Exponentially Weighted Moving Average Chart Research Articles

Adaptive sample size strategies for process dispersion using EWMA control chart

AbstractMonitoring the stability of measured dispersion in a process quality parameter is one of the critical steps in statistical process control (SPC) and is a paramount activity carried out by specialists. For this purpose, the current research presents a promising and computationally effective tool where the exponentially weighted moving average (EWMA) statistic utilizes the quadratic sigmoid function by incorporating sample size leading to the Quadratic Sigmoid Exponential weighted Moving Average (QSEWMA) control chart. The adaptive sample size is a strategy that increases or decreases the sample size according to the conditions of the process and accordingly, increases the sensitivity and response of the control chart. The proposed strategy improves the detection of process dispersion changes, making it a more efficient and effective way of quality assurance. Our proposed control chart is superior to the EWMA‐S2 chart, HHW‐EWMA chart CUSUM‐S2chart, and parametric free adaptive EWMA (AEWMA) charts for dispersion through extensive Monte Carlo simulations. The proposed QSEWMA chart has better detection sensitivity and fewer false alarms, and is, in general, more effective comparatively. At last, a real data application test is done to measure the practicability and highest accuracy of the above chart.

Efficient ARL estimation for general control charts using censored run lengths

The average run length (ARL), the average number of in-control signals before an out-of-control signal occurs, is a critical measure of the performance of a control chart. Various methods have been explored for estimating the ARL, including Markov chain-based approximation, integration method, and the Monte Carlo method, especially for time-dependent charting statistics. Although computationally expensive, the Monte Carlo method is generic and applicable to all control charts. Typically, it involves choosing the maximum run length in advance and discarding iterations that fail to produce out-of-control signals to accommodate some unusually lengthy runs. However, these discarded runs provide information for estimating the run length. This paper introduces a new Monte Carlo approximation that retains failed iterations as Type I censored observations for ARL estimation. Our approach relies on the memoryless expectation of the run length distribution to have a simple and efficient ARL estimator. This assumption is necessary for mean estimation in Type I censored data and aligns with existing ARL approximations used in popular control charts like Shewhart, exponentially weighted moving average (EWMA), and cumulative sum (CUSUM) charts. Numerical simulations demonstrate our method’s computational and statistical efficiency across the mean, median, EWMA, and CUSUM charts.

Exponentially weighted moving average chart using zero-inflated negative binomial distribution

Zero-inflated models are frequently used to deal with data having many zeros. A commonly used model for over-dispersed data containing zeros is known as the zero-inflated Poisson model. However, to account for the heterogeneity of counts that leads to excess variance besides inflation of zeros in the data using a more flexible model than the zero-inflated Poisson model, a zero-inflated negative binomial (ZINB) is suggested. In the present study, Shewhart and exponentially weighted moving average (EWMA) control charts are suggested to monitor the ZINB data. The charts are compared using the average run length and standard deviation of run length by using extensive Monte Carlo simulations. Besides a comprehensive simulation study assuming different settings of parameters of ZINB, a real data set is used to show the practicality of the proposed charts. The results indicate that the EWMA chart is better than the Shewhart chart.

Extended GWMA control charts: A critical evaluation

AbstractWe demonstrate that the recently introduced triple generally weighted moving average (GWMA) chart and its counterpart, the double GWMA chart, incorporate sub‐optimal weighting patterns that may assign more weight to certain historical data points at the expense of more recent ones. Moreover, these control charts, when compared to the exponentially weighted moving average (EWMA) chart, exhibit a substantial computational burden. Our findings underscore that a well‐designed EWMA chart offers superior overall performance in comparison to these control charts.

New extended exponentially weighted moving average control chart for monitoring process mean

In this article, we develop a new control chart based on the Exponentially Weighted Moving Average (EWMA) statistic, termed the New Extended Exponentially Weighted Moving Average (NEEWMA) statistic, designed to recognize slight changes in the process mean. We derive expressions for the mean and variance of the NEEWMA statistic, ensuring an unbiased estimation of the mean, with simulation results showing lower variance compared to traditional EWMA charts. Evaluating its performance using Average Run Length (ARL), our analysis reveals that the NEEWMA control chart outperforms EWMA and Extended EWMA (EEWMA) charts in swiftly recognizing shifts in the process mean. Illustrating its operational methodology through Monte Carlo simulations, an illustrative example using practical data is also provided to showcase its effectiveness.

