ†2)and the initial value of the extended EWMA control chart (𝐸0 = 𝜈).  The exponential white noise parameter for in-control state, 𝛼0.  The shifts value, 𝛿 = 0.005, 0.01, 0.025, 0.05, 0.1, 0.25, 0.5, 1 to determine the out-of-control state parameter 𝛼1 = (1 + 𝛿)𝛼0.  An acceptable ARL0 = 370 for in-control state and the lower control limit π‘Ž. Step 2: Compute the upper control limit, 𝑏 by Equation 11 that yield the desire average run length for in - control process. Step 3: Compute a solution of ARL1 for the specific shift in process where 𝛼1 = (1 + 𝛿)𝛼0 by the equation of explicit formula (12) and the NIE method (17) with different quadrature rules and set the numbers of subinterval, π‘š = 600. The CPU time of each method is also collected. Step 4: Compute the absolute percentage difference, APD (%) which is defined as: 𝐴𝑃𝐷(%) = |πœ•(𝜈)βˆ’οΏ½ΜƒοΏ½(𝜈)| πœ•(𝜈) Γ— 100 (18) Table 2 presents the ARL values of the extended EWMA control chart, calculated using an explicit formula and four composite quadratic rules for the NIE method. This analysis was conducted on various ARX(p,r) processes, specifically the ARX(1,2), ARX(2,1), and ARX(3,2) models when π‘Ž = 0, 𝝁 = 𝟏 and specific π€πŸ = 𝟎. πŸŽπŸ“ and πœ†2 = 0.025.The results indicate that the ARL values obtained from the derived explicit formula are very similar to those approximated by the NIE method. In fact, the small APD (%) suggests that the proposed explicit formula can accurately evaluate the ARL when compared to the NIE method. In addition, the CPU time shows that the explicit formula takes less than 0.001 seconds to compute the ARL, whereas the NIE method takes approximately 3.1 to 3.5 seconds. Notably, the composite Bool’s rule is the fastest among the other rules. The advantage of explicit formula in this work which rapidly calculating the accurate ARL values is similar to the previous studies [20-25] that derived the explicit formula for other control charts with various pattern of time series model with exponential white noise. HighTech and Innovation Journal Vol. 5, No. 4, December, 2024 907 4.2. The Performance of Extended EWMA Control Chart The ARL of the extended EWMA control chart has been studied under different conditions of the relevant parameters to assess the sensitivity in detecting the process change. The overall performance measures namely the average extra quadratic loss (AEQL), the performance comparison index (PCI) and the relative mean index (RMI) are used to compare the efficiency of the extended EWMA control chart and the classical EWMA control chart and defined as follows: 𝐴𝐸𝑄𝐿 = 1 π›₯ βˆ‘ (𝛿𝑖 2 Γ— 𝐴𝑅𝐿(𝛿𝑖)) π›Ώπ‘šπ‘Žπ‘₯βˆ‘ 𝛿𝑖=π›Ώπ‘šπ‘–π‘› (19) where 𝛿𝑖 is the value of change in the process mean at each level i, 𝐴𝑅𝐿(𝛿𝑖) is the ARL value of the control chart for the change level 𝛿𝑖and π›₯ is the number of shift levels from π›Ώπ‘šπ‘–π‘› to π›Ώπ‘šπ‘Žπ‘₯. In this study, the increments π›₯ =9 from π›Ώπ‘šπ‘–π‘›=0 to π›Ώπ‘šπ‘Žπ‘₯=1. The control chart with the smallest value of AEQL is implied to be the most effective one. The PCI is the ratio of the AEQL of a control chart and the AEQL of the most effective control chart denoted as AEQLbase defined as, 𝑃𝐢𝐼 = 𝐴𝐸𝑄𝐿 π΄πΈπ‘„πΏπ‘π‘Žπ‘ π‘’ (20) The RMI is calculated as: 𝑅𝑀𝐼 = 1 𝑛 βˆ‘ 𝐴𝑅𝐿(𝛿𝑖)βˆ’π΄π‘…πΏπ‘ π‘šπ‘Žπ‘™π‘™π‘’π‘ π‘‘(𝛿𝑖) π΄π‘…πΏπ‘ π‘šπ‘Žπ‘™π‘™π‘’π‘ π‘‘(𝛿𝑖) 𝑛 𝑖=1 (21) where 𝑛 is the number of the shifts, 𝐴𝑅𝐿(𝛿𝑖), 𝑖 = 1, . . . , 𝑛is the ARL of a control chart for a shift 𝛿𝑖 and π΄π‘…πΏπ‘ π‘šπ‘Žπ‘™π‘™π‘’π‘ π‘‘(𝛿𝑖) is the smallest ARL among the competing control charts for the shift 𝛿𝑖. It is similar to the AEQL which can implies that a control chart with a lowest value of RMI has the most powerful detection ability. Table 3 reveals the ARL values of ARX(1,1) model when πœ†1= 0.05, 0.10, 0.15 and πœ†2= 0.015, 0.025, 0.035, 0.045. The result shows that the extended EWMA charts with different values of πœ†2 obtained the lower ARL than the EWMA charts for all magnitudes of change. However, it can be seen that the ARL values of the extended EWMA and classical EWMA chart are hardly different when the shift sizes are larger. The AEQL, PCI and RMI values indicate that the performance of the extended EWMA control chart slightly improved when πœ†2 increased. Moreover, we consider the AEQL values of each fixed πœ†2and found that the control chart showed the better performance when πœ†1 is increased. Similary, Table 4 illustrate the ARL values of ARX(2,2) model when the smoothing parameterπœ†1= 0.05, 0.10, 0.15 and πœ†2=0.3πœ†1, 0.5πœ†1, 0.7πœ†1, 0.9πœ†1. The results confirm that the extended EWMA control charts quicklier detecting changes than the EWMA control chart and also show the better performance when