Detection Sensitivity of a Modified EWMA Control Chart with a Time Series Model with Fractionality and Integration

Among the many statistical process control charts, the modified exponentially weighted moving average (EWMA) control chart has been designed to swiftly detect a small shift in a process parameter. Herein, we propose two explicit formulas for the average run length (ARL) for integrated moving average (IMA) and fractional integrated moving average (FIMA) models combined with the modified EWMA control chart for time series prediction. The application of the suggested control chart procedures depends on the residuals of the IMA and FIMA models. The performance of the control chart with both models is evaluated by using the ARL. Explicit formulas for the ARL for the two models with the modified EWMA statistic are derived and their precision is compared with the numerical integral equation method. The explicit formulas could accurately predict the true ARL while markedly decreasing the computational processing time compared to the numerical integration method. The capabilities of the IMA and FIMA models with the modified EWMA control chart were studied by varying g times the last term and exponential smoothing parameter λ, with the relative mean index being used to evaluate these situations. The results show that the modified EWMA control chart with either model performed better than the original EWMA control chart. Furthermore, the modified EWMA control chart with either model was highly efficient when g increased and λ was small. Two applications involving energy commodity prices are used to illustrate the efficacies of the proposed approaches, the results of which were in accordance with the experimental study.


2-Literature Review
Many researchers have proposed explicit formulas for the ARL on control charts with time series models. Petcharat et al. [26] presented explicit formulas for the ARL of random observations from a MA process with exponential white noise running on a CUSUM control chart; they compared the computational times of the explicit formulas and the NIE method and found that using the former was much faster. Sunthornwat et al. [27] proposed explicit formulas for the analytical ARL on an EWMA control chart with a long-memory ARFIMA model by using a solution for the integral equation and compared them with the NIE method; once again, the computational time for the former was much lower. Sunthornwat and Areepong [28] derived explicit formulas for the ARL for seasonal and non-seasonal MA processes with exogenous variables running on a CUSUM control chart and found their optimal parameters. When comparing the solution for a CUSUM control chart with that for an EWMA control chart, the former detected small process shifts better but moderate-to-large process shifts worse than the latter.
Phanthuna et al. [29] introduced process shift detection for the modified EWMA control chart with an AR(1) process and exponential white noise. Explicit formulas for the ARL were derived and checked with the NIE method in terms of accuracy and computational time. Although both produced ARLs similar to the exact ARL, the explicit formulas method was much faster than the NIE method in obtaining an accurate value. Moreover, the authors reported an efficiency comparison of the CUSUM, EWMA, and modified EWMA control charts; their proposed control chart provided the best performance for small-to-intermediate shifts in the process parameter. Later, Phanthuna et al. [30] improved the explicit formulas for computing ARL solutions for the modified EWMA scheme in that they could be used for any order p of an AR(p) model, where p is a positive integer. Supharakonsakun et al. [31] studied a detection procedure for the EWMA and modified EWMA control charts with an ARMA(1,1) process and exponential white noise. Explicit formulas for the exact ARL were provided and their accuracy was compared with the NIE method. Supharakonsakun et al. [32] suggested explicit formulas for the ARL with observations from a MA model on the modified EWMA control chart and compared their capability with the same process on the standard EWMA scheme for monitoring PM2.5 and carbon monoxide air pollution data; their results show that the modified EWMA control chart was much better at detecting shifts in the process parameter than the standard EWMA control chart. Supharakonsakun [33] designed the modified EWMA control chart for a seasonal MA model and evaluated its efficacy by using the ARL calculated via explicit formulas and the NIE method. The efficiencies of the method for the EWMA, CUSUM, and modified EWMA control charts for a seasonal MA process with exponential white noise support the superiority of the latter.

