Analytical Solutions of ARL for SAR(p) L Model on a Modified EWMA Chart

A modified exponentially weighted moving average (EWMA) scheme expanded from an EWMA chart is an instrument for immediate detection on a small shifted size. The objective of this research is to suggest the average run length (ARL) with the explicit formula on a modified EWMA control chart for observations of a seasonal autoregressive model of order p th (SAR(p) L ) with exponential residual. A numerical integral equation method is brought to approximate ARL for checking an accuracy of explicit formulas. The results of two methods show that their ARL solutions are close and the percentage of the absolute relative change (ARC) is obtained to less than 0.002. Furthermore, the modified EWMA chart with the SAR(p) L model is tested to shift detection when the parameters c and λ are changed. The ARL and the relative mean index (RMI) results are found to be better when c and λ are increased. In addition, the modified EWMA control chart is compared to performance with the EWMA scheme and such that their results encourage the modified EWMA chart for a small shift. Finally, this explicit formula can be applied to various real-world data. For example, two data about information and communication technology are used for the validation and the capability of our techniques.


Introduction
A control chart is one of the instruments for Statistical Process Control (SPC). The ordinary control charts are well known such as a Shewhart control chart [1], an exponentially weighted moving average (EWMA) chart [2], a cumulative sum (CUSUM) scheme [3]. A Shewhart chart is the basis of a control chart that detected a large shift on 3-sigma control limits speedily. Next, EWMA and CUSUM control charts are developed to a small shift detection appropriately. For the EWMA control chart, many recent works of literature were found to this chart usage, for example, Li et al. [4] introduced the EWMA control chart to invent the transient patterns of motivation and potential of specifying seasonal faster motivation. Moreover, Nawaz et al. [5] presented the EWMA control chart by integrating multiscale principal component analysis for improving the monitoring efficiently and detecting an online multiscale mistake. Recently, Hu and Liu [6] detected the positive shifts of zero-inflated poisson models by using a weighted score test statistic on an upper-sided exponentially weighted moving average control chart.
Meanwhile, the EWMA control chart is improved to a performance by many researchers. One of them is the modified EWMA control chart which was originally presented by Patel and Divecha [7] and developed by Khan et al. [8]. The modified EWMA statistic is expanded by adding a multiple of a previous shift term and a constant c for an abrupt detection of autocorrelated observations. The modified EWMA control chart was continuously studied in various literature [9][10][11].
For the ability comparison of control charts, one of the well-known measurements is the average run length (ARL) [12] presented by using numerous calculations such as a Markov chain [13], a Monte Carlo simulation [14], a numerical integral equation (NIE) method [15], and an explicit formula such that the later one can be found to exact solutions. For example, Petcharat [16] suggested the explicit formula of the ARL for the SAR(p) L process in the case of an exponential error on the EWMA chart. Next, Sukparungsee and Areepong [17] presented ARL solutions by using the explicit formula on an EWMA control chart of the AR(p) model for white noise with exponential distribution. Recently, Phanthuna et al. [18] introduced the modified EWMA control chart on the AR(1) process in the case of exponential white noise by using the explicit formula for solving the ARL.
In this research, the modified EWMA chart of the SAR(p) L model with exponential white noise is proposed. A SAR(p) L model is an autoregressive model added to consider a seasonal component in time series analysis. A seasonal autoregressive model can be applied to many fields such as engineering [19], environment [20] and communication [21].
Nowadays, global communication has rapidly and continuously entered into the digital age. Therefore, information and communication technology plays a crucial role as a tool for accessing information in a digital platform such that the use of the internet is essential for supporting many areas such as educations, businesses and commerces, entertainment, etc. For this case study, the ARL explicit formula is applied to the percentages of internet users by news and business website categories for preparing to support network applications in the future.
In section 2, the modified EWMA chart of a seasonal autoregressive model is displayed. Moreover, the explicit formula is derived and the NIE method is presented for ARL evaluations. In section 3, the ARL results of two techniques are compared for checking preciseness. Next, the capability of the EWMA and modified EWMA control chart is compared. Afterward, the explicit formula of ARL is applied to real data of time series. Discussion and Conclusion are described in sections 4 and 5, respectively. Finally, the future research is suggested in section 6.

Materials and Methods
This section suggests designing a modified EWMA statistic together with observations of a SAR(p) L model. Next, explicit formulas of the ARL are derived and compared with the NIE method.

The Modified EWMA Scheme with a SAR(p) L Model
Patel and Divecha [7] and Khan et al. [8] expanded a modified exponentially weighted moving average (EWMA) control chart originated from the classical EWMA scheme. For a random variable sequence, X t is an observation of a SAR(p) L model with average µ and variance 2 , σ where t is a positive integer. SAR(p) L model: where λ is an exponential smoothing parameter with 0 1, an EWMA statistics is found in (2), when c = 0. Control bounds of a modified EWMA chart: where W s is suitable to control width limits, and λ is an exponential smoothing parameter. In similar ways, the control bounds of a classical EWMA chart can be looked for (3), as c = 0. If Equation (1) of a SAR(p) L model is substituted into (2), then the modified EWMA statistic with a SAR(p) L model can be written as: For detection of an out-of-control process, the corresponding stopping time on SAR(p) L model of the modified EWMA control chart, where Y t = u is the initial value, a and b are lower and upper control limits, is defined as: On the other hand, Y t is in an in-control process can be reorganized in the error term t ε as: An initial value of the ARL for the SAR(p) L model on the modified EWMA control chart is determined as:

Analytical Solutions of ARL
For the modified EWMA control chart of the SAR(p) L model, ARL is solved by using the method of the Fredholm integral equation of the second kind [22]. ARL(u) recalled for an initial ARL can be described as: After that, the integral variable is changed for an easier way such that In the next step, the integral equation in (9) is proved by using Banach's fixed point theorem [23] for checking the persistence and uniqueness of ARL solutions on the SAR(p) L model of the modified EWMA scheme such that this procedure is presented in Appendix.

