ARL Evaluation of a DEWMA Control Chart for Autocorrelated Data: A Case Study on Prices of Major Industrial Commodities

The double exponentially weighted moving average (DEWMA) control chart, an extension of the EWMA control chart, is a useful statistical process control tool for detecting small shift sizes in the mean of processes with either independent or autocorrelated observations. In this study, we derived explicit formulas to compute the average run length (ARL) for a moving average of order q (MA(q)) process with exponential white noise running on a DEWMA control chart and verified their accuracy by comparison with the numerical integral equation (NIE) method. The results for both were in good agreement with the actual ARL. To investigate the efficiency of the proposed procedure on the DEWMA control chart, a performance comparison between it and the standard and modified EWMA control charts was also conducted to determine which provided the smallest out-of-control ARL value for several scenarios involving MA(q) processes. It was found that the DEWMA control chart provided the lowest out-of-control ARL for all cases of varying the exponential smoothing parameter and shift size values. To illustrate the efficacy of the proposed methodology, the presented approach was applied to datasets of the prices of several major industrial commodities in Thailand. The findings show that the DEWMA procedure performed well in almost all of the scenarios tested.


1-Introduction
Statistical process control (SPC) is very important for verifying the quality of a product and also monitoring and improving the process for its manufacturing.One of the tools used to achieve this is the control chart.They have been applied in various fields to monitor a process and detect changes therein.Well-known ones include the Shewhart, cumulative sum (CUSUM) [1], and exponentially weighted moving average (EWMA) [2] control charts.The Shewhart control chart is good for detecting large changes in the process mean, whereas the CUSUM and EWMA control charts are better at detecting small-to-moderate shifts.The double EWMA (DEWMA) control chart is an extension of the EWMA control scheme carried out by using exponential smoothing parameters [3,4].The DEWMA control chart is good for detecting small sustained shifts in the mean of a process with normally distributed observations and performs better than the EWMA control chart in detecting small shifts in the process mean ranging from 0.1 to 0.5 of the process standard deviation [5].
Comparing its performance with that of the standard EWMA control chart can be evaluated based on the zero-stat performance [6].It can be said that the DEWMA control procedure with the larger exponential smoothing parameter values ( > 0.05 ) performs better than the EWMA control chart in detecting very small sustained shifts of the process mean.Besides, the DEWMA control chart was used to monitor Poisson data.The simulation results show that the DEWMA control chart is more sensitive to small downward process mean shifts than the EWMA control chart [7].The modified EWMA (MEWMA) control chart is highly effective at detecting small and abrupt changes in the mean of processes comprising independent normally distributed or autocorrelated observations.Various experimental studies on processes involving non-normal distributions [10][11][12] and real data [13][14][15] indicate that the MEWMA control chart is more effective at detecting changes in the process mean than the EWMA control chart.Time-series data are habitually autocorrelated, which can be serially correlated over a long period of discretely timed observations.The degree of correlation of the same variable over two successive time intervals can be measured by how much the lagged version of the observations of the variable is related to the original observations.For various situations, the analysis of autocorrelation helps to determine repeating periodic patterns in the form of a mathematical representation of the degree of similarity between the original and lagged versions over a long time period.Autocorrelation can provide information about short-term trends to support the prediction of future points with a short holding period.The autoregressive moving average (ARMA) model [16][17][18] is an amalgamation of autoregressive (AR) [19][20][21] and moving average (MA) [22][23][24] models.
Control charts have been employed for monitoring and detecting processes in which the observations are independently normally distributed [25][26][27], non-normally distributed [11,28,29], or autocorrelated [30][31][32].The measurement of a control chart's performance can be undertaken to ensure that it is appropriate for a particular process, for which the average run length (ARL) is the most often used evaluation method.It represents the average number of observations until an out-of-control signal occurs.Monte Carlo simulation [33,34] and the numerical integral equation (NIE) method [35][36][37] are the most often used schemes to compute the ARL.Moreover, the exact ARL has been employed in several studies [38][39][40][41][42]. From the outcomes of comparative studies, the EWMA control chart is more powerful than the CUSUM control chart for monitoring and detecting small and abrupt changes in the mean of a process involving autocorrelated observations [43,44], while the MEWMA control chart is more effective than either of them in this scenario [14,18,20,31,32].However, deriving the ARL for an autocorrelated process with exponential white noise running on a DEWMA control chart using explicit formulas has not previously been reported.Thus, this became the focus of the present study.A comparison of explicit formula-derived ARLs of MA(q) processes with exponential white noise running on DEWMA, MEWMA, and EWMA control charts is also presented.