3-1-The Modified EWMA Control Chart
The modified EWMA control chart can quickly detect a small shift in the process mean. In the EWMA statistic, the term g(Xt -Xt-1) from the modified EWMA control chart has been assigned to g = 0, and thus the modified EWMA statistic can be defined as: where is an exponential smoothing parameter where 0 < < 1, Xt is the data sequence at t = 1, 2, 3,... with mean and variance 2 , and g is a suitable constant. The initial values of sequences Yt and Xt (t = 0) are determined for and the asymptotic variance of Yt as ( + 2 + 2 2 ) 2 (2 − ) ⁄ . Thus, the bounds of the control limits of the modified EWMA control chart can be constructed for , , and suitable control width limit C as follows:

3-2-The IMA-Modified EWMA Control Chart
For time series, the ARIMA (p, d, q) model, where p is the order of the autoregressive process, d is the order of differencing, and q is the order of the MA process, is widely applied to real data from many fields. When there is no autoregression in the time series, the IMA model is equivalent to the ARIMA (0, d, q) model. The IMA (d, q) model, denoted as Mt, can be defined by using backward shift operator B as follows: is defined by using the Binomial Theorem [34]. Therefore, the IMA model can be rewritten as: where, θ0 is the process average, θi (i = 1,2,…,q) are the coefficients of the IMA model (-1< θi <1) and εt-i is exponential white noise at time t.

3-2-1-The Explicit Formula for the ARL on the Modified EWMA Control Chart with the IMA Model
The ARL, which is the average number of observations before an alarm occurs, is a commonly used measure for the sensitivity of a control chart. In this section, the explicit formula for the ARL on the modified EWMA control chart with the IMA model is solved. From Equations 1 and 3, the modified EWMA statistic for the IMA model can be derived as: The lower and upper bounds of the control limits are l and h, respectively. For the in-control process, the interval of Yt can be written as:  The integral equation for the ARL on the modified EWMA control chart with an IMA (d, q) process and initial value Y0 = u is derived by using the Fredholm integral equation [35] as follows: By solving the integral equation, the integral variable can be adjusted by setting = ( + ) − . Thus, TARL can be reworked as: Since the function of is exponential, then TARL(u) can be rewritten as: After checking the uniqueness of the ARL solution by using Banach's fixed point theorem [36], the explicit formula for the ARL of the IMA model on the modified EWMA control chart is derived by setting = ∫ Afterward, integral equation P can be obtained as: Tu e e e e e

3-2-2-The NIE Method for the ARL on the Modified EWMA Control Chart with the IMA Model
The NIE method is a technique for accurately approximating the ARL that can be used to determine the efficacy of explicit formula derivations. In this section, NTARL(u) is defined by using the NIE method with Simpson's quadrature rule [37] to estimate the ARL on the modified EWMA control chart with the IMA model and exponential white noise. The integral equation in Equation 4 is solved by using a 2m +1 linear equation system on interval [l, h] with length 2m. After that, the weight of each point is determined as follows: the start and end points (wj = vj/3) and the even and odd points within the interval (wj = 4vj/3 and wj = 2vj/3, respectively) such that vj = (hl)/2m; j = 0, 1, 2, …, 2m. and xj+1 = jwj+1 + l. Finally, the NIE method provides:

3-3-The FIMA-Modified EWMA Control Chart
For some cases, parameter d of the IMA model must be expressed as a fraction (where − 1 2 ≤ ≤ 1 2 ) rather than an integer, thereby providing the fractional integrated MA (FIMA) model. For the pattern of backward shift operator B, the FIMA (a/b, q) model or Ft is defined as: where, a and b are constants ( < ). By applying the generalized Newton binomial theorem [38], , the FIMA model can be solved as: where, θ0 is the process average, θi (i = 1,2,…,q) are the coefficients of the FIMA model (-1< θi <1) and εt-i is exponential white noise at time t.