πœ†2increased and close to πœ†1. Table 2. The ARL from explicit formula against NIE method using four quadrature rules for the extended EWMA control chart on ARX(p,r) model given 𝒂 = 𝟎, 𝝁 = 𝟏, π€πŸ = 𝟎. πŸŽπŸ“, π€πŸ = 𝟎. πŸŽπŸπŸ“ and ARL0=370 ARX πœ™π‘– , 𝛽𝑗 , 𝑏 𝛿 Explicit (CPU Time) NIE (CPU Time in seconds, APD(%)) Midpoint Trapezoidal Simpson’s Bool’s ARX(1,2) πœ™1= -0.2 𝛽1= 0.25 𝛽2= 0.10 b = 0.00029919 0.000 370.79588139338 (<0.001) 370.7958813921 (3.437, 3.500Γ—10-10) 370.7958813968 (3.422, 9.304Γ—10-10) 370.7958813937 (3.375, 7.687Γ—10-11) 370.7958813937 (3.360, 7.714Γ—10-11) 0.005 138.81636527871 (<0.001) 138.8163652788 (3.453, 9.797Γ—10-11) 138.8163652806 (3.484, 1.335Γ—10-9) 138.8163652794 (3.438, 4.963Γ—10-10) 138.8163652794 (3.437, 5.108Γ—10-10) 0.010 84.385613287935 (<0.001) 84.38561328855 (3.453, 6.699Γ—10-10) 84.38561328957 (3.484, 1.940Γ—10-9) 84.38561328889 (3.453, 1.131Γ—10-9) 84.38561328889 (3.438, 1.132Γ—10-9) 0.025 37.566999643581 (<0.001) 37.56699964348 (3.563, 2.705Γ—10-10) 37.56699964391 (3.484, 8.853Γ—10-10) 37.56699964362 (3.437, 1.147Γ—10-10) 37.56699964362 (3.438, 1.147Γ—10-10) 0.050 18.548715459259 (<0.001) 18.54871545920 (3.547, 2.971Γ—10-10) 18.54871545940 (3.453, 7.720Γ—10-10) 18.54871545927 (3.438, 5.930Γ—10-11) 18.54871545927 (3.437, 5.930Γ—10-11) 0.100 8.5084421809348 (<0.001) 8.508442180907 (3.437, 3.267Γ—10-10) 8.508442180984 (3.469, 5.807Γ—10-10) 8.508442180933 (3.453, 2.434Γ—10-11) 8.508442180933 (3.438, 2.421Γ—10-11) 0.250 2.8749912056601 (<0.001) 2.874991205656 (3.532, 1.537Γ—10-10) 2.874991205670 (3.485, 3.656Γ—10-10) 2.874991205660 (3.453, 1.948Γ—10-11) 2.874991205660 (3.437, 1.948Γ—10-11) 0.500 1.4971503983684 (<0.001) 1.497150398367 (3.500, 2.177Γ—10-9) 1.497150398370 (3.468, 1.994Γ—10-9) 1.497150398368 (3.438, 2.116Γ—10-9) 1.497150398368 (3.437, 2.116Γ—10-9) 1.000 1.1054759084698 (<0.001) 1.105475908470 (3.484, 1.087Γ—10-11) 1.105475908470 (3.484, 1.900Γ—10-11) 1.105475908470 (3.438, 9.039Γ—10-13) 1.105475908470 (3.438, 9.039Γ—10-13) HighTech and Innovation Journal Vol. 5, No. 4, December, 2024 908 ARX(2,1) πœ™1= 0.1 πœ™2= -0.2 𝛽1= 0.5 b = 0.004929596 0.000 370.00748430131 (<0.001) 370.0074838708 (3.453, 1.163Γ—10-7) 370.0074851591 (3.422, 2.318Γ—10-7) 370.0074843002 (3.375, 2.876Γ—10-10) 370.0074843002 (3.391, 2.873Γ—10-10) 0.005 190.55222061678 (<0.001) 190.5522204017 (3.468, 1.129Γ—10-7) 190.5522210465 (3.469, 2.254Γ—10-7) 190.5522206167 (3.438, 7.872Γ—10-11) 190.5522206167 (3.438, 7.872Γ—10-11) 0.010 127.43166667364 (<0.001) 127.4316665325 (3.469, 1.107Γ—10-7) 127.4316669558 (3.484, 2.215Γ—10-7) 127.4316666736 (3.453, 1.724Γ—10-11) 127.4316666736 (3.422, 1.648Γ—10-11) 0.025 62.634879012605 (<0.001) 62.63487894622 (3.531, 1.060Γ—10-7) 62.63487914542 (3.453, 2.120Γ—10-7) 62.63487901262 (3.437, 2.699Γ—10-11) 62.63487901262 (3.422, 2.699Γ—10-11) 0.050 32.758067253609 (<0.001) 32.75806722111 (3.438, 9.919Γ—10-8) 32.75806731864 (3.500, 1.986Γ—10-7) 32.75806725362 (3.453, 5.709Γ—10-11) 32.75806725362 (3.453, 5.709Γ—10-11) 0.100 15.823474648979 (<0.001) 15.82347463517 (3.438, 8.739Γ—10-8) 15.82347467659 (3.500, 1.744Γ—10-7) 15.82347464898 (3.438, 1.416Γ—10-10) 15.82347464898 (3.422, 1.416Γ—10-10) 0.250 5.5103356099647 (<0.001) 5.510335606715 (3.469, 5.897Γ—10-8) 5.510335616464 (3.500, 1.179Γ—10-7) 5.510335609965 (3.453, 2.354Γ—10-12) 5.510335609966 (3.438, 2.354Γ—10-12) 0.500 2.5546720917965 (<0.001) 2.554672091019 (3.515, 3.044Γ—10-8) 2.554672093352 (3.468, 6.087Γ—10-8) 2.554672091796 (3.437, 3.824Γ—10-13) 2.554672091795 (3.438, 3.824Γ—10-13) 1.000 1.4769505548513 (<0.001) 1.476950554717 (3.453, 9.083Γ—10-9) 1.47695055512 (3.484, 1.817Γ—10-8) 1.476950554851 (3.453, 6.765Γ—10-13) 1.476950554851 (3.453, 6.765Γ—10-13) ARX(3,2) πœ™1= -0.1 πœ™2= 0.2 πœ™3= -0.3 𝛽1= 0.5 𝛽2=-0.25 b = 0.000347603 0.000 370.02826148995 (<0.001) 370.0282614895 (3.516, 1.311Γ—10-10) 370.0282614959 (3.437, 1.597Γ—10-9) 370.028261491595 (3.422, 4.446Γ—10-10) 370.028261491598 (3.422, 4.454Γ—10-10) 0.005 140.84047889377 (<0.001) 140.840478892 (3.484, 1.267Γ—10-9) 140.8404788943 (3.532, 4.040Γ—10-10) 140.840478892769 (3.422, 7.100Γ—10-10) 140.84047889277 (3.500, 7.093Γ—10-10) 0.010 85.969167561932 (<0.001) 85.96916756157 (3.516, 4.169Γ—10-10) 85.96916756298 (3.516, 1.222Γ—10-9) 85.9691675620432 (3.484, 1.292Γ—10-10) 85.9691675620434 (3.469, 1.295Γ—10-10) 0.025 38.427560279249 (<0.001) 38.42756027909 (3.547, 4.205Γ—10-10) 38.42756027969 (3.547, 1.141Γ—10-9) 38.427560279287 (3.516, 9.966Γ—10-11) 38.4275602792872 (3.485, 9.992Γ—10-11) 0.050 19.019467342452 (<0.001) 19.01946734238 (3.515, 3.801Γ—10-10) 19.01946734265 (3.532, 1.065Γ—10-9) 19.01946734247 (3.484, 1.105Γ—10-10) 19.01946734247 (3.469, 1.105Γ—10-10) 0.100 