2-1-EWMA Control Chart
The EWMA control chart was the first in the quality control literature proposed by Roberts (1959) [2].It has been widely applied in statistical process control (SPC) due to their performance is more powerful than exist charts in continually monitors and detects small changes in the process mean.The past and current observations are considered to create the control statistic with their weighted average.The EWMA control statistic can be written by the recursive equation as, where  is an exponential smoothing parameter, which is 0 <  < 1.The initial value is defined by  0 =  0 with the target value  0 ,   is the process with mean  and variance  2 .
The variance of ).If  gets large, the term (1 − ) 2 converge to 0. Therefore, the general upper control limit (UCL), center line (CL) and lower control limit (LCL) to detect the sequence   are given by where  0 is the target mean,  is the process standard deviation, and  1 is the appropriate control width limit.

2-2-Double EWMA Control Chart
A DEWMA control chart was the initially introduced by Brown (1962).He indicated a different context to forecast future time series observations.Later, Shama & Shamma (1992) developed and evaluated the idea of using the increase of the sensitivity of the EWMA control chart to smaller shifts in the process mean via a double exponential weighting of moving averages implementation.The recursive control statistic of the DEWMA chart is defined as [45]: where  1 and  2 is an exponential smoothing parameter, which is 0 <  1 ,  2 ≤ 1.The initial value is defined by  0 =  0 =  0 with the target value  0 ,   is the process with mean  and variance  2 .

The variance of 𝑊
).
For large values of , the asymptotic variance becomes to ]. Therefore, the general upper control limit (UCL), center line (CL) and lower control limit (LCL) to detect the sequence   are given by: ], (4) where  0 is the target mean,  is the process standard deviation, and  2 is the optimal control width limit.

3-1-Explicit Formula
There are many methods to evaluate the ARL, one of them is explicit formula which is provide the exact ARL values.Various studies have been implemented under different situations of their works both model is based on assumption and violate process.
Autocorrelation is a characteristic of data which shows the degree of similarity between values of the same variables over successive time intervals where the basic assumption of instance independence underlines most conventional models is infringed.Multitudinous processes of time series modeling have been employed in numerous applications.
The observation of this study is the moving average in the general order note that by MA(q) process.It can be described as follows: where  is the mean of the process,   is a white noise which is assumed to be the exponential distribution,   is a coefficient which is |  | < 1;  = 1,2,3, . . ..
Therefore, the Double EWMA statistic for the MA(q) process can be written as: where  = 1, 2, 3, … , .We use one side of the control limit (i.e.,  = 0and  = ).Then obtain: If  1 causes the out-of-control stat for  1 with the starting value  0 = , then: If  1 causes the in-control stat for  1 , then: If can be written in the form as: .
The probability that  1 satisfies the bounds mentioned above for probability distribution function  1 is derived in the form: According to the method of Champ and Rigdon [46], the ARL of Double EWMA control chart for the MA(q) model can be written in the form of the integral equation as: Changing the integral variable, we obtain: In this study,   is defined to be the exponentially distributed with parameter .Therefore, a Fredholm integral equation of the second kind for the ARL can be written as: where and; Finally, the equation 10 is replaced by equation 11.The ARL of the Double EWMA control chart for the MA(q) process with exponential white noise is provided by deriving a Fredholm integral equation of the second kind as follows: It can be written in the form: when  0 ,  and  0 are the starting value of the process for MA(q) model.The in-control process of ARL corresponds to ARL0 when  0 = 1.An out-of-control process when  1 > 1, provide the ARL1 values.