3-3-1-The Explicit Formula for the ARL on the Modified EWMA Control Chart with the FIMA Model
For the FIMA model, the modified EWMA statistic created by combining Equations 1 and 10 can be rewritten as: The interval for modified EWMA statistic Yt is defined under the lower (r) and upper (s) bounds before an observation that is out-of-control occurs as follows: The interval of can be derived as: The integral equation of the second kind is used to find the explicit formula for the ARL on a modified EWMA control chart with the FIMA model given initial value Y0 = u as follows: Moreover, EARL(u) can be rearranged by using the exponential distribution of the error term as: The uniqueness of the ARL formula was verified by using Banach's fixed point theorem. Equation 12 is solved by Subsequently, EARL(u) can be rewritten as: is put into Q as follows: After Q has been obtained and substituted into 13, then the explicit formula for ARL on the modified EWMA control chart with the FIMA (a/b, q) model called EARL(u) can be defined as: Eu e e e e e

4-Research Methodology
In this research, the explicit formulas used to calculate the ARL on the modified EWMA control chart with either the IMA or FIMA model are obtained from Equations 7 and 14, respectively, while the ARLs for the NIE method are obtained from Equations 8 and 15, respectively. Meanwhile, m = 500. The bound control limits are studied on exponential distribution with interval [0, ∞), where the lower control limit is 0 and the upper control limit is discovered for ARL0 = 370; the latter is assigned by using exponential parameter 0 for the in-control process and process mean u = 0 , Mt-1, Mt-2,..., Mt-d = 0 , Ft-1, Ft-2, Ft-3,...= 0 . After a change in the process mean, 1 = (1 + ) 0 becomes the exponential parameter for ARL1 (the out-of-control process), where is the mean shift size. In the two models, 1 is assigned with as 0.01, 0.02, 0.05, 0.10, 0.20, 0.50, 1.00, 1.50, or 2.00. The most efficient control chart achieves the lowest value of ARL1.
The difference between the ARLs using the two techniques can be expressed as the absolute percentage relative change (APRC) [39] as follows: In addition, the relative mean index (RMI) [40] can also be used and is computed as: where ARLi(x) is the ARL of a control chart for order i and ARLi(min) is the minimum ARL of all of the control charts for order i. The control chart obtaining the smallest RMI is the best at detecting a shift in the process mean The procedure shown in Figure 1 can be used to find solutions.

5-Results
The solutions were tested both experimentally and with real data. Experimentally, the explicit formulas were used to compute the ARL on the modified EWMA control chart with either an IMA or FIMA model. The ARL results from the NIE method were used to confirm the results attained by using the explicit formulas. After that, the explicit formulas were used to compute the ARL under various sets of conditions. Afterward, the proposed control chart with an IMA or FIMA model was used to detect shifts in the mean of real datasets.

5-1-Experimental Study
In Table 1, the results for the explicit formula and the NIE methods for the ARL on the modified EWMA control chart with various IMA and FIMA models for g = 1 are reported for = 0.05, 0.10, or 0.20, various ( = 1,2, … , ), and ARL0 = 370. Since all of the APRC results for both methods are almost 0, their accuracies are almost identical. In addition, the explicit formulas were suddenly calculated. Meanwhile, the computation time for calculating the ARL by using the NIE method was 13-15 seconds with the IMA model and 32-40 seconds with the FIMA model. Therefore, the explicit formulas can be used to effectively and speedily detect a shift in the process mean on the modified EWMA control chart with either the IMA or FIMA model and exponential while noise. The performance of the modified EWMA control chart with the IMA or FIMA model was studied with various values of g and (Tables 2 and 3, respectively). ARL1 was calculated for each shift size . Note that for g = 0, the modified EWMA control chart is the same as the original EWMA scheme. The ARL results for the two models in one direction show that as the value of g was increased, the performance of the control chart improved. Moreover, for each value of g, the modified EWMA control chart was more effective than the original EWMA scheme for a small shift size. When was increased, the modified EWMA control chart produced a smaller ARL. The original EWMA scheme at = 0.20 provided a good performance when was small and, for intermediate-to-large shift sizes, it was more efficacious at = 0.05.   Tables 4 and 5 provide the RMI computations using the results from Tables 2 and 3, respectively. The RMI solutions in Table 4 when varying g while was fixed show that the modified EWMA control chart for the largest value of g attained the smallest RMI at each value of for both models, thereby signifying its excellent efficiency in both cases. In addition, for all values of g, the modified EWMA control chart performed better than the original EWMA scheme except for when g = 0.2 and = 0.05. In Table 5, when varying while g is fixed, the RMI for the original EWMA scheme was the lowest at = 0.05 and is thus the most suitable under these conditions, which is in accordance with previous research [41] for the EWMA control chart with an IMA (1,1) model. The modified EWMA control chart with g = 0.2, 0.5, 1, or 2 produced lower RMI values when was higher. However, the RMI for a high value of g (5) was similar for all values of . Therefore, the modified EWMA control chart can be recommended when g is large and with any value of . The RMI results are graphically displayed in Figure 2.