8.7437261931818 (<0.001) 8.743726193149 (3.640, 3.704Γ—10-10) 8.743726193257 (3.531, 8.588Γ—10-10) 8.7437261931852 (3.469, 3.923Γ—10-11) 8.74372619318519 (3.469, 3.935Γ—10-11) 0.250 2.9548002268956 (<0.001) 2.954800226896 (3.485, 2.427Γ—10-10) 2.95480022691 (3.500, 4.680Γ—10-10) 2.954800226896 (3.484, 5.756Γ—10-12) 2.954800226896 (3.469, 5.756Γ—10-12) 0.500 1.5261219828111 (<0.001) 1.526121982810 (3.500, 8.845Γ—10-11) 1.526121982814 (3.515, 1.684Γ—10-10) 1.526121982811 (3.484, 3.274Γ—10-12) 1.526121982811 (3.453, 3.274Γ—10-12) 1.000 1.1140149698577 (<0.001) 1.1140149698575 (3.578, 1.437Γ—10-11) 1.114014969858 (3.562, 2.872Γ—10-11) 1.1140149698578 (3.484, 2.872Γ—10-11) 1.114014969858 (3.484, 2.872Γ—10-11) Table 3. The ARL for the extended EWMA control chart on ARX(1,1) model compare with the EWMA chart when 𝒂 = 𝟎, π“πŸ = 𝟎. πŸ‘ and 𝜷𝟏 = 𝟎. πŸ“ are given π€πŸ Shift π€πŸ EWMA (𝒃′= 0.02128157) 0.015 (𝒃 = 0.00856734) 0.025 (𝒃 =0.004688704) 0.035 (𝒃 =0.002569308) 0.045 (𝒃 =0.001408792) 0.05 0.000 370.0230749 370.020397 370.0022024 370.0094036 370.0106203 0.005 202.5551198 189.3755089 177.7365048 167.4941544 225.9518908 0.010 138.6254961 126.3611946 116.0110431 107.2478077 161.9505653 0.025 69.96172424 61.9557122 55.51111554 50.26046147 86.40129664 0.050 37.18100648 32.35704228 28.56364099 25.52882316 47.48474047 0.100 18.23027117 15.61062999 13.58536505 11.98626866 23.98344592 0.250 6.461339461 5.429878966 4.652293688 4.051757178 8.80793257 0.500 2.979979859 2.520221731 2.185067614 1.93488656 4.069266324 1.000 1.650467505 1.463568934 1.334456146 1.243486237 2.121210512 AEQL 0.348578213 0.302898171 0.26994368 0.245605819 0.458157095 PCI 1.419258775 1.233269522 1.09909318 1 1.865416289 RMI 0.370374045 0.214558988 0.094476877 0 0.713546025 HighTech and Innovation Journal Vol. 5, No. 4, December, 2024 909 0.10 (𝑏 =0.0274078) (𝑏 =0.02021706) (𝑏 =0.014933349) (𝑏 =0.01104089) (π‘βˆ—=0.04343651) 0.000 370.08679 370.0148561 370.0069575 370.0150284 370.0130927 0.005 126.6270088 119.6525792 113.4260721 107.8149375 138.9463404 0.010 76.35379692 71.31980306 66.91788227 63.02495293 85.53792815 0.025 34.8301204 32.20631679 29.95111527 27.98707599 39.74672435 0.050 18.27777373 16.8175237 15.57170341 14.49398222 21.04399329 0.100 9.419977003 8.63595675 7.970708904 7.39822652 10.91527025 0.250 4.005211661 3.668840714 3.386090563 3.145134846 4.652867167 0.500 2.285739767 2.112181538 1.968257802 1.847385373 2.624587041 1.000 1.524507577 1.437949538 1.367737098 1.310163906 1.69751911 AEQL 0.279859148 0.261550343 0.246479494 0.23392091 0.315900212 PCI 1.196383633 1.118114422 1.053687307 1 1.350457349 RMI 0.204350975 0.125239921 0.057967885 0 0.354353665 0.15 (𝑏 =0.04833234) (𝑏 =0.03935632) (𝑏 =0.03208744) (𝑏 =0.02618639) (π‘βˆ—=0.06598734) 0.000 370.0397208 370.0157474 370.0168905 370.0286903 370.0338381 0.005 105.0220357 100.3838646 96.18482371 92.34918578 113.0463707 0.010 61.35845538 58.22081974 55.41829121 52.88969654 66.89597728 0.025 27.50341639 25.94585264 24.56893211 23.33825561 30.29427181 0.050 14.51714039 13.66250439 12.90998692 12.23993223 16.0568046 0.100 7.667982244 7.208446946 6.804820954 6.446376029 8.498012692 0.250 3.482624435 3.279985683 3.102726461 2.946075936 3.849774279 0.500 2.122980254 2.01370871 1.918774969 1.835538708 2.322185838 1.000 1.492216472 1.43421367 1.384430089 1.341369114 1.599268701 AEQL 0.264394363 0.252603169 0.242407716 0.233514582 0.285999763 PCI 1.132239199 1.081744732 1.038083849 1 1.2247619 RMI 0.144727436 0.090399477 0.042560489 0 0.242358726 The higher performance of the extended EWMA chart is consistent with the study of Karoon et al [21] which reported that when smoothing parameter πœ†2is increasing. In addition, the findings that the adjusted EWMA-type are consistent show more effective in detecting changes the classical EWMA chart with previously presented such as studies showing in previous studies [24, 25]. According to the results from Tables 3 to 5 illustrate the study of the extended EWMA chart and original EWMA when choosing πœ†1 = 0.15 and πœ†2 = 0.9πœ†1 to evaluated the ARL and overall performance criteria on ARX(3,1) model with varying the values of LCL from 0 to 0.075. The results insist that the extended EWMA control charts are more effective than the classical EWMA control chart for every different value of π‘Ž, furthermore, the AEQL value indicate that the control charts are more slightly sensitivity when the LCL value increase. Table 4. The ARL for the extended EWMA control chart on ARX(2,2) model compare with the EWMA chart given π“πŸ = π“πŸ = 𝟎. πŸ‘, 𝜷𝟏 = 𝟎. πŸ“ and 𝜷𝟐 = 𝟎. πŸπŸ“ π€πŸ 𝜹 π€πŸ EWMA (𝒃′=0.00210744) 𝟎. πŸ‘π€πŸ (𝒃 =0.000469435) 𝟎. πŸ“π€πŸ (𝒃 =0.000172602) 𝟎. πŸ•π€πŸ (𝒃 =0.00006347) 𝟎. πŸ—π€πŸ (𝒃 =0.00002334) 0.05 0.000 370.0623806 370.0851232 370.2705782 370.303918 370.0557168 0.005 143.6099899 131.3741982 121.3469874 112.9932666 167.4925777 0.010 88.1180071 78.84564694 71.50919257 65.57709639 107.3486736 0.025 39.62228502 34.65638219 30.84069842 27.82653203 50.49432756 0.050 19.70209443 16.97689337 14.90780681 13.28804074 25.81118976 0.100 9.109697307 7.729595114 6.692445983 5.887764573 12.25996381 0.250 3.095444374 2.614765165 2.265882038 2.005240113 4.241481434 0.500 1.582397851 1.404972604 1.285218258 1.202671002 2.041950164 1.000 1.132053547 1.079399244 1.048080721 1.029242898 1.289990658 AEQL 0.210959608 0.194070285 0.182739658 0.174901114 0.255464393 PCI 1.206165032 1.109600048 1.044817003 1 1.460621874 RMI 0.336424167 0.191309977 0.083133016 0 0.669446511 HighTech and Innovation Journal Vol. 5, No. 4, December, 2024 910 0.10 (𝑏 =0.000949982) (𝑏 =0.0003493386) (𝑏 =0.0001284868) (𝑏 =0.00004726) (𝑏′=0.004266463) 0.000 370.0087502 370.0120179 370.0252132 370.0816262 370.0300218 0.005 71.649121 63.8472325 57.75480129 52.88951613 88.24019188 0.010 39.57608587 34.83510149 31.20579234 28.35098335 50.02999584 0.025 16.82029518 14.65876241 13.02550689 11.75257144 21.71070307 0.050 8.564602417 7.429641125 6.5759909 5.912616074 11.15848394 0.100 4.361126688 3.781726754 3.348530314 3.013730289 5.698686323 0.250 1.957676502 1.73723013 1.577566986 1.458460132 2.48618116 0.500 1.303114058 1.210685405 1.148358179 1.105417891 1.543269823 1.000 1.08108828 1.048751844 1.029524129 1.01795948 1.178254014 AEQL 0.178945097 0.170070318 0.164204698 0.160209916 0.202791007 PCI 1.116941452 1.061546765 1.024934671 1 1.265783113 RMI 0.29562144 0.167910647 0.072906246 0 0.590748109 0.15 (𝑏 =0.0014279593) (𝑏 =0.0005250648) (𝑏 =0.0001931210) (𝑏 =0.0000710357) (𝑏′=0.006418365) 0.000 370.0007716 370.0137482 370.2646943 370.0054564 370.0056041 0.005 54.72707532 48.48968749 43.67986131 39.86641746 68.30247889 0.010 29.6429765 26.02645315 23.27907858 21.12757265 37.74195095 0.025 12.59578915 10.99351383 9.787432007 8.849442004 16.25336403 0.050 6.55216269 5.714438353 5.085790438 4.597862738 8.478559573 0.100 3.494654209 3.063447519 2.741505761 2.492863279 4.494553824 0.250 1.735702035 1.566190128 1.443509752 1.352019619 2.143298599 0.500 1.243300745 1.169081181 1.119055437 1.084595034 1.436557432 1.000 1.068790317 1.04135321 1.025043217 1.015233965 1.151365586 AEQL 0.172403186 0.165235199 0.160521447 0.157327506 0.191804795 PCI 1.095823545 1.050262621 1.020301219 1 1.219143424 RMI 0.27876938 0.158247137 0.068716666 0 0.558574232 Table 5. The ARL of the EWMA and the extended EWMA control charts on ARX(3,1) model when the lower control limits (𝒂) are varied and π€πŸ = 𝟎. πŸπŸ“, π€πŸ = 𝟎. πŸ—π€πŸ, π“πŸ = 𝟎. 𝟏, π“πŸ = π“πŸ‘ = 𝟎. 𝟐, 𝜷𝟏 = 𝟎. πŸπŸ“ are given 𝒂 Control chart 𝜹 AEQL PCI RMI 0 0.005 0.01 0.025 0.050 0.100 0.250 0.500 1.000 0 Extended 𝑏 =0.00125924 370 56.45 30.62 12.99 6.729 3.561 1.744 1.241 1.066 0.200 1 0 EWMA 𝑏′=0.1205195 370 134.9 82.73 38.55 20.65 10.975 4.932 2.904 1.912 0.407 2.034 1.403 0.025 Extended 𝑏 =0.026263382 370 49.25 26.52 11.28 5.918 3.211 1.654 1.217 1.061 0.197 1 0 EWMA 𝑏′=0.1488308 370 124.4 75.052 34.61 18.55 9.938 4.575 2.766 1.867 0.392 1.986 1.433 0.05 Extended 𝑏 =0.051267388 370 42.84 22.94 9.807 5.220 2.909 1.574 1.194 1.057 0.195 1 0 EWMA 𝑏′=0.17723 370 114.3 67.93 31.04 16.66 9.007 4.251 2.638 1.825 0.378 1.940 1.462 0.075 Extended 𝑏 =0.07627128 370 37.19 19.84 8.539 4.619 2.647 1.504 1.175 1.052 0.192 1 0 EWMA 𝑏′=0.2057207 370 104.7 61.37 27.83 14.97 8.173 3.956 2.519 1.785 0.364 1.894 1.489 4.3. Applications Typically, data in economics and financial applications is collected over specific periods, such as daily, weekly, or monthly. Current observations often depend on previous data, leading to correlations within the data itself. In this context, the SCB stock price (measured in THB) and the exchange rate (USD/THB) are considered exogenous variables. Data was collected daily from January 4, 2022, to June 28, 2022. The second application examines the Thailand GDP percentage expansion (%YoY) incorporating two exogenous variables: the exports (%) and the imports (%). This data HighTech and Innovation Journal Vol. 5, No. 4, December, 2024 911 was collected quarterly from 2001 to 2020 and is of great interest for study. In this section, the explicit formula for evaluating the ARL of the ARX(p,r) model on the extended EWMA control chart are applied to the real datasets. We also compare the efficiency of detecting process changes with the traditional control chart. The two applications, SCB stock price and GDP percentage expansion are tested for autocorrelation using the Box- Jenkins