Explicit Formula
After checking a unique solution of ARL, Equation (9) can be converted by setting new variables as: then this equation can be rewritten as: Next step, IE is discussed and ARL(z) is replaced by (10) as follows: Thus, the IE of (11) is substituted into (10), then ARL(u) can be rearranged as: Finally, Equation (12) is the explicit formula to solve the ARL on the modified EWMA control chart of the SAR(p) L model.

NIE Method
Otherwise, the NIE method can be used to calculate ARL solutions to the SAR(p) L model on the modified EWMA control chart. From Equation (8), the ARL of the NIE method or ARL N (u) is estimated by using the n linear equation systems with the composite midpoint quadrature rule [18] on the interval [a, b] such that given the distance of n equal separated intervals to be d j = (b -a)/n and the intermediate value of the j th interval to be z j = (j -0.5)d j + a. Finally, the NIE method can be solved to the ARL as follows:

Results
For ARL results of an in-control process, ARL 0 is defined when the exponential parameter of the ARL is set to 0 . β β = Otherwise, the 1 β is assigned as: 1 0 = (1 ) , β δ β + where 1 0 > β β and δ is the size of mean shift for an out-of-control process such that this ARL situation is called to be ARL 1 .
ARL solutions of the NIE method and the explicit formula are compared on the modified EWMA chart for observations of the SAR(p) L model with the absolute relative change (ARC) [24] computed as: Moreover, each control chart can be compared to perform by measuring the relative mean index (RMI) [25] defined as: where ARL i (c) is the ARL of row i on the tested control chart, ARL i (s) is the lowest ARL of row i from all the control charts such that a control chart is more effective if the RMI value is lower.

Experimental Results
For this section, a simulation of the in-control process is typically given ARL 0 = 370 such that the initial parameters are set u = 1, X 0 , X t-L ,..., X t-pL = 1, 0 β =1. For the out-of-control process, ∞ , the upper bound b is found by using the least a to be 0.
In Tables 1 and 2, the ARL 0 of the modified EWMA control chart at c = 1 is computed by using two techniques to be the explicit formula and the NIE method with various λ and φ for the SAR(1) 12 model and the SAR(2) 12 model, respectively. The results of two methods are compared with the ARC for checking the precision of solutions such that all of them are of little value, less than 0.002, correspondingly. The capability of control charts can be compared by calculating the ARL in Table 3 and Fig. 1A for the SAR(1) 12 model and Table 4 and Fig. 1B for the SAR(2) 12 model. The results of the modified EWMA control charts for all c can be effectively detected to be faster than the EWMA scheme for a small shift size. Correspondingly, the RMI of the modified EWMA control charts is lower than the EWMA chart. Moreover, the modified EWMA chart is more performance when c is increased.  For Fig. 2A and Fig. 2B, three different , λ 0.05, 0.10 and 0.20, are compared to ARL 1 of the modified EWMA chart with c = 1 such that higher λ can be detected to shift faster than on observations of the SAR(1) 12 model and the SAR(2) 12 model, respectively.

Application for Real Data
Two practical datasets are used in this case study which are the percentages of internet users by news and business website categories in Thailand [26]. First, the autocorrelation of the observations is assessed by using the Box-Jenkins technique to determine the fitting of forecast time series data models. Next, applying the t-statistic is proved that the datasets are suitable for a SAR(p) L model. Researchers verified that the white noise is followed by an exponential distribution.
Dataset 1 is the percentage of internet users by news category collected monthly from January 2015 to December 2020. This dataset proves an autocorrelated time series suitable for the SAR(1) 12  For Dataset 1, the ARL of control charts is evaluated in Table 5 and Fig. 3A for the SAR(1) 12 model. Accordingly, Dataset 2 of the SAR(2) 12 model is presented to the ARL results on control charts in Table 6 and Fig. 3B. The results of both datasets similarly appear to simulated data such that the modified EWMA control charts are found to have higher performance than the EWMA chart when a shift detection is of small size and these charts adjusted c enlarge.

Discussion
For the proposed explicit formula, their ARL 0 and ARL 1 results are closed to the NIE method such that the percentage of the absolute relative change (ARC) is obtained to lower than 0.002. Moreover, this explicit formula of ARL is used for comparing the EWMA and the modified EWMA control charts adjusted c such that the modified EWMA control charts get less ARL and RMI value than the EWMA chart for a small shift and a high c, then they summarized that modified EWMA control charts can be detected more quickly. Afterward, the modified EWMA control chart is experimented to vary λ such that the results are presented to better efficacy for higher .
λ In addition, this explicit formula can be applied with real data such as the percentages of internet users by news and business website categories in Thailand, then these results are agreeable to simulated situations. For this research, the explicit formula can be used under the conditions of the SAR(p)L model in the case of exponential white noise.

Conclusions
In this research, the modified EWMA control chart is presented for observations of the SAR(p) L model with exponential residual. The explicit formula of the ARL is derived to measure the efficiency of this control chart and compared to the NIE method for checking the exactness of this explicit formula by using the ARL and ARC solutions such that the results of two methods are not much different. Furthermore, the modified EWMA control charts adjusted care compared to the EWMA by using the ARL and RMI calculations. Finally, this process can be applied for observing real-life situations.

Future Research
For future studies, we will develop the explicit formula of the ARL on this control chart for other models and construct explicit formulas for modern control charts. Other techniques will be suggested for calculating the ARL such as the Markov chain approach, Monte Carlo simulation, and Martingale approach.