Modification of ARL for detecting changes on the double EWMA chart in time series data with the autoregressive model

This research aims to derive the average run length (ARL) evaluation of the double exponentially weighted moving average (double EWMA) control chart for observation data that follows exponential white noise in a time series model with an autoregressive model. Since most real-world data is automatically correlated, autoregressive models are available. Comparisons were made between the ARLs obtained using the explicit formula and the numerical integral equation (NIE) approach. The results showed that the explicit formula's use of the ARL outperformed the NIE approach in terms of computation time. After that, the efficacy of the exponentially weighted moving average (EWMA) and double EWMA charts is then compared using the suggested explicit ARL formula. The ARL of the double EWMA chart was found to perform better than the ARL of the EWMA chart in all situations. It also uses natural gas and diesel prices on stock exchanges around the world as cases studies. The results show that the double EWMA chart has better detection sensitivity than the EWMA chart, and the results are consistent with the experimental results. As a result, the sensitivity of the double EWMA chart in detecting changes makes it a good alternative for monitoring processes with real-world data.


Introduction
Time-series data analysis has been the subject of extensive research.Currently, it is utilised for predicting a variety of circumstances, including stock trends epidemic diseases, export sales and cost prediction (Cheng et al., 2020;Satrio et al., 2021).There are common uses for these models as an autoregressive (AR), a moving average (MA), and an autoregressive moving average (ARMA).Statistical process control (SPC) has been widely used in the industrial sector to monitor, control, and improve operations.Many researchers have studied SPC using real-world data and case studies.For instance, Tegegne et al. (2022) suggested a new kind of multivariate control chart for the manufacturing process, which was utilised to better decide on quality control during the production process.They were conducted to validate the control chart by using a case study from the cement industry.Yeganeh et al. (2023) employed SPC techniques, particularly control charts, to track the probability of connections between network surveillance system nodes using control charts based on machine learning.
The control chart is widely used in the fields of finance, business, engineering, healthcare, and others to track processes and detects sudden changes in the mean or variance of those processes, especially in the field of industries (Fugen, 2019;Zhang et al., 2022).Using control charts, it is possible to distinguish between common and unique sources of process variation and handle each case separately.The control limits on control charts are determined by common factors.Control charts provide precise instructions on how and when to modify a process.Nowadays, control charts are an essential tool in SPC, and their industrial significance lies in their ability to monitor and control process variability, improve process efficiency, reduce costs, ensure product quality, and support decision-making.There are many types of research that were presented that utilised the control chart in the field of industries.Shrestha (2021) used a control chart to control the sausage manufacturing process.Tegegne et al. (2022) used a control chart to decide on quality control during the production process in their case study of the Ethiopian cement industry, and so on.
The concept of the monitoring process was initially introduced in Shewhart's control chart (1931).The quality of the Shewhart control chart is only effectively achieved by identifying large shifts in the process parameters.There are many viable alternatives to wellknown and efficient technology for detecting and monitoring slight changes in processes.The cumulative sum (CUSUM) control chart was created by Page (1954), and Montgomery (2009) provided more information on the chart's specifics.The CUSUM control chart was employed in the current investigation because of its greater capacity to identify minor parameter adjustments and to benefit from the advantages of employing an upper-sided control chart.The exponentially weighted moving average (EWMA) control chart was suggested by Roberts (1959), and it is generally recognised that it may be used to monitor and identify slight changes.After that, several researchers proposed a variety of different control charts utilising the EWMA control chart development to