Exact Run Length Sensitivity of DEWMA Control Chart Based on Quadratic Trend Autoregressive Model

The double exponentially weighted moving average (DEWMA) chart is a control chart that is a vital analytical tool for keeping track of the quality of a process, and the sensitivity of the control chart to the process is evaluated using the average run length (ARL). Herein, the aim of this study is to derive the explicit formula of the ARL on the DEWMA chart with the autoregressive with trend model and its residual, which is exponential white noise. This study shows that this proposed method was compared to the ARL derived using the numerical integral equation (NIE) approach, and the explicit ARL formula decreased the computing time. By changing exponential parameters that were relevant to evaluating in various circumstances, the sensitivity of AR(p) with the trend model with the DEWMA chart was investigated. These were compared with the EWMA and CUSUM charts in terms of the ARL, standard deviation run length (SDRL), and median run length (MRL). The results indicate that the DEWMA chart has the highest performance, and when it was small, the DEWMA chart had high sensitivity for detecting processes. Digital currencies are utilized to demonstrate the efficacy of the proposed method; the results are consistent with the simulated data. Doi: 10.28991/ESJ-2023-07-06-03 Full Text: PDF

The primary objective of this research is to propose explicit formulas for the Average Run Length (ARL) of the Double Exponentially Weighted Moving Average control chart (DEWMA) for the Seasonal Moving Average process (SMA (Q)L) with exponential white noise. The Numerical Integral Equation by the midpoint rule is employed to compare the results derived from the formulas and evaluate their accuracy using the percentage of accuracy (%Acc). The DEWMA control chart's efficacy is measured by calculating the average run length (ARL), median run length (MRL), and standard deviation of run length (SDRL). Significant agreement was observed between the numerical approximations and the explicit formulations for SMA(2)4 and SMA(3)12 processes. This finding indicates the formulations are sufficiently precise. A comparison of Exponentially Weighted Moving Average (EWMA) and DEWMA control charts relating to mean process variations is performed. For practical data, WTI oil prices are used to determine the efficacy of the explicit formula approach.

Control charts are frequently used instruments for process quality monitoring. Another name for the NEEWMA control chart is the new extended exponentially weighted moving average (new extended EWMA) control chart. The lower control limit (LCL) and upper control limit (UCL) are equally spaced from the center line, giving it a symmetrical design. Because of its symmetry, the NEEWMA chart is very good at identifying even the tiniest changes in operation by detecting deviations from the target in both upward and downward directions. This study derives an explicit formula for the average run length (ARL) of the NEEWMA control chart based on the autoregressive (AR) model with exponential white noise. The focus is on the zero-state performance of the NEEWMA control chart, which is derived using explicit formulas. Banach’s fixed-point theorem was used to prove existence and uniqueness of this formula. The accuracy of this formula is validated by comparing it to the numerical integral equation (NIE) method using percentage accuracy (%Acc). The results show that the NEEWMA control chart is more efficient than the ARL evaluated by the NIE method, particularly regarding computation time. The performance of the NEEWMA control chart is compared with the EWMA and extended EWMA control charts by evaluating both the ARL and standard deviation run length (SDRL). The NEEWMA control chart outperforms the others in detection performance, followed by the extended EWMA and EWMA control charts. Further verification of its superior performance is provided through comparisons using the average extra quadratic loss (AEQL) and the performance comparison index (PCI), which confirm that it outperforms both the EWMA and extended EWMA control charts across various parameters and shift sizes. Finally, an illustrative example using real-life economic data demonstrates its efficiency.

This research aims to investigate a Homogenously Weighted Moving Average (HWMA) control chart for detecting minor and moderate shifts in the process mean. A mathematical model for the explicit formulae of the average run length (ARL) of the HWMA control chart based on the autoregressive (AR) process is presented. The efficacy of the HWMA control chart is evaluated based on the average run length, the standard deviation of run length (SDRL), and the median run length (MRL). As illustrations of the design and implementation of the HWMA control chart, numerical examples are provided. In numerous instances, a comparative analysis of the HWMA control chart relative to the Extended Exponentially Weighted Moving Average (Extended EWMA) and cumulative sum (CUSUM) control charts with mean process shifts is performed in detail. Additionally, the relative mean index (RMI), the average extra quadratic loss (AEQL), and the performance comparison index (PCI) are utilized to evaluate the performance of control charts. For various shift sizes, the HWMA control chart is superior to the Extended EWMA and CUSUM control charts. This study applies empirical data from the area of economics to validate the explicit formula of ARL values for the HWMA control chart. Doi: 10.28991/HIJ-2024-05-01-02 Full Text: PDF