The evaluation of the long-term drift of electrical capacitance measurement standards by two methods

Measurements of electrical capacitance are essential in various fields of electrical engineering, electronics and other fields. Capacitance measurements are necessary for correct design of electrical circuits and devices. Such measurements help ensure the stability and reliability of electrical systems such as power supplies, filters, capacitors, etc. In some applications, such as radio transmitters, filters, and other electronic devices, it is needed to maintain certain frequency characteristics. It is critical and appropriate to consider various factors related to the drift of the measuring instrument or measurement standard when performing measurements to ensure the required accuracy and reliability of the measurements. The study of the time drift of the measurement standard is mandatory when carrying out comparisons of national measurement standards. The types of the drift and main methods of its evaluation for measuring instruments and measurement standards between their calibrations were analysed. A conventional method of the long-term drift analysis involves the use of regression models followed by their detailed analysis. Such models are specific mathematical functions that describe theoretical values that best represent the underlying bias in the time series for the long-term drift. Exponentially Weighted Moving Average (EWMA) charts reduces the lag inherent in conventional moving averages by giving more weight to recent observations. The results of the evaluation of the long-term drift of measurement standards of electrical capacitance for high-precision calibration of measurement standards by polynomial regression diagrams and EWMA charts are given. Polynomials of the 2nd degree were sufficient to approximate the drift of electrical capacitance measurement standards under consideration. The application of EWMA charts showed greater sensitivity to drift changes over the past few years of observations compared to the regression analysis. Consistent results were obtained.

Pollution concentration monitoring using a new Birnbaum–Saunders control chart

AbstractAir pollution monitoring is an important issue in environmental science. The Birnbaum–Saunders (BS) distribution, originally applied to describe product failure time distribution to fatigue failures and general random wear failures, is also well to describe the pollutant concentration data due to accumulations of various pollutants in the air over time. Sulfur dioxide (SO2) is a critical factor in air pollution. Hence, it is important to monitor its concentration variation for air pollution prevention. Due to the complexity of its distribution form, there is no reliable and easy‐to‐use control chart for monitoring pollutant concentrations based on the BS distribution. We found that the SO2 concentration data follows the BS distribution. In this study, we propose a new median control chart based on the exact sampling distribution of the monitoring statistic to detect shifts in the median of BS distribution. Thus, given the false alarm rate, the control limits for such control charts can be obtained precisely satisfying a preset in‐control average run length using Monte Carlo simulations. The out‐of‐control average run lengths are calculated by simulation to evaluate the detection performance of the proposed chart when the median shifts occur. We further compare the detection performance of the proposed chart and those of the existing control charts based on asymptotic sampling distributions. In order to improve the detection ability of the proposed chart for small median shifts, an exponentially weighted moving average (EWMA) control chart is constructed. The results of numerical analyses demonstrated that the proposed EWMA chart performs much better than all existing control charts for monitoring the median of BS distribution. Finally, the proposed control charts are applied to monitor the median of SO2 concentrations for air pollution control, showing that both charts can effectively detect a shift in the median of SO2 concentrations. The proposed EWMA control chart even detects out a small shift in the median of SO2 concentrations. The results provide a continuous monitoring solution for air pollution prevention.

Shewhart-EWMA chart for monitoring binomial data subject to shifts of random amounts

Attribute charts are used widely for monitoring binary events in manufacturing, service, and healthcare processes. While Shewhart type charts are efficacious in detecting large or sudden shifts in a nonconforming rate p, exponentially weighted moving average (EWMA) type charts tend to be more powerful for detecting smaller or gradual shifts. Since many processes can shift by various amounts, this paper proposes a combined optimized Shewhart-EWMA dual chart that combines the strengths of both charts to quickly detect shifts of any random magnitude within a defined range. The performance of this combined Shewhart-EWMA chart is optimized by allocating total detection power between the Shewhart chart element and EWMA chart element in a manner that yields the best overall effectiveness while maintaining the false alarm rate at a specified desired level. In numeric analysis under a range of settings, the detection speed of this combined Shewhart-EWMA chart is found to outperform individual Shewhart and EWMA charts, with up to 499% and 31% fewer average number of defective items respectively until process shifts of random sizes are detected. It also demonstrates enhanced overall efficacy when compared to synthetic, cumulative sum, combined Shewhart-synthetic, and combined Shewhart-CUSUM charts.