improve the ability to detect slight changes.The modified exponentially weighted moving average (modified EWMA) control chart was devised by Khan et al. (2017) after being initially proposed by Patel and Divecha (2011).The extended exponentially weighted moving average (extended EWMA) control chart was suggested by Naveed et al. (2018).The extended EWMA statistics, which can detect minor changes faster than the EWMA control chart shown in Karoon et al. (2022) and Karoon et al. (2023), were created by adding the EWMA statistics with the exponential smoothing parameters.This approach was shown to be more successful at recognising tiny changes.Additionally, Shamma and Shamma (1992) introduced the double exponentially weighted moving average (double EWMA) control chart before Mahmoud and Woodall (2010) offered it.The double EWMA control chart performs better than the EWMA control chart when the monitor process is utilised with small shift sizes rapidly.It is now generally acknowledged that a reliable control chart is essential for quickly detecting subtle process changes.
The average run length (ARL) is frequently utilised to assess control chart performance.The ARL is separated into two states; the ARL is ARL 0 , which is described by the value as it happens from the in-control state and should be large, and ARL 1 , which is explained by the value as it occurs from the out-of-control stage and should be as tiny as feasible.
A variety of methods can be employed to estimate the ARL of the control chart, including Monte Carlo and Markov Chain.Moreover, there is Numerical Integral Equations (NIE) method (see Karoon et al., 2021).Also, Areepong mentioned the NIE method for SPC in 2009.For quality control, it was used to identify the distribution change point.To evaluate the principal components of EWMA control charts for various distributions, NIE was proposed to derive a numerical algorithm and infer precise analytical formulas.In addition, ARLs were calculated using explicit formulas.There is a large body of literature that cites the explicit ARL formulas for control charts with time series models.Due to reality, the control chart's performance may suffer if the observations are serially associated.For this reason, the ARL has been suggested by several researchers when there is a serial correlation in the observations.Areepong and Sukparungsee (2016) proposed an explicit formula for the ARL on the EWMA control chart with the ARIMA model.After that, Sunthornwat et al. (2017) derived the analytical ARL on the EWMA control chart for a long-memory autoregressive fractionally integrated moving average (ARFIMA) model.On the EWMA control chart, Peerajit et al. (2017) proposed the explicit formulas for the derivation of the ARL using the data are the ARFIMA model on the CUSUM chart.Later, The capacity of the ARL was put out by Supharakonsakun et al. (2020), and it was obtained from an explicit formula for the MA(1) model of the modified EWMA control chart.Supharakonsakun (2021) proposed an explicit formula of the ARL with observations of the MA(p) model.After that, Phanyaem (2022) presented an explicit formula and NIE method of the ARL for the SARX(p,r) L model based on the CUSUM control chart.Petcharat (2022) proposed the ARL of the CUSUM control chart when the data are SARX(p) L model.Moreover, Karoon et al. (2022) proposed an explicit formula of ARL on the extended EWMA control chart when the observations are the AR(p) model.Recently, Peerajit and Areepong (2023) proposed the ARL of an autoregressive fractionally integrated process with exponential white noise running on the modified EWMA control chart.
As was already indicated, these results show that the ARL may be used to compare efficiency with control charts and that explicit formulas require considerably less calculation time to assess the ARL than the other approaches.However, the explicit formulas for the double EWMA chart have not yet been released.As a result, the focus of this research was on developing an ARL for using an AR model to monitor change in a time series.This method is based on the double EWMA chart with underlying exponential white noise in different scenarios.The ARL that was obtained by an explicit formula was provided.And then, that is expanded for comparison with the typical EWMA control chart.Besides, the ARL for them is also extended to compare processes with the diesel and natural gas price datasets.Roberts (1959) created the first suggestion for the EWMA chart.It is a tool that is typically used to monitor and identify slight changes in the process.The equation below can be used to express the statistics of the EWMA control chart:

EWMA control chart
where the EWMA chart parameter Y t is a sequence of autoregressive AR(p) model and a sequence data at t = 1, 2, 3, . . .with exponential white noise, λ is an exponential smoothing parameter (0 < λ ≤ 1), Y t (t = 0) is the initial value of the EWMA statistics that is equal to μ and the variance of Y t is λσ 2 /(2 − λ).Therefore, the upper control limit (UCL) and lower control limit (LCL) of the EWMA chart can be established from the mean μ, the standard deviation σ , and a suitable control width limit Q E as follows below: The stopping time of the EWMA control chart is expressed as: where h is the upper control limit, a constant parameter.

Double EWMA control chart
The double EWMA control chart was proposed by Shamma and Shamma (1992) and Mahmoud and Woodall (2010) made it available.It was adjusted by two exponential smoothing and then extended from the EWMA chart.The equation below may be used to describe the statistics of the double EWMA chart.
where the double EWMA control chart parameter Y t is a sequence of autoregressive AR(p) model and a sequence data at t = 1, 2, 3, . . .with exponential white noise, λ 1 and λ 2 is an exponential smoothing parameter 0 < λ 1 , λ 2 ≤ 1, Y t (t = 0) is the initial value of the double EWMA statistics that is equal to μ and the variance of Y t is . Therefore, the upper control limit (UCL) and lower control limit (LCL) of the EWMA control chart can be established from the mean μ, the standard deviation σ , and a suitable control width limit Q D as follows below: . ( 5) The stopping time of the EWMA control chart is expressed as: where h is the upper control limit, a constant parameter.However, λ 1 values of the double EWMA statistics were replaced by 1, and the double EWMA statistics is the EWMA statistics.Page (1954) introduced the CUSUM control chart in quality control for identifying minor variations in process mean.The equation below can be used to express the statistics of the CUSUM chart.

Cumulative Sum control chart (CUSUM)
where a is non-zero constant, C 0 = u is the initial value of CUSUM with u ∈ [0, h], h is upper control limit (UCL), and the stopping time of the CUSUM chart is defined as

The ARL on double EWMA control chart for time series with AR model
The time series is a collection of chronologically organised data.The time series is observed and studied in order to identify its patterns of change and development and forecast its future trend.Time-series data can be divided into two categories: stationary data and nonstationary data.The time-series data collection with stationary data does not show a trend or a seasonal influence.The data collection solely contains random error as a cause of variation, whereas non-stationary time-series data is a time-series data set that exhibits a trend or a seasonal effect.There are other sources of variation in the data set besides just random error.The Box-Jenkins method is currently the most complete and accurate algorithm for analysing and predicting time series data.Methods for measuring timed data are referred to as times series.Autoregression (AR), Moving Average (MA), Autoregressive Moving Average (ARMA), Autoregressive Integrated Moving Average (ARIMA), and Seasonal Autoregressive Integrated Moving-Average (SARIMA) are examples of common types.
One of the most commonly used models is the autoregressive model, which was also used in this study.Regression models, known as autoregressive (AR) models, have a dependent or response variable that is a linear function of its previous values.The autoregressive model has an important parameter r, p, which is the order of an autoregressive term and is often used for actual data.The AR(p) model is defined by the following equation: where ω is the constant of the model, where ARL 0 refers to the in-control ARL with θ = ∞, which denotes no change in the statistical process.On the other hand, ARL 1 refers to the out-of-control ARL with θ = 1, which designates the first-time point in the statistical process at which a change occurs place from β 0 to β 1 .

Explicit formulas of ARL on Double EWMA control chart for AR model
In this section, the explicit formula of the ARL is solved on the double EWMA chart with the AR model.Let's start by substituting Equation ( 9) into Equation ( 4) as follows: The double EWMA statistics with AR(p) can be described as: In control process, the interval of D 1 between the lower and upper bound control limits are expressed to be l and h can be written as follows below.
On the variable ζ 1 , this interval can be rewritten as: The Fredholm integral equation is used to express the integral equation of the ARL on the double EWMA control chart for the AR(p) model with an initial value D 0 = ψ.The equation represented is as follows: obtained by substituting the integration variable, L(ψ) is restructured as follows: As the function of ζ 1 has an exponential distribution, L(ψ) can be expressed as follows: The ARL solution is verified by Banach's fixed-point theorem.This is considered an ARL solution in terms of its existence and uniqueness (see Sofonea et al., 2005), from Equation ( 11), suppose that Consequently, the ARL solution that is provided in Equation ( 11) can be reformulated by establishing additional variables as follows: After that, the integral equation ρ, which can be written as: Finally, Equation ( 12) is substituted into the solution of ρ as shown in Equation ( 13), which is obtained as: Since the AR(p) model is used with the double EWMA chart, the explicit ARL formula is given in Equation ( 14).Moreover, the in-control process is replace β with β 0 , whereas the out-ofcontrol process is replace β with β 1 , as well as β 1 = (1 + δ)β 0 , and δ denotes the shift size in the monitoring process.