The main goal of this study is to establish explicit solutions for the average run length (ARL) of the Homogenously Weighted Moving Average control chart when subjected to autoregressive with trend process. The accuracy of the explicit formula for the ARL is evaluated in comparison to the numerical integral equation method. To evaluate the two approaches, the accuracy percentage was employed. A determination is carried out of the HWMA control chart’s effectiveness using the median run length (MRL), the standard deviation of run length (SDRL), and the average run length (ARL). A comprehensive comparison is performed between the HWMA control chart, the Extended Exponentially Weighted Moving Average (EEWMA), and the cumulative sum (CUSUM) control charts with mean process shifts to illustrate the design and implementation of the HWMA control chart. As criteria for various values of design parameters, the performance of these control charts can also be evaluated using the relative mean index (RMI), the average extra quadratic loss (AEQL), and the performance comparison index (PCI). To evaluate the effectiveness of our explicit formula approach, we employ this formula on copper price data.

In this paper, two single control charts, i.e. the Maximum Double Exponentially Weighted Moving Average (Max-DEWMA) and the Sum of Squares Exponentially Weighted Moving Average (SS-DEWMA) control charts, which can simultaneously monitor the process mean and/or variability, are compared. The existing comparison based on average run length (ARL) criterion reveals that the optimal SS-DEWMA chart generally gives more favorable results compared to the optimal Max-DEWMA chart for detecting shifts in the process mean and/or variability. However, an interpretation solely based on ARL can be misleading. Therefore, the main objective of this study is to compare the performances of the two charts by evaluating their median run lengths (MRLs), when the process mean and/or variability shift at various magnitudes. A Monte Carlo simulation is conducted using the Statistical Analysis Software (SAS). Overall, the MRL results are in accordance with the ARL findings.

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. Doi: 10.28991/ESJ-2023-07-05-020 Full Text: PDF

Traditional control charts usually fail to perform well in the case of outliers, or when the mean and variability of a process vary at the same time in quality control. In this paper, a very powerful exponentially weighted moving average (EWMA) control chart using the robust coefficient of variation (CV) has been proposed to overcome these hurdles. Unlike approaches that require data transformation or adaptive tuning, the scale‐invariance of the robust CV‐statistic can be used to stabilize the proposed RCV‐EWMA chart, keeping it sensitive in a polluted data setting. Through extensive Monte Carlo simulations, the capabilities of the chart are analyzed in different conditions of contamination levels and shift level. The total key performance indicators, such as Average Run Length (ARL), Standard Deviation of Run Length (SDRL), and Median Run Length (MRL) are given. The findings show that the suggested chart has good in‐control characteristics (ARL = 370) and is a worthwhile chart to detect small‐to‐still shifts despite up to 25% contamination. Its applicability in practice is also confirmed by a real data application that demonstrates how the chart can determine the abnormalities in the processes and be resistant to outliers. The results place the RCV‐EWMA control chart as an effective and dependable modern tool of monitoring processes especially those in very noisy or highly variable conditions.

The mixed control chart is proposed to improve detection performance with fewer process shifts. In this study, we proposed the modified exponentially weighted moving average - moving average control chart (MMEM), a new mixed control chart for observing the changes in the process mean. Average run length, standard deviation of run length, and median run length can be used to examine the effectiveness of detecting changes in the proposed chart with Shewhart, Moving Average (MA), Modified Exponentially Weighted Moving Average (MEWMA), and Mixed Moving Average - Modified Exponentially Weighted Moving Average (MMME) control charts in parametric and nonparametric distributions that use Monte Carlo simulation. The results demonstrate that the proposed chart outperforms other control charts mostly in the detection of small-to-moderate shifts. To illustrate the application of the proposed chart, chemical process temperature data and dataset on survival times of a group of patients suffering from head and neck cancer disease and treated with radiotherapy were provided, and it was discovered that the proposed chart performs better than other control charts.

Quality-control charts are widely used to monitor and detect shifts in the process mean and dispersion. Abbasi and Miller [MDEWMA chart: an efficient and robust alternative to monitor process dispersion, J Stat Comput Simul 2013;83:247–268] suggested a robust mean deviation exponentially weighted moving average (MDEWMA) control chart for monitoring process dispersion under simple random sampling. In this study, an improved MDEWMA (IMDEWMA) control chart is proposed under ranked set sampling to monitor process dispersion. Detailed Monte Carlo simulations are performed from symmetric and asymmetric populations to investigate the performances of the proposed and existing control charts in terms of average run length (ARL), median run length and standard deviation of run length. An application to real-life data is also presented to illustrate the use of the IMDEWMA control chart. It is observed that the IMDEWMA control chart indicates a shift in process dispersion substantially quicker than the MDEWMA control chart, while maintaining comparable ARLs when the process is in control.

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.