3-2-Numerical Integral Equation
The numerical integral equation (NIE) is a method of estimation the ARL approach.In this study, it is used to compare the accuracy of ARL obtain by explicit formulas method.According to the integral equation in ( 9), the Gauss-Legendre quadrature rule approach is approximated the integral by finite sum of areas of rectangles with base   and heights chosen as values of  midpoints of the one-side interval which divide [0, ] into a partition from 0 to u.It can be derived the ARL calculation of the integral equation as follows.

𝐴𝑅𝐿(𝑎
where ) and   = In order to compute the ARL values, the relation of a matrix form can be rewritten in a form as: where Hence, the approximation of ARL by using NIE method for the Double EWMA control chart of MA(q) process is expressed in the form as follows:

3-3-Existence and Uniqueness of ARL Demonstrate
Here, Banach's Fixed-point Theorem is used to show the uniquely exists solution of the integral equation for explicit formula.In this section, let be an operation in the class of all continuous functions expressed by: As stated by Banach's Fixed-point Theorem, if an operator  is a contraction, then the fixed-point equation () =  has a unique solution.To prove that Equation 16exists and has a unique solution, the following theorem can be claimed.
Proof: To show that T in ( 16) is a contraction mapping for 12 where From Banach's Fixed-point Theorem, we get the following result approved to existence and uniqueness of a solution of the ARL for MA(q) process on the DEWMA control chart.

4-Experimental Results
In this section, the results for evaluating the ARL obtained by explicit formulas and NIE method which is accurate method (viewed as the most accurate method) are presented in the term of the absolute percentage relative error (APRE) (a measure of the exactness of the ARL), which is given by: where and  ̃ are exact and approximation of ARLs derived by explicit formulas and NIE method, respectively, as provided by MATHEMATICA.The experimental results are reported in Tables 1 to 3.   The methodology of our process of computation the ARL values by using the explicit formulas and NIE method running on the double EWMA control chart for the MA(q) process with exponential white noise is presented in Figure 1.

Figure 1. The diagram of the research methodology
Table 1 shows the ARL values for an MA(1) process with exponential white noise running on the DEWMA control chart when  = 3; = −0.1,-0 .6, or 0.7;  1 = 0.05; and  2 = 0.05 for ARL0 = 370 or 500 .The numerical results were obtained after setting in-control process parameter  0 = 1and out -of-control process parameter  1 > 1 ( i.e.,  1 = (1 + ) 0 )for shift size  = 0.0001, 0.0003, 0.0005, 0 .001, 0.005, 0.01, 0 .03, 0.05, 0.1, or 0.5 and where u1 and u2 are the upper control limits (UCLs) for ARL0 = 370 and 500, respectively .The results using the two methods indicate that they are in excellent agreement since the APRE values were less than 3 .5×10 -3 . 1 =  2 = 0.05, as is usually recommended for the EWMA control chart, was used for computing the ARL .The ARL results show that the DEWMA control chart is very sensitive for detecting changes in the process mean when the shift size was very small for  1 =  2 = 0.1.
The ARL results for MA(2) and MA(3) processes,  = 0.0001, 0 .0005, 0 .001, 0 .005, 0.01, 0.03, 0.05, 0.1, or 0 .5 and ARL0 =370 or 500 are given in Tables 2 and 3, respectively .The other settings for the simulation were  = 3; 1 = −0.1,0.2, or -0.3; 2 = 0.6, -0.7, or 0.8;  1 = 0.05; and  2 = 0.05for the MA(2) process and  = 3; 1 = −0.3,-0.4, or 0.2;  2 = 0.5, -0 .6, or 0 .4; 3 = −0.7,-0.8, or 0.7;  1 = 0.05; and  2 = 0.05for the MA(3) process .Moreover, u1 and u2 are the UCLs for ARL0 = 370 and 500, respectively.The results in Tables 2 and 3 indicate that the exact ARL provided by the explicit formulas and the estimated ARLs provided by the NIE method are in good agreement, as indicated by the APRE results being close to zero for all of the cases studied.In addition, the ARL1 values were sensitive to small changes in the process mean.Consequently, the exact ARL values provided by the explicit formulas can be utilized to efficiently and rapidly detect changes in the mean of an MA)q( process with exponential white noise running on a DEWMA control chart. Tables 4 to 6 report the ARLs for MA(1), MA(2), and MA(3) processes running on the DEWMA, EWMA, and MEWMA control charts, respectively .For these experiments,  = 0.0001, 0.0005, 0.001, 0 .005, 0 .01, 0.03, 0.05, 0.1, or 0 .and u1 and u2 are the UCLs for the DEWMA control chart for ARL0 = 370 and 500, respectively.The experimental results show that the ARL values provided by the DEWMA control chart for the MA(1) process with  1 =  2 = 0.05 were much lower than the others for both ARL0 = 370 and 500, which was also the case for the MA(2) and MA(3) processes, indicating that it could detect changes more quickly and more sensitively than the other two control charts .These results are graphically presented in Figures 2 to 4.