Numerical integral equation of ARL on double EWMA control chart for AR model
The numerical integral equation method, or NIE method, of the ARL is used to compare with the explicit formula in terms of the computational time spent.Let N(ψ) denote the ARL of the double EWMA control chart on the AR(p) model with an exponential white noise, with the ARL solution being computed according to the midpoint quadrature rule.Specifically, after selecting a quadrature rule, the interval In accordance with the quadrature rule, the approximation for an integral is assessed as follows: Let N(v i ) represent a numerical approximation to the integral equation.It is the result of solving the following linear equations: The m linear equation system is shown as exists, the unique solution is shown as: where ) is the unit matrix order m, 1 m×1 = [1, 1, . . ., 1] is a column vector of N(v i ), and R m×m is a matrix, the definition of (m,mth) elements of R matrix is expressed as Finally, v i is instead of ψ into N(v i ), the approximation of the NIE method of the ARL is represented as follows: where η j is a set of the division point within the interval η j = (j − 0.5)w j + l for j = 1, 2, . . ., m and w j is a weight of the composite midpoint formula; w j = (h − l)/m.

The existence and uniqueness of the explicit formula for the ARL based on the double EWMA control chart
In this section, the existence and uniqueness of the ARL solution are confirmed using Banach's fixed-point theorem.Banach's Fixed-point Theorem, sometimes referred to the contraction theorem, is concern with certain mappings of a complete metric space into itself.As we shall see, a fixed point is a point that is mapped to itself, and this statement establishes requirements adequate for the existence and uniqueness of a fixed point.

The performance measurement of control chart
The performance of control charts is commonly evaluated in terms of the average run length (ARL).It is used to detect changes in the mean of the monitoring process.As a result, the performances of the ARL, which are produced by explicit equations, and the NIE technique, which is used to monitor shifts with an autoregressive model, or AR model, on the double EWMA chart, were compared.The absolute percentage relative change (%APRC) indicates the relative performance of two methods of the ARL (Phanthuna & Areepong, 2022), which is defined as follows: After that, the performance of the ARL is then evaluated with different parameter values based on the double EWMA chart.And then it is extended for comparison with the EWMA chart.The ARL measurement is appropriate to evaluate the effectiveness of control charts in the process parameters when shift sizes are different.The performance of a control chart throughout a range of shifts (δ min ≤ δ ≤ δ max ) can be evaluated using overall performance measures, which are recommended by several researchers.Some of them have metrics that are used to assess their performance, such as the average extra quadratic loss (AEQL) and the performance comparison index (PCI) (see Alevizakos et al., 2021).
The AEQL is described mathematically as     Table 3. Comparing the ARL values on the double EWMA and EWMA control charts for the AR(2) model with different λ 1 and determined λ 2 = 0.1 and ω = 0.  where the specific of shift in the process is represented by δ, and then is the sum of number of divisions from δ min to δ max .In the study, = 11 is determined from δ min = 0 to δ max = 1.The most effective control chart is those with the lowest AEQL values.The PCI measurement is the ratio of the AEQL of the control chart and the AEQL of the most effective control chart (shown as the AEQL lowest ).It is described mathematically as The most effective control chart's PCI value is 1, whereas the values of the other control charts in competition are more than 1.
Table 4. Comparing the ARL values on the double EWMA and EWMA control charts for the AR(3) model with different λ 1 and determined λ 2 = 0.1 and ω = 0.

The results
A common criterion for assessing the efficacy of a control chart is the ARL for identifying alterations in the process.That is an evaluation of its practicality and sensitivity.
In this study, the efficiency of the explicit formula and the NIE method for determining the ARL for monitoring shifts were compared.These were evaluated utilising the double EWMA chart and the autoregressive (AR) model.The AR(1), AR(2), and AR(3) autoregressive models were expressed.The number of division points m = 500 is determined for the approximation of the ARL by the NIE method.The solutions have been split into two categories: simulated and real-world data.For an initial ARL, In the in-control process scenario, ARL 0 is followed by exponential white noise with a mean of β = β 0 = 1.
In the out-of-control process scenario, ARL 1 was computed with parameter values as exponential white noise with a mean of β = β 1 , as β 1 = β 0 (1 + δ), and then δ equals 0.001, 0.002, 0.003, 0.005, 0.01, 0.03, 0.05, 0.1, 0.5, 1.As previously stated, "exponential white noise" is defined as the residual of an exponential distribution with uncorrelated data.As a result, δ = 0 indicates that the process is in-control, whereas δ > 0 indicates that the process is out-of-control.ARL 1 could have a low value on the efficacy control chart.Also, ARL 0 was determined at 370, and the Mathematica, spec Intel(R) Xeon(R) CPU X5680 @ 3.33 GHz 3.33 GHz (2 processors) RAM 32.0 GB was used to evaluate through study.
The process can be succinctly stated as follows: Step 1: Input parameters for example, the coefficients of autoregressive; φ i , the initial values of the autoregressive; Y t−1 , Y t−2 , . . ., Y t−p , and the control chart parameters; λ 1 , λ 2 .
Step 2: Impose the parameter of exponential white noise;β 0 for the in-control process and assign the parameter of exponential white noise;β 1 = (1 + δ)β 0 for the out-ofcontrol process.Step 3: Determine the initial value of known parameter; ω and the initial value of the double EWMA statistic; ψ.
Step 4: Specify the lower control limit (l) equals 0 and fixed the initial value ARL 0 for computing compute the upper control limit (h).
Step 5: Compare the ARL of the explicit formula and the NIE method.Step 6: Compute the solution of ARL 1 for shift sizes in the monitoring process where β 1 = (1 + δ)β 0 , and then the upper control limit (h) is the result of Step 4.

