Exponentially weighted moving average—Moving average charts for monitoring the process mean

This research aimed to propose a newly-mixed control chart called the Exponentially Weighted Moving Average—Moving Average Chart (EWMA-MA) to detect the mean change in a process underlying symmetric and asymmetric distributions. The performance of the proposed control chart are compared with Shewhart, MA, EWMA, MA-EWMA and EWMA-MA control charts by using average run length (ARL), standard deviation of run length (SDRL), and median run length (MRL) as the criteria for measuring efficiency which evaluated by using Monte Carlo simulation (MC), Moreover, the proposed control chart will be applied to real data. The results of performance comparison showed that the presented control charts performed better detection than the Shewhart, MA, and EWMA charts. However, the results of detection tended to be slower than those for the MA-EWMA chart. The value of ARL1 for the mixed control chart depends on the parameters of the statistics for such control chart. The EWMA-MA chart is a variable following λ and the MA-EWMA chart is varied according to w. From applying the proposed control chart to the data for flow in the Nile River and data of the real GDP growth (%) in the Lebanese economy, it was found to be in accordance with the research results.


Introduction
Control charts are a fundamental tool of SPC, control charts are now widely used, not only in industry, but also in many other areas with real applications, such as health care [1], manufacturing processes [2], environmental sciences [3], etc. Shewhart [4] developed the first control chart considered as the main tool of SPC using statistical principles in generating. It is sometimes called the Shewhart chart, which is a chart using the data of the previous production process to scatter a plot and consider the production process. Thus, the pattern of the scatter plot cannot be seen if the production process does not change significantly. For this reason, the Shewhart chart is good at detecting larger shifts in the process. Later, Page [5] and Roberts [6] invented a control chart that could detect changes in the production process, even if the change was only slight, called the cumulative sum (CUSUM) chart and the exponentially a1111111111 a1111111111 a1111111111 a1111111111 a1111111111 that the CUSUM-EWMA chart had better efficiency for detection than the control charts of Shewhart, CUSUM-S 2 , S 2 -EWMA, CS-EWMA, floating T-S 2 , floating U-S 2 , classical EWMA, and CUSUM charts. Many authors designed MEC and MCE control charts for various situations including, for example, Aslam [20], Zaman et al. [21], Lu [22], Osei-Aning et al. [23] and Riaz et al [24]. Recently, Aslam et al. [25] proposed the DMA-EWMA chart with the data of exponential distribution. The results found that the proposed chart has the efficiency in detecting changes better than the proposed chart in the research of Khoo and Wang [26].
Therefore, the researcher decided to study for proposed a new control chart by combining the EWMA chart with the MA chart, called the EWMA-MA chart, which is used for detecting the mean changes of the process by comparing the efficiency of the EWMA-MA chart with the MA-EWMA, Shewhart, EWMA, and MA charts. If any chart gives the lowest ARL, it means such a chart has the best efficiency to detect changes. Moreover, this can also be applied to the real data for flow in the Nile River and data of the real GDP growth (%) in the Lebanese economy, which is a normal distribution.

Moving Average (MA), Exponentially Weighted Moving Average (EWMA), mixed EWMA-MA and performance measures evaluation
The control chart was first used in 1924. The first person who initiated and applied the control chart for controlling the production process was Dr. Walter Andrew Shewhart. The control chart performs three main functions. The first function is to define the production standards and the second function is to facilitate the production process to achieve the goal. The last function is to improve the production process. A control chart could be classified into 2 types: variables and attribute control charts. A variable chart is used in controlling the production process and has measurable features of attributes, i.e., � X chart, R chart, and S charts. An attribute chart is used for controlling the production process with the measurement of product quality by counting, including the p chart, c chart, np chart, etc. In this research, the related control charts are as follows:

Moving Average chart (MA chart)
For the MA chart [8], the mean is found following each w of time. Assuming that we have k random samples of size n�1, and suppose that � X 1 ; . . . ; � X k are independent and identically distributed (i.i.d.) in the time domain are the average of the i th -sample for i =1,. . .,k, the value can be found as follows: At time i, the statistics of the MA i control chart are calculated from finding the mean of each w time by calculating the means of the sub-sample � X i ; � X iÀ 1 ; . . . which can be divided into two cases as follows: where w is the width of the MA control chart, the mean and variance of statistics MA i are: Therefore, the control limits of the MA control chart are following where H 1 is a coefficient of control limit of MA control chart, μ is the mean and σ 2 is the variance of the process under control.