time series technique. The exogenous variables also incorporate to the prediction models for test the significant effect in the model. Table 6 reveals t-test statistic, the coefficient estimation and the Root Mean Square Error (RMSE) and Normalized Bayesian Information Criterion (BIC) values from the fitted ARX(p,r) model. The result indicates that ARX(1,1) model is the most suitable model for describing the pattern of the first application due to the lowest RMSE and Normalized BIC values. For GDP percentage expansion observation, the prediction model ARX(1,2) has the lowest of RMSE and normalized BIC values. To apply the explicit formula to the practical situation, the residuals between the actual values and the prediction assumed as the exponential white noise are determined by Kolmogonov-Smirnov testing. Table 7 shows that the residuals of the optimal model of two applications are all exponentially distributed. Therefore, the prediction model, ARX(1,1) for the SCB stock price (π‘Œπ‘‘)with exchange rate (USD/THB) as input variable (𝑋1𝑑)can be assigned as π‘Œπ‘‘ = 0.956π‘Œπ‘‘βˆ’1 + 3.432𝑋1𝑑 + νœ€π‘‘ where the in-control parameter, 𝛼0= 2.334. For the second dataset, the residuals follow the exponential distribution, νœ€π‘‘ ∼ 𝐸π‘₯𝑝(1.426) when the process is an in- control state. Thus, the prediction model ARX(1,2) for GDP percentage expansion (π‘Œπ‘‘)with the export (𝑋1𝑑)and import (𝑋2𝑑) can be written as π‘Œπ‘‘ = 0.819π‘Œπ‘‘βˆ’1 + 0.255𝑋1𝑑 + 0.062𝑋2𝑑 + νœ€π‘‘ Tables 8 and 9 demonstrate the ARL values calculated form the explicit formula of ARX(1,1) and ARX(1,2) for the two real-word data when the smoothing parameter πœ†1= 0.05, 0.10, 0.15 and πœ†2=0.3πœ†1,0.5πœ†1,0.7πœ†1,0.9πœ†1was set. The results confirm that the extended EWMA control chart quicklier detecting changes than the EWMA control chart and also show the better performance when πœ†2increased and close toπœ†1, The overall performance measure AEQL, PCI and RMI also present in Figures 1 and 2 for stock price and GDP, respectively Table 6. The ARX(p,r,) estimation and the model fit for applications Data Model Variables Coefficient Std. t Sig Model fit RMSE Normalized BIC SCB stock price ARX(1,1) AR(1) (οΏ½Μ‚οΏ½1) 0.956 0.029 33.041 0.000 3.915 2.813 Exchange rate(οΏ½Μ‚οΏ½1) 3.432 0.207 16.566 0.000 ARX(2,1) AR(1) (οΏ½Μ‚οΏ½1) 0.867 0.095 9.133 0.000 3.930 2.863 AR(2) (οΏ½Μ‚οΏ½1) 0.094 0.096 0.979 0.330 Exchange rate(οΏ½Μ‚οΏ½1) 3.419 0.233 14.650 0.000 GDP percentage expansion ARX(1,2) AR(1) (οΏ½Μ‚οΏ½1) 0.819 0.067 12.238 0.000 1.807 1.347 Export(οΏ½Μ‚οΏ½1) 0.255 0.037 6.806 0.000 Import(οΏ½Μ‚οΏ½2) 0.062 0.030 2.075 0.041 ARX(2,2) AR(1) (οΏ½Μ‚οΏ½1) 0.902 0.116 7.793 0.000 1.811 1.407 AR(2) (οΏ½Μ‚οΏ½1) -0.103 0.115 -0.901 0.370 Export(οΏ½Μ‚οΏ½1) 0.254 0.036 7.044 0.000 Import(οΏ½Μ‚οΏ½2) 0.072 0.029 2.468 0.016 Table 7. Exponential white noise testing Data Model Mean (𝜢𝟎) Kolmogorov-Smirnov Z Sig. SCB stock price ARX(1,1) 2.334 0.854 0.460 GDP percentage expansion ARX(1,2) 1.426 0.917 0.370 HighTech and Innovation Journal Vol. 5, No. 4, December, 2024 912 Table 8. The ARL for ARX(1,1) applying to SCB stock price data π€πŸ 𝜹 π€πŸ EWMA (𝒃′=0.0317708) 𝟎. πŸ‘π€πŸ (𝒃 =0.018917715) 𝟎. πŸ“π€πŸ (𝒃 =0.013406835) 𝟎. πŸ•π€πŸ (𝒃 =0.0095070806) 𝟎. πŸ—π€πŸ (𝒃 =0.00674426) 0.05 0.000 370.0040501 370.0007067 370.0004582 370.0599657 370.3089563 0.005 106.820121 100.4375606 94.729478 89.60972967 117.9773533 0.010 62.42901753 58.09904009 54.29967473 50.94787977 70.20205248 0.025 27.81192489 25.66414057 23.80794348 22.19160901 31.75271609 0.050 14.49887529 13.32686357 12.32088143 11.4500065 16.67061524 0.100 7.489519175 6.867501306 6.336730702 5.879612506 8.65150742 0.250 3.257345392 2.993835405 2.771643106 2.582446113 3.757137073 0.500 1.935603515 1.801737894 1.690857261 1.598142672 2.195037645 1.000 1.365284958 1.300277016 1.247971345 1.205547605 1.495656746 AEQL 0.243356 0.229353 0.217862 0.208346 0.270805 PCI 1.168041 1.10083 1.045677 1 1.299786 RMI 0.201751 0.123876 0.057337 0 0.346859 0.10 𝑏 = 0.038019014 𝑏 = 0.0269074 𝑏 = 0.019065803 𝑏 = 0.013519388 𝑏′= 0.0641 0.000 370.0008847 370.0180957 370.002168 370.0111296 370.1931347 0.005 78.12655773 72.89852825 68.31871872 64.27717034 87.58161942 0.010 43.89350325 40.63849202 37.82925685 35.38170207 49.90909725 0.025 19.21316807 17.69225938 16.39399277 15.2734093 22.06984454 0.050 10.14435448 9.327202978 8.633014535 8.036299121 11.68975246 0.100 5.448509626 5.013056812 4.644729563 4.329359697 6.276711769 0.250 2.614675638 2.423520999 2.263449008 2.127751184 2.982606894 0.500 1.707551758 1.605294058 1.521063555 1.45087754 1.908159639 1.000 1.296473483 1.243334201 1.200767108 1.166335662 1.404112067 AEQL 0.220553 0.209613 0.200696 0.193346 