Simulated data
The aforementioned algorithm was used to determine ARL 0 and ARL 1 , as well as to evaluate the efficiency of the proposed ARL by the explicit formula and the ARL by the NIE method in situations where the in-control and out-of-control processes were present.The results computed based on specified parameters such as φ i and ω for ARL 0 = 370 using the ARL on the double EWMA chart with the AR(p) model as AR(1), AR(2), and AR(3) models obtained from two techniques are reported in Table 1.The coefficient parameters for the AR(1), AR(2), and AR(3) models were set as φ The ARL values of the explicit formula technique denote as L(ψ), and it is computed by using Equation ( 14).And then, the NIE method denotes as N(ψ), it is calculated by Equation ( 17).The %APRC of all situations is extremely low, nearly equal to 0. However, the computational times for the ARL values calculated using the NIE technique are about 3.7-4.3s, while the explicit formula appears practically instantly for all situations.Therefore, using these explicit formulas as the next step is a good idea.
Next, the effectiveness of the explicit ARL based on the double EWMA chart running on the AR(p) model is compared with the EWMA and CUSUM charts and then investigated using various λ.Moreover, the double EWMA chart serves as the EWMA chart if instead of λ 1 equals 1. ARL 0 value was fixed as 370 for in-control situation.ARL 1 values were used to compare the efficiency of the EWMA, double EWMA, and CUSUM charts and were set to values mentioned above.The results in Tables 2-4 show that the lower

Real-world data
The prices of diesel and natural gas are shown in Figure 1, with the time series brought up for study in this part using the AR(1) and AR(2) models, respectively.Two datasets, which comprise quarterly price data, were selected as the models by SPSS.The results are in Table 5 shows that two datasets are suitable parameters for the AR model.The Kolmogorov-Smirnov test was then used to determine that white noise significantly fits the exponential mean, as shown in Table 6.For the diesel price, it was fitted to the AR(1), which expressed as Y t = 3.772 + 0.943Y t−1 + ζ t , ζ t ∼ Exp(0.2194).For the natural gas price, it was fitted to the AR(2), which expressed as Y t = 16.473+ 0.373Y t−1 − 0.256Y t−2 + ζ t , ζ t ∼ Exp(2.8641).
Tables 7-8 show that the results of the comparison of control charts between the double EWMA and EWMA charts and the results show the double EWMA chart with the lower λ 1 had a lower ARL 1 and was more effective than the EWMA chart in all situations.However, the performance of CUSUM charts is not compared in this section because CUSUM charts give ARL1 values that are much higher than EWMA and EWMA charts.Moreover, the results show the outcomes of two datasets in a way similar to simulated data, as illustrated in Figure 2. The AEQL and PCI values supported the control chart's effectiveness.The results verify that the double EWMA chart with λ 1 = 0.05 had the lowest AEQL and PCI equals 1 outperformed the double EWMA chart with higher λ 1 and the EWMA chart, all of which have higher AEQL and PCI > 1, as illustrated in Figure 3.
Additionally, Figures 4 and 5 also show how well the control chart performs to detect shift changes in the monitoring process.For the AR(1) model in Figure 4, the double EWMA control chart (λ 1 = 0.05) recognises shifts as being out of control at the first observation, whereas the EWMA control chart (λ 1 = 1) does so at the 15th observation.For the AR(2) model in Figure 5, the double EWMA control chart (λ 1 = 0.05) recognises shifts as being out of control at the first observation, whereas the EWMA control chart (λ 1 = 1) does so at the 34th observation.The results show that in the monitoring process, the double EWMA control chart may detect shift changes faster than the EWMA control chart.Besides, An excellent option for spotting change is the double EWMA control chart, which performs better than the EWMA control chart.And then, the exponential smoothing parameter λ 1 = 0.05 is recommended for the double EWMA control chart.