Exponentially Weighted Moving Average chart (EWMA chart)
The EWMA control chart was introduced by Roberts [6] (see also Lucas and Saccucci [27]), which is suited to detect a small change in process parameters. An EWMA control chart for monitoring the mean of a process is based on the statistic.
where λ is the weighing parameter of the data in the past having the values from 0 to 1, and � X i is the mean of the process at time i. At the very first time point Z 0 = μ 0. (the steady and initial value), where X i (i = 1,2,. . .) are independent and normally distributed observations, the statistic Z i for sampling means, � X i should be used, instead of X i in Eq (4), and s � X ¼ s= ffi ffi ffi n p should be used instead of σ in Eqs (5) and (7), then the mean and variance of Z i are: From Eq (5), when i!1, then the asymptotic variance is Therefore, the control limits of the EWMA control chart are following ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi where H 2 is a coefficient of control limit of EWMA control chart, μ 0 is the mean of the process and variance is s � X 2 :

Mixed Moving Average-Exponentially Weighted Moving Average Chart (MA-EWMA chart)
The MA-EWMA chart was presented by Taboran et al. [28]. The chart was the combination of MA and EWMA control chart. In the mathematical model developed for the MA-EWMA chart design, the plot statistic Z i of the EWMA chart is used as an input to the MA chart (Eq (1)). Therefore, the statistic of MA-EWMA chart as follows: Thus, the asymptotical control limit of MA-EWMA control chart is as follows: ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi where H 3 is a coefficient of control limits of MA-EWMA control chart, μ Z is the mean of the process and variance is s 2 Z :

Mixed Exponentially Weighted Moving Average-Moving Average Chart (EWMA-MA chart)
The EWMA-MA chart was generated from combining the EWMA chart with the MA chart. Those charts were effective alternatives to EWMA and MA charts. The statistics still belong to the EWMA chart, as shown in Eq (4).
where λ is the weighing parameter of the data in the past having the values from 0 to 1, Z 0 is the starting value and is set to be equal to the target mean μ 0 , then the UCL and LCL of the EWMA-MA chart are the expected values for the data, which will be the same value of the MA chart. Variance will be applied between the EWMA and MA charts, as shown in Eq (2) and Eq (6). The control limits are as follows: ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi where H 4 is the coefficient of the control limits for the EWMA-MA chart, μ MA is the mean of the process and variance is s 2 MA :

Performance measurement methodology
There are several methods for measuring the efficiency of control charts. The most popular measures of the performance are average run length (ARL). ARL is the sample of points under the control limit prior to the signaling process to access the control limit for the first time. ARL is considered in 2 cases. ARL 0 is used in considering the in-control process, while ARL 1 is used in considering the out-control process. The mean of the RL distribution is the ARL, the standard deviation of the run length (SDRL) is also computed. The control charts having the best efficiency will give the least ARL 1 of the control charts. That means such a control chart can detect changes in the mean of the process the soonest. However, the disadvantage of the ARL is the skewness of the run length distribution changes from highly skewed when the process is in-control to approximately symmetric when the process mean shift is large, interpretation based on ARL alone could be erroneous [29]. Therefore, the MRL are used as the criteria for measuring the efficiency of the non-normality cases, which is more credible since it is less affected by the skewness of the run length distribution [30,31]. In this research, the ARL, SDRL, and MRL are used as the criteria for measuring the efficiency of the control charts which evaluated by using Monte Carlo simulation (MC), MC is simple to program and is adapted for controlling and testing accuracy, which are the estimation of ARL, SDRL, and MRL generated from the creation of the program simulated for finding ARL, SDRL, and MRL which can be found as follows: ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi where RL t is the number of samples before the out-control process being detected for the first time

Evaluation methods
This research studied the performance of the proposed control chart compared with the Shewhart, MA, EWMA, and MA-EWMA charts when the process is not under control. The study was conducted with 6 distribution processes divided into symmetric distributions, which are Normal(0,1), Laplace(0,1), Logistic(6,2), and Student t 10 distributions and asymmetric distributions with skew to the right, which are Exponential(1) and Gamma(4,1). The determination of the moving average period (w) of the MA chart equal to 5. When the data has Normal(0,1), Laplace (0, 1), Logistic(6,2), Student t 10 , Gamma(4,1) and Exponential (1) distribution, we have used location shifts in this format: μ 1 = μ 0 +δσ 0 where δ refers to the amount of shift, μ 1 is the shifted mean, μ 0 is the in-control mean and, σ 0 is the controlled value of process standard deviation. The parameters for each chart were defined such that the ARL when the process is under control was equal to 370. Set the sample size of each round of experiment at 10,000 and the number of experiment repeat above process for the Monte Carlo simulation (MC) at 200,000 cycles for finding the ARL, SDRL, and MRL. The programs used to process the results are R program.