0.242281 PCI 1.140715 1.084136 1.038015 1 1.253096 RMI 0.194682 0.119009 0.05491 0 0.338152 0.15 𝑏 = 0.057220513 𝑏 = 0.04044011 𝑏 = 0.02863122 𝑏 = 0.02029285 𝑏′= 0.096866815 0.000 370.0002681 370.00073 370.0373006 370.0491258 370.0012379 0.005 70.05726393 65.15864916 60.92132591 57.63997126 79.10781293 0.010 38.96391285 35.98501512 33.44259984 31.24396259 44.5831598 0.025 17.01228401 15.64200086 14.48366842 13.49029437 19.6358306 0.050 9.040053548 8.306713536 7.689271449 7.161623511 10.45229959 0.100 4.928824771 4.537551996 4.209261916 3.929640297 5.685608497 0.250 2.446797359 2.273286471 2.128957938 2.007123417 2.785598961 0.500 1.64587503 1.551618886 1.474405078 1.410285066 1.833044172 1.000 1.277067435 1.227051647 1.187164634 1.15498856 1.379405859 AEQL 0.214403 0.204243 0.196007 0.189245 0.234829 PCI 1.132942 1.079251 1.035732 1 1.240876 RMI 0.192439 0.116833 0.05331 0 0.338134 HighTech and Innovation Journal Vol. 5, No. 4, December, 2024 913 (a) (b) (c) Figure 1. AEQL, PCI and RMI value on the control charts for SCB stock price application where (a) π€πŸ = 𝟎. πŸŽπŸ“, (b) π€πŸ = 𝟎. 𝟏𝟎 and (c) π€πŸ = 𝟎. πŸπŸ“ 0.346859349 1.299785727 0.27080488 0 1 0.208345787 0.057336695 1.04567671 0.217862337 0.12387632 1.100829824 0.229353256 0.2017514 1.168041196 0.243356462 0 0.5 1 1.5 RMI PCI AEQL Series5 Series4 Series3 Series2 Series1 2 10.3  2 10.5  2 10.7  2 10.9  EWMA 0.338151651 1.253095763 0.242281168 0 1 0.193346091 0.054909888 1.038015141 0.20069617 0.119008864 1.038015141 0.20069617 0.194681653 1.140715172 0.22055282 0 0.5 1 1.5 RMI PCI AEQL Series5 Series4 Series3 Series2 Series1 2 10.3  2 10.5  2 10.7  2 10.9  EWMA 0.338133715 1.786753915 0.23482912 0 1 0.189244703 0.053310471 0.28170126 0.196006795 0.116832845 0.617363886 0.204242533 0.192439483 1.016881737 0.214403323 0 0.5 1 1.5 2 RMI PCI AEQL Series5 Series4 Series3 Series2 Series1 2 10.3  2 10.5  2 10.7  2 10.9  EWMA HighTech and Innovation Journal Vol. 5, No. 4, December, 2024 914 Table 9. The ARL for ARX(1,2) applying to GDP percentage expansions data π€πŸ 𝜹 π€πŸ EWMA (𝒃′=0.01069878) 𝟎. πŸ‘π€πŸ (𝒃 =0.00229284) 𝟎. πŸ“π€πŸ (𝒃 =0.000823678) 𝟎. πŸ•π€πŸ (𝒃 =0.000296076) 𝟎. πŸ—π€πŸ (𝒃 =0.00010644541) 0.05 0.000 370.0195125 370.0168014 370.0111423 370.0010579 370.0193069 0.005 176.1708882 158.5753877 144.2945208 132.6327563 210.3047398 0.010 114.6481409 99.88600383 88.54410867 79.67349984 146.0956492 0.025 54.67975265 46.01742504 39.69272597 34.92894616 74.99295654 0.050 28.07796619 23.1364354 19.61045931 16.99686804 40.2267433 0.100 13.32666035 10.75927865 8.958461228 7.640384668 19.86256417 0.250 4.553212746 3.613581107 2.977569056 2.528109829 7.079400055 0.500 2.142852181 1.762440126 1.520724827 1.361474675 3.247087222 1.000 1.318633449 1.18534282 1.10951606 1.065340391 1.757527932 AEQL 0.265825 0.228883 0.205742 0.190636 0.375299 PCI 1.394412 1.200629 1.079238 1 1.968667 RMI 0.482399 0.263632 0.110906 0 1.040855 0.10 𝑏 =0.00464517 𝑏 =0.0016683882 𝑏 =0.0005998031 𝑏 =0.0002156943 𝑏′=0.021751807 0.000 370.0844572 370.004395 370.0081256 370.0006901 370.0002399 0.005 93.258375 80.48276256 70.90113885 63.53864909 121.8164902 0.010 53.2575307 45.03837717 39.07747821 34.60957764 72.87038299 0.025 23.20433801 19.30221432 16.54093729 14.50635031 33.00876762 0.050 11.91624534 9.836255509 8.379050339 7.312193249 17.26214393 0.100 6.056588517 4.981442311 4.235229765 3.692728425 8.873488972 0.250 2.602744369 2.175942576 1.888509094 1.685962798 3.769792565 0.500 1.587676052 1.391521285 1.267234871 1.18545813 2.163745959 1.000 1.193841436 1.112669882 1.066563681 1.039715824 1.463351631 AEQL 0.207328 0.187726 0.175636 0.167864 0.266973 PCI 1.235092 1.11832 1.046298 1 1.590409 RMI 0.434137 0.236377 0.09922 0 0.947839 0.15 𝑏 =0.006987309 𝑏 =0.002508225 𝑏 =0.00090163 𝑏 = 0.0003242357 𝑏′=0.032870324 0.000 370.0015582 370.0069603 370.0544269 370.0026892 370.0008461 0.005 72.02856861 61.51285853 53.79130086 47.9445093 96.72326424 0.010 39.99670149 33.62557532 29.07021455 25.68715058 55.77959201 0.025 17.25654106 14.34934962 12.30871606 10.81213801 24.75080983 0.050 8.984280125 7.44793 6.377499842 5.596034753 13.01337024 0.100 4.736081026 3.937189241 3.384884769 2.984071306 6.863927909 0.250 2.225599379 1.898323953 1.678483015 1.523718346 3.131910337 0.500 1.469854915 1.312777862 1.213434874 1.148109584 1.935891627 1.000 1.163970193 1.095243656 1.056258866 1.033566465 1.39405184 AEQL 0.195216 0.179327 0.169584 0.163354 0.244267 PCI 1.19505 1.097785 1.038144 1 1.49533 RMI 0.412786 0.224275 0.094051 0 0.910455 HighTech and Innovation Journal Vol. 5, No. 4, December, 2024 915 (a) (b) (c) Figure 2. AEQL, PCI and RMI value on the control charts for GDP percentage expansions application where (a) π€πŸ = 𝟎. πŸŽπŸ“, (b) π€πŸ = 𝟎. 