5-Application of Real Data
It is well-known that commodity prices are subject to volatility over time, which depends on demand .When the prices of energy products including oil and natural gas rise, the prices of agricultural and other essential products also rise due to increased transportation costs .Thus, commodity prices play a major role in the economy .Thus, using a control chart to anticipate future market changes provides an interesting opportunity.
The real datasets used in this study comprise the price data for major industrial commodities )diesel, gasoline 95, and NGV )natural gas for vehicles( prices; unit: Baht( in Thailand.The Box-Jenkins technique was used to determine whether the observations were autocorrelated, while the t-statistic was used to assess whether the time-series data follow an MA)q( process.Dataset 1 comprises the prices of diesel collected every three months from the third quarter of 2011 to the fourth quarter of 2022.The 39 observations were proved to be autocorrelated and suitable for the MA)1( model written as follows: The ARLs on the MA)1(, MA)2(, and MA)3( processes comprising diesel, gasoline 95, and NGV prices running on a DEWMA control chart for  1 =  2 =  = 0.1are reported in Tables 7 to 9, respectively.The performance evaluation of the proposed scheme was also conducted to assess the ARLs of the processes derived by using explicit formulas running on EWMA [30] and MEWMA [23,31] control charts, where h1 and h2 are the UCLs for the EWMA control chart, b1 and b2 are the UCLs for the MEWMA control chart, and u1 and u2 are the UCLs for the DEWMA control chart for ARL0 = 370 and 500, respectively.The ARL results for the real processes in Tables 7 to 9 are in good agreement with those from the experimental study.The DEWMA control chart was exceptionally good at detecting shifts in the process mean for any magnitude of shift size for all of the scenarios tested except for  = 0.5 with the MA)3( process )NGV price(, for which the MEWMA control chart provides the smallest ARL1. that ‖( 1 ) − ( 2 )‖ ≤ ‖ 1 −  2 ‖ for all  1 ,  2 ∈ [0, ] with 0 ≤  < 1 under the norm ‖‖ ∞ =  ∈(0,) ||.From Equations 10 and 16: 5 .Meanwhile,  1 =  2 =  = 0.05 for the MA(1) model and  1 =  2 =  = 0.1 for the MA(2) and MA(3) models .Moreover, h1 and h2 are the UCLs for the EWMA control chart, b1 and b2 are the UCLs for the MEWMA control chart, Input parameters Start Set the initial value ARL0=370, 500 for calculation the UCL Compare the ARLs of the explicit formula with NIE method Compute ARL values End Compare the capability of the DEWMA chart and existing the control charts Print results