A control chart is a powerful statistical process monitoring tool that is frequently used in many industrial and service organizations to monitor in‐control and out‐of‐control performances of the manufacturing processes. Cumulative sum (CUSUM) and exponentially weighted moving average (EWMA) control charts have been recognized as potentially powerful tool in quality and management control. These control charts are sensitive to both small and moderate changes in the process. In this paper, we propose a new CUSUM (NCUSUM) quality control scheme for efficiently monitoring the process mean. It is shown that the classical CUSUM control chart is a special case of the proposed controlling scheme. The NCUSUM control chart is compared with some of the recently proposed control charts by using characteristics of the distribution of run length, i.e. average run length, median run length and standard deviation of run length. It is worth mentioning that the NCUSUM control chart detects the random shifts in the process mean substantially quicker than the classical CUSUM, fast initial response‐based CUSUM, adaptive CUSUM with EWMA‐based shift, adaptive EWMA and Shewhart–CUSUM control charts. An illustrative example is given to exemplify the implementation of the proposed quality control scheme. Copyright © 2013 John Wiley & Sons, Ltd.

The main objective of this study is to find an explicit formula for the average run length (ARL) of a Homogenously Weighted Moving Average control chart (HWMA) for an autoregressive process with a quadratic trend under zero state. The two-sided HWMA control chart construction procedure is proposed, and the performance of the control chart is measured using the average run length (ARL), standard deviation run length (SDRL), and mean run length (MRL). In addition, the accuracy of the explicit formula for ARL is compared with the accuracy of the numerical integral equation method. In this research, the performance of the HWMA and modified exponential weighted moving average control charts (MEWMA) for quadratic trend AR(1) and AR(3) models at different levels of process average change is compared. The efficacy of these control charts can additionally be assessed by the EARL, ESDRL, and EMRL metrics. The proposed control chart is applied to simulated and actual data, namely the finished goods inventory index of PCBA.

ABSTRACTConventional control chart schemes often assume normality when monitoring processes, a condition that is not consistently applicable. The aforementioned constraint is notably apparent in a variety of engineering dealings, especially when characterized by the prevalence of the inverse Maxwell process. An important development in the field of process monitoring is the growing importance of an adaptive exponentially weighted moving average () control chart. This study proposed an control chart for inverse Maxwell processes, abbreviated as . The chart is evaluated using several criteria, including average run length, median run length, and standard deviation run length. The comprehensive analysis includes extra quadratic loss, relative average run length, and performance comparison index. A two‐step optimization procedure was utilized to identify the optimal design parameters, ensuring that the monitoring scheme is capable of effectively detecting both large and small shifts. To evaluate the effectiveness of the chart, it is compared to other charts in the same family, for inverse Maxwell distribution, such as control chart, and exponentially weighted moving average ( control chart. The results indicate that the proposed chart exhibits higher efficiency compared to its competitors. To demonstrate the practical application of the proposed control chart, two real‐life examples are provided, using brake pad failure data and the COVID‐19 estimated reproduction number to demonstrate its utilization in an actual data set. The findings of our study reveal the significant role of the proposed chart in improving process monitoring. Its effectiveness in detecting shifts in different scenarios has been established, as evidenced by its successful use in practical settings.

Among various statistical process -control (SPC) methods, control charts are widely employed as essential instruments for monitoring and improving process quality. This study focuses on a new modified exponentially weighted moving-average (New Modified EWMA) control chart that enhances detection capability under integrated and fractionally integrated time-series processes. Special attention is given to the effect of symmetry on the chart structure and performance. The proposed chart preserves a symmetric monitoring configuration, in which the two-sided design (LCL>0) establishes control limits that are equally spaced around the center line, enabling balanced detection of both upward and downward shifts. Conversely, the one-sided version (LCL=0) introduces a deliberate asymmetry to increase sensitivity to upward mean shifts, which is particularly useful when downward deviations are physically implausible or less critical. The efficacy of the control chart utilizing both models is assessed through Average Run Length (ARL). Herein, the explicit formula of ARL is derived and compared to the ARL obtained from the Numerical Integral Equation (NIE) in terms of both accuracy and computational time. The accuracy of the analytical ARL expression is validated by its negligible percentage difference (%diff) in comparison to the results derived using the NIE approach, and the display processing time not exceeding 3 s. To confirm the highest capability, the suggested method is compared to both the classic EWMA and the modified EWMA charts using evaluation metrics such as ARL and SDRL (standard deviation run length), as well as RMI (relative mean index) and PCI (performance comparison index). Since asset values are volatile due to positive and negative market influences, symmetry is crucial in financial monitoring. Thus, symmetric control-chart structures reduce directional bias and better portray financial market activity by balancing upward and downward movements. Finally, examination of US stock prices illustrates performance, employing a symmetrical two-sided control chart for the rapid detection of changes through the new modified EWMA, in contrast to standard EWMA and modified EWMA charts.