Discussions and conclusions
The explicit formula for the ARL on the double EWMA control chart was put out in this study.It is faster than the NIE method in computing the ARL for the AR(p) model with exponential white noise.The proposed ARL is compared with the ARL on the EWMA control chart based on in-control and out-of-control scenarios.It was found that the double EWMA control chart's ARL consistently outperformed the traditional control chart's ARL.The double EWMA chart with the lower λ 1 had a lower ARL 1 and was superior to the EWMA chart in terms of effectiveness for all cases.The exponential smoothing parameter λ 1 = 0.05 is recommended.The efficacy of the control chart was validated by the AEQL and PCI values.Moreover, the performance of the ARL was expanded to include use with real-world datasets.The prices of diesel and natural gas were used as real-world datasets.The outcomes of two datasets are displayed in the findings in a style equivalent to simulated data.However, the double EWMA control chart can be expanded for future study in order to integrate with more models and cover more distributions.
and then initial values of the autoregressive AR(p) model; Y t−1 , Y t−2 , . . ., Y t−p are equal to 1.In earlier studies, ARL formulae for monitoring change processes using data that were characterised as the AR model were developed utilising the equation of the AR model in Equation (9).For instance, Phanthuna et al. (2021), and Karoon et al. used the AR model in Equation (9) to fit with ARL for the EWMA type control chart.For the double EWMA control chart running an autoregressive model under exponential white noise in this study, let L(ψ) represent the ARL conditional on the initial value.The initial value for monitoring the double EWMA statistic D 0 = ψ is established at ψ ∈ [0, h].Let P θ denote the induced expectation and probability measure associated with ψ at change-point (θ ) under the density function f (y, β).The ARL for AR(p) model on the double EWMA control chart defined with L(ψ) can be described as follows:

Theorem 3. 1 (
Banach's Fixed-point Theorem:): Let (Y, d) be a complete metric space and let T : Y → Y be a contraction mapping on Y. Then T has a unique fixed point y ∈ Y(such that T(y) = y) with contraction constant r ∈ [0, 1)

Figure 1 .
Figure 1.The time series of datasets (a) Diesel price with AR(1) and (b) Natural gas price with AR(2).

Figure 2 .
Figure 2. ARL values of double EWMA and EWMA charts for (a) the AR(1) and (b) the AR(2) models.

Figure 3 .
Figure 3.The AEQL and PCI values of two models: (a) and (b) for the AR(1), (c) and (d) for the AR(2).

Figure 4 .
Figure 4.The effectiveness of detecting shift change in the monitoring process on the (a) EWMA and (b) double EWMA charts for the AR(1) model.

Figure 5 .
Figure 5.The effectiveness of detecting shift change in the monitoring process on the (a) EWMA and (b) double EWMA charts for the AR(2) model.

Table 1 .
Comparing the ARL values of the explicit formula against the NIE method for AR(p) models on the double EWMA control chart with the initial ARL 0 = 370 and ω = 0 based on different situations.

Table 2 .
Comparing the ARL values on the double EWMA and EWMA control charts for the AR(1) model with different λ 1 and determined λ 2 = 0.1 and ω = 0.

Table 5 .
The AR(p) coefficients for two datasets from the first quarter 2000 to the second quarter 2022.

Table 6 .
Testing the datasets for the exponential white noise of the exponential distribution.

Table 7 .
Comparing the ARL values on the double EWMA and EWMA control charts for the AR(1) model of the diesel price ($/gallon) with different λ 1 and given λ 2 = 0.1 and ω = 0.

Table 8 .
Comparing the ARL values on the double EWMA and EWMA control charts for the AR(2) model of the natural gas price ($/mcf) with different λ 1 and given λ 2 = 0.1 and ω = 0.