𝟏𝟎 and (c) π€πŸ = 𝟎. πŸπŸ“ 1.040854743 1.968666973 0.375299169 0 1 0.190636189 0.110905591 1.079237677 0.205741758 0.263631984 1.200628644 0.228883269 0.482399238 1.394411554 0.265825305 0 0.5 1 1.5 2 2.5 RMI PCI AEQL Series5 Series4 Series3 Series2 Series1 2 10.3  2 10.5  2 10.7  2 10.9  EWMA 0.947838828 1.590409323 0.266972593 0 1 0.167864077 0.099219742 1.046297915 0.175635834 0.236377116 1.118319787 0.187725719 0.43413716 1.235092125 0.207327599 0 0.5 1 1.5 2 RMI PCI AEQL Series5 Series4 Series3 Series2 Series1 2 10.3  2 10.5  2 10.7  2 10.9  EWMA 0.910454635 1.495329956 0.24426746 0 1 0.163353552 0.094050669 1.038143679 0.169584458 0.224275198 1.097785063 0.17932709 0.412786054 1.195050102 0.19521568 0 0.5 1 1.5 2 RMI PCI AEQL Series5 Series4 Series3 Series2 Series1 2 10.3  2 10.5  2 10.7  EWMA 2 10.9  HighTech and Innovation Journal Vol. 5, No. 4, December, 2024 916 5. Conclusion The capacity of the control charts in capturing changes in the process is usually assessed by the general characteristic, the ARL, which can be described through the Fredholm integral equation of the second kind. This research focused on the alternative methodology in calculating the ARL of the extended EWMA control chart for the ARX model with exponential white noise. The explicit formula derived from the ARL integral equation is proposed and proved the existence and uniqueness by applying the condition of Banach’s fixed point theorem. The accuracy of the exact solutions is verified by NIE methods with four different composite quadrature rules. The result indicates that the ARL from two methods is close, and the computation time of the proposed explicit formulas is less than 0.001 second. The second purpose of this study is to compare the sensitivity of the extended EWMA and the classical EWMA control charts under various situations and also examine the optimal condition of the smoothing parameter of the EWMA-type charts. It can be seen that the extended EWMA control chart shows better performance in detecting process mean changes, especially small shift sizes, as confirmed by overall performance criteria such as AEQL, PCI, and RMI values. Moreover, the result indicated that the extended EWMA control chart has higher efficiency when the smoothing parameter π€πŸ is almost equal to π€πŸ. The two real datasets, namely SCB stock price and GDP percentage expansions with external factors, are applied to demonstrate the performance of the relevant control charts when the residuals of the forecasting model are exponentially distributed. However, the proposed procedure, an explicit formula, works in some conditions, in particular, when data is autocorrelated with exponential white noise. For future study, an explicit formula for the ARL will be developed for other time series models running on extended EWMA or the new adjusted control charts. 6. Declarations 6.1. Author Contributions Conceptualization, T.M., Y.A., and S.S.; methodology, T.M.; software, T.M.; validation, T.M., Y.A., and S.S.; formal analysis, T.M.; investigation, Y.A.; resources, T.M.; data curation, Y.A.; writingβ€”original draft preparation, T.M.; writingβ€”review and editing, T.M.; visualization, Y.A.; supervision, Y.A.; project administration, T.M.; funding acquisition, Y.A. All authors have read and agreed to the published version of the manuscript. 6.2. Data Availability Statement The SCB stock price and the GDB percentage expansions datasets can be found here: https://investing.com and https://www.nesdc.go.th, respectively. 6.3. Funding This research was funded by Thailand Science Research and Innovation Fund (TSRI), 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. 7. References [1] Shewhart, W. A. (1930). Economic quality control of manufactured product 1. Bell System Technical Journal, 9(2), 364-389. doi:10.1002/j.1538-7305.1930.tb00373.x. [2] Page, E. S. (1954). Continuous Inspection Schemes. Biometrika, 41(1/2), 100-115. doi:10.2307/2333009. [3] Roberts, S. W. (1959). Control Chart Tests Based on Geometric Moving Averages. Technometrics, 1(3), 239. doi:10.2307/1266443. [4] Patel, A. K., & Divecha, J. (2011). Modified exponentially weighted moving average (EWMA) control chart for an analytical process data. Journal of Chemical Engineering and Materials Science, 2(1), 12–20. https://investing.com/ https://www.nesdc.go.th/ HighTech and Innovation Journal Vol. 5, No. 4, December, 2024 917 [5] Khan, N., McClean, S., Zhang, S., & Nugent, C. (2023). Performance evaluation of multivariate statistical techniques using edge- enabled optimisation for change detection in activity monitoring. Journal of Cloud Computing, 12(1), 