Modified exponentially weighted moving average control chart for monitoring process dispersion

The exponentially weighted moving average ( EWMA ) charts are widely used memory-type charts for monitoring small to moderate shifts in process parameters. However, the performance of the EWMA chart has been improved over time due to various modifications and enhancements. This paper proposes two modified EWMA ( M ‐ EWMA ) charts to monitor process dispersion. The upper-sided modified EWMA chart is denoted by M ‐ EWMA U , while a lower-sided modified EWMA chart is symbolized as the M ‐ EWMA L chart. Monte Carlo simulations determine the proposed schemes’ run length (RL) properties in zero- and steady-state scenarios. The proposed upper- and lower-sided M ‐ EWMA charts are compared to the upper- and lower-sided CH ‐ EWMA and SJ-EWMA charts. The comparisons reveal that the proposed M-EWMA charts have better detection ability than the upper- and lower-sided CH-EWMA, and SJ-EWMA charts, respectively. Finally, a real-life application is provided to illustrate the implementation of the upper-sided M-EWMA and CH-EWMA charts practically in both zero-state and steady-state cases.

Developing Bayesian EWMA chart for change detection in the shape parameter of Inverse Gaussian process.

Bayesian Control charts are emerging as the most efficient statistical tools for monitoring manufacturing processes and providing effective control over process variability. The Bayesian approach is particularly suitable for addressing parametric uncertainty in the manufacturing industry. In this study, we determine the monitoring threshold for the shape parameter of the Inverse Gaussian distribution (IGD) and design different exponentially-weighted-moving-average (EWMA) control charts based on different loss functions (LFs). The impact of hyperparameters is investigated on Bayes estimates (BEs) and posterior risks (PRs). The performance measures such as average run length (ARL), standard deviation of run length (SDRL), and median of run length (MRL) are employed to evaluate the suggested approach. The designed Bayesian charts are evaluated for different settings of smoothing constant of the EWMA chart, different sample sizes, and pre-specified false alarm rates. The simulative study demonstrates the effectiveness of the suggested Bayesian method-based EWMA charts as compared to the conventional classical setup-based EWMA charts. The proposed techniques of EWMA charts are highly efficient in detecting shifts in the shape parameter and outperform their classical counterpart in detecting faults quickly. The proposed technique is also applied to real-data case studies from the aerospace manufacturing industry. The quality characteristic of interest was selected as the monthly industrial production index of aircraft from January 1980 to December 2022. The real-data-based findings also validate the conclusions based on the simulative results.

A critique on the use of the belief statistic for process monitoring

AbstractRecently, several control charts have been proposed in the statistical process monitoring literature based on a belief statistic. In this article we compare the zero‐state average run‐lengths and conditional expected delay profiles of the exponentially weighted moving average (EWMA) and belief statistic‐based (BSB) charts. We show that an ordinary EWMA chart is far better than the BSB chart when detecting delayed shifts in the mean of a normally distributed process.

Monitoring gamma type-I censored data using an exponentially weighted moving average control chart based on deep learning networks

In recent years, deep learning methods have been widely used in combination with control charts to improve the monitoring efficiency of complete data. However, due to time and cost constraints, data obtained from reliability life tests are often type-I right censored. Traditional control charts become inefficient for monitoring this type of data. Thus, researchers have proposed various control charts with conditional expected values (CEV) or conditional median (CM) to improve efficiency for right-censored data under normal and non-normal conditions. This study combines the exponentially weighted moving average (EWMA) CEV and CM chart with deep learning methods to increase efficiency for gamma type-I right-censored data. A statistical simulation and a real-world case are presented to assess the proposed method, which outperforms the traditional EWMA charts with CEV and CM in various skewness coefficient values and censoring rates for gamma type-I right-censored data.