Expansion of the MA-EWMA control chart
From the MA-EWMA chart was presented by Taboran et al. [28], the researcher expanded on the work by changing w and λ, as shown in Tables 5-8. When the distributions are Normal (0,1) and Exponential(1), MA-EWMA will present a decreasing ARL 1 as w increases. When changing λ = 0.05, 0.10, 0.25, 0.50 and 0.75, ARL 1 is almost the same without much difference, as shown in Fig 3, it can be seen that the obtained ARL 1 is less if parameter w is greater.

Practical applications
In this case study, the real data of the flow rate of Nile river between 1871-1930 [32] and real GDP growth (%) in the Lebanese economy data from International Monetary Fund (IMF) between 1970-2003 [33] can be found in S1 Table, based on the time-series data used in this study, therefore, the basic alternate is that the time series is stationary (or trend-stationary). In consequence, the concept and theory relating to the classical time series to test the stationary  [34]. The result shows that the studied data do not have a unit root and are stationary. According to the statistical assumptions to construct variables quality parametric control charts [35], they were found that the studied data agree to the assumptions which contains normal distribution and independence. We have considered two applications: the data for flow in the Nile River and data of the real GDP growth (%) in the Lebanese economy.

Application I: the Nile river flow rate between 1871-1930
From the data of the Nile river flow rate between 1871-1930 with normal distribution when the process had not changed and the mean of process at 1,100 m 3 /seconds had not changed. In 1900, the process changed, so the mean decreased to 850 m 3 /second with standard deviation of 125m 3 /seconds. The data generated the Shewhart, MA, EWMA, MA-EWMA and EWMA-MA charts as Eqs (3), (7), (9) and (11)    The bold is minimal of ARL, SDRL and MRL.

Conclusions, discussions and recommendations
In this research, a new control chart is proposed, called the EWMA-MA chart to detect changes in the mean of the process in cases where the change process is under control ARL 0 = 370. From research, the results showed that the proposed chart has better detection efficiency than the Shewhart, MA, and EWMA charts for all levels of changes when studying under asymmetric distributions with right skew. When studying under a process with symmetric distributions, there are some distributions that give minimally different results. Overall, however, the proposed chart exhibited better performance compared to the Shewhart, MA, and EWMA charts. When comparing the proposed chart with the MA-EWMA chart under a process with asymmetric and symmetric distributions, it is found that the MA-EWMA chart has better efficiency in detecting changes of parameters than the EWMA-MA chart for all levels of changes. However, from the results of comparison, it is found that the MA-EWMA and EWMA-MA charts have ARL 1 depending on the parameters of statistics of such control chart. That is MA-EWMA chart, when varying span size w, ARL 1 will be different. However, ARL 1 will not be different if λ has been changed. On the contrary, ARL 1 will be different for the EWMA-MA chart when changing λ. However, if span size w is changed, ARL 1 will not be different. From applying the proposed control chart to the data for flow in the Nile River and the real GDP growth (%) in the Lebanese economy, it is found to be in accordance with the research results.
In addition, the researchers compared the performance of MA-EWMA are compared with DEWMA mean [36], GWMA-CUSUM [37], and Progressive Mean [38] control charts. The results found that the MA-EWMA chart with that of the DEWMA mean chart by simulating the observations to have normal distribution (μ = 0, σ = 1) when the in-control ARL is 200. It was found that the DEWMA mean is more sensitive than the MA-EWMA chart when δ�1.00 but if the shifts size of δ >1.00, the two control charts will share the similar efficiency in detecting the change. Besides, the comparison between the MA-EWMA control chart, the GWMA-CUSUM control chart at the in-control ARL is 500, and the Progressive Mean (PM) control chart at the in-control ARL is 370, found that the MA-EWMA chart performed better than the GWMA-CUSUM and PM charts for all magnitudes of change, except for at δ = 0.25, the GWMA-CUSUM and PM charts performed better than the MA-EWMA chart. However, it depends on the different parameters of each control chart due to the fact that each charts, for example, if setting w of the parameter in a large number, the ARL 1 of the MA-EWMA control chart will be lower. The results of the real GDP growth (%) in the Lebanese economy were consistent with Harvie et al. [39] study that the timing of major structural breaks in the Lebanese economy by applying the Zivot and Andrews (ZA) procedure, using annual time series data spanning the years from 1970 through 2003. the timing of the structural breaks for the real GDP growth (%) occurred in the years 1987, which are also the years when the country experienced a significant degree of macroeconomic and political instability. These findings therefore confirm the proposed charting method can used to for monitoring, controlling and can be applied to other fields such as health care, epidemiology, environmental sciences, etc. However, the adapted data must be in accordance with the charting statistic and the control limits depend on this assumption and as such the properties of parametric control chart.
In future studies, the scope of the study may be extended in terms of sample size, the method used in determining the ARL, and the process under control in other cases including the application to real data with other distributions, such as asymmetric distributions. Supporting information S1 Table. This is the S1 Table The real