91. doi:10.1186/s13677- 023-00467-x. [6] Abbas, N., Riaz, M., & Does, R. J. M. M. (2013). Mixed exponentially weighted moving average-cumulative sum charts for process monitoring. Quality and Reliability Engineering International, 29(3), 345–356. doi:10.1002/qre.1385. [7] Naveed, M., Azam, M., Khan, N., & Aslam, M. (2018). Design of a Control Chart Using Extended EWMA Statistic. Technologies, 6(4), 108–122. doi:10.3390/technologies6040108. [8] Zahid, R., Noor-ul-Amin, M., Khan, I., AlQahtani, S. A., Pathak, P. K., & Rahimi, J. (2023). Combination of memory type ratio and product estimators under extended EWMA statistic with application to wheat production. Scientific Reports, 13(1), 13547. doi:10.1038/s41598-023-40687-4. [9] Jawad Mirza, M., Hashmi, S., Thomas Mwakudisa, M., Safariyan, A., Naghmi Habibullah, S., & Noor-ul-Amin, M. (2024). Performance evaluation of extended EWMA control chart in the presence of measurement error. Communications in Statistics: Simulation and Computation, 1–16. doi:10.1080/03610918.2024.2408624. [10] Alwan, L. C. (1992). Effects of autocorrelation on control chart performance. Communications in Statistics - Theory and Methods, 21(4), 1025–1049. doi:10.1080/03610929208830829. [11] Reynolds, M. R., Arnold, J. C., & Baik, J. W. (1996). Variable sampling interval X charts in the presence of correlation. Journal of Quality Technology, 28(1), 12–30. doi:10.1080/00224065.1996.11979633. [12] Zhang, N. F. (1997). Detection capability of residual control chart for stationary process data. Journal of Applied Statistics, 24(4), 475–492. doi:10.1080/02664769723657. [13] Tyagi, D., & Yadav, V. (2024). Combined Quality Control Scheme for Monitoring Auto correlated Process. Thailand Statistician, 22(4), 986–1005. [14] MaΓ§aira, P. M., Tavares ThomΓ©, A. M., Cyrino Oliveira, F. L., & Carvalho Ferrer, A. L. (2018). Time series analysis with explanatory variables: A systematic literature review. Environmental Modelling and Software, 107, 199–209. doi:10.1016/j.envsoft.2018.06.004. [15] Suparman. (2018). A new estimation procedure using a reversible jump MCMC algorithm for AR models of exponential white noise. International Journal of GEOMATE, 15(49), 85–91. doi:10.21660/2018.49.3622. [16] Champ, C. W., & Ritrdon, S. E. (1991). A Comparison of the Markov Chain and the Integral Equation Approaches for Evaluating the Run Length Distribution of Quality Control Charts. Communications in Statistics - Simulation and Computation, 20(1), 191– 204. doi:10.1080/03610919108812948. [17] Paichit, P. (2016). Average run length of control chart for ARX(1) process with exponential white noise. Global Journal of Pure and Applied Mathematics, 12(3), 2143–2153. [18] Phanyaem, S. (2022). Explicit Formulas and Numerical Integral Equation of ARL for SARX(P,r)L Model Based on CUSUM Chart. Mathematics and Statistics, 10(1), 88–99. doi:10.13189/ms.2022.100107. [19] Suriyakat, W., & Petcharat, K. (2022). Exact Run Length Computation on EWMA Control Chart for Stationary Moving Average Process with Exogenous Variables. Mathematics and Statistics, 10(3), 624–635. doi:10.13189/ms.2022.100319. [20] Supharakonsakun, Y. (2021). Comparing the effectiveness of statistical control charts for monitoring a change in process mean. Engineering Letters, 29(3), 1108–1114. [21] Karoon, K., Areepong, Y., & Sukparungsee, S. (2023). Trend Autoregressive Model Exact Run Length Evaluation on a Two- Sided Extended EWMA Chart. Computer Systems Science and Engineering, 44(2), 1143–1160. doi:10.32604/csse.2023.025420. [22] Zhang, L., Suraphee, S., & Busababodhin, P. (2023). Derivation of Explicit Formulae for Performance Measures of CUSUM Control Chart for SMA(Q)s Model with Exponential White Noise. Lobachevskii Journal of Mathematics, 44(9), 3902–3913. doi:10.1134/S1995080223090457. [23] Peerajit, W. (2024). Determining the ARL for a Shift in the Mean of a Long-Memory ARFIMA(1, d, 1)(1, D, 1)s Process with Exponential White Noise Running on a CUSUM Control Chart. Thailand Statistician, 22(2), 407–429. [24] Sunthornwat, R., Sukparungsee, S., & Areepong, Y. (2024). The Development and Evaluation of Homogenously Weighted Moving Average Control Chart based on an Autoregressive Process. HighTech and Innovation Journal, 5(1), 16–35. doi:10.28991/HIJ-2024-05-01-02. [25] Phanthuna, P., Areepong, Y., & Sukparungsee, S. (2024). Performance Measurement of a DMEWMA Control Chart on an AR(p) Model with Exponential White Noise. Applied Science and Engineering Progress, 17(3), 7088. doi:10.14416/j.asep.2023.10.005. [26] Phanyaem, S. (2024). Precise Average Run Length of an Exponentially Weighted Moving Average Control Chart for Time Series Model. Thailand Statistician, 22(4), 909–925.