5-2-Real Data Study
Oil and natural gas affect people in countless ways across the globe. They are used to fuel cars, heat homes, cook food, and generate electricity. Meanwhile, fuel consumption is steadily increasing while production is not. The price of oil and natural gas is influenced by the global stock market. For this research, the natural gas and WTI crude oil prices were used to analyze the IMA and FIMA models on the modified EWMA control chart: Dataset 1 is appropriate for the IMA model (the natural gas price from 1 March 2021 to 30 April 2021 [42]) while Dataset 2 is suitable for the FIMA model (the price of WTI crude oil from 1 Jan 2021 to 31 March 2021 [43]). For Dataset 1, the fitted equation for the IMA (2,1) model was = 0.0496 + − 0.925 −1 + 2 −1 − −2 and for Dataset 2, the approximated FIMA (1/2,2) model was = 1.248 + + 0.304 −1 + 0.342 −2 + 0.5 −1 + 0.125 −2 + 0.0625 −3 + ⋯. After that, the residuals of two observations were tested by using a statistical hypothesis for an exponential distribution. The results show that Datasets 1 and 2 attained ~(0.0496) and ~(1.248), respectively.
The ARLs on the modified EWMA control charts for the natural gas price dataset using the IMA (2,1) model with various values of and g are given in Table 6 while those for the WTI crude oil price dataset using the FIMA (1/2,2) model are provided in Table 7. The results using the two datasets are in the same direction (Figure 3). For a small shift, the modified EWMA control chart for both models and datasets was more effective than the original EWMA scheme for all g except for = 0.05 and g = 0.2. In addition, the modified EWMA control chart preformed the best when g was large for all .     The RMIs for the results in Tables 6 and 7 are reported in Tables 8 and 9, respectively, with a summary for both datasets being shown in Figure 4. In the first case, the modified EWMA control chart with the IMA or FIMA model for stable λ maintained their efficiency when g was enlarged. Moreover, the modified EWMA control chart could detect a shift in the process mean more quickly than the original EWMA scheme in most situations. Meanwhile, the modified EWMA control chart with the IMA or FIMA model produced better performances when λ was increased. However, for g = 5, the proposed chart and the original EWMA scheme attained the best efficacy at λ = 0.05. These results confirm those from the experimental study.

6-Conclusion
Herein, we present the modified EWMA control chart with IMA or FIMA models and exponential white noise and evaluate its performance by using the ARL. The equations for the IMA and FIMA models were rearranged by using a backward shift operator, after which they were merged with the modified EWMA statistic. The NIE method of the ARL for both models was derived under the control limits of the residuals to estimate the ARL, while explicit formulas for the two models were derived to solve the exact ARL. Afterward, the results obtained by using both techniques were compared to check their accuracy and computational speed. The original and modified EWMA control charts with the IMA or FIMA model were compared in terms of efficiency by using the results of ARL and RMI calculations. The modified EWMA control chart with either model was studied while varying the values of g and λ, and it was found that its efficacy improved as g was increased. Besides, increasing the value of λ enabled faster detection of process mean shifts on the modified EWMA control chart. Last, natural gas and WTI crude oil price datasets were used for the IMA and FIMA models, respectively. The results confirmed those obtained experimentally. These formulas could be applied to other real-life data following IMA and FIMA models, albeit the explicit formulas are limited to the residuals of an exponential distribution. Future studies will be conducted on explicit formulas for the ARL on modern control charts to improve their efficiency for parameter shift detection.

7-2-Data Availability Statement
These dataset to be oil and natural gas prices can be found here: https://investing.com/commodities/natural-gashistorical-data and https://investing.com/commodities/crude-oil-historical-data.