Quasi-online failure times identification of mobile heat sources in 2D geometry

Identifying the failure instants in thermal systems subject to 2D parabolic partial differential equations presents a significant challenge, especially when the systems involve mobile heat sources. In the context of this study, mobile heat sources are examined, along with a set of stationary sensors, while assuming known and constant-velocity trajectories for the heat sources. This research introduces a quasi-online methodology that incorporates Exponentially Weighted Moving Average (EWMA) charts for immediate failure detection. When a failure is detected via the EWMA charts, the Conjugate Gradient Method, traditionally developed for offline applications, is activated. This method is adapted to a quasi-online framework, facilitating a more rapid and precise identification of malfunctioning heat sources, the exact time of their failures, and the possibility of restoring normal operations. To assess the performance and reliability of this approach, it is compared with a Bayesian filter-based method, particularly using the Kalman filter for this purpose. Monte Carlo simulations are employed to evaluate the resilience and effectiveness of the quasi-online method, focusing on the system’s sensitivity to the accuracy of sensor measurements.

On the performance and comparison of various memory-type control charts

Several versions of the exponentially weighted moving average (EWMA) control chart, such as the generally weighted moving average (GWMA), the double, triple, and quadruple EWMA (DEWMA, TEWMA, and QEWMA), and the homogeneously weighted moving average (HWMA) charts, have been developed to enhance its detection ability, especially for small and moderate shifts. These charts have been criticized as it is considered that an appropriately designed EWMA chart provides as good or better zero-state and steady-state average run-length (ARL) performance. In this article, we perform a careful comparison study of the above control charts under similar in-control (IC) run-length properties. We show that the “new” memory-type control charts have better out-of-control (OOC) run-length characteristics, especially for small to moderate shifts in the process mean for both the zero-state and steady-state cases, except for the HWMA chart where its steady-state OOC performance is very poor. We also present a comparison study based on the worst-case ARL measure. Finally, the extensions of the EWMA chart have better IC robustness ability than the EWMA and HWMA charts.

Reassessment of performance evaluation of EWMA control chart for exponential process

AbstractMemory control charts or charts, such as cumulative sums and exponentially weighted moving average (EWMA), are widely used for tracking minor to moderate changes in process parameters. The literature presented an EWMA chart for monitoring exponential processes, assuming probability limits to determine both in‐ and out‐of‐control average run length (ARL). The use of probability limits for memory charts is an incorrect method. So, we are aiming to correct the EWMA chart by Aslam et al. and the proposed EWMA chart (represented by ) for the exponential process using the run‐length (RL) method based on the simulation method instead of the probability‐based method. The ARL, standard deviation of RL, and median of RL are used to evaluate the proposed chart's performance. For practical consideration of the proposed chart, a real‐world example of a urinary tract infection is also presented.

Designing a Log-Logistic-Based EWMA Control Chart Using MOPSO and VIKOR Approaches for Monitoring Cardiac Surgery Performance

This study aims to develop a risk-adjusted Exponentially Weighted Moving Average (EWMA) control chart for computer-based performance monitoring in cardiac surgery. Patients have distinct risk factors that impact the surgical process before it even begins. As a result, risk adjustment is carried out utilizing the Accelerated Failure Time (AFT) model to consider these factors. Before using the risk-adjusted EWMA chart, the optimal parameter design should be established, considering the required statistical and economic factors. A model for Multi-Objective Decision Making (MODM) with multiple assignable causes has been proposed to accomplish this. The model is solved using a two-stage methodology based on the Multi-Objective Particle Swarm Optimization (MOPSO) method and the VIekriterijumsko KOmpromisno Rangiranje (VIKOR) method. An actual case study for cardiovascular patients has been undertaken to demonstrate the performance and effectiveness of the suggested model. The multi-objective and pure economic models have been thoroughly compared. The economic model with statistical constraints and the multi-objective model have also been compared again. The findings suggest that the multi-objective design of the risk-adjusted EWMA chart exhibits higher statistical performance in both cases against a small augment in cost.

THE PERFORMANCES OF MIXED EWMA-CUSUM CONTROL CHARTS BASED ON MEDIAN-BASED ESTIMATORS UNDER NON-NORMALITY

Exponentially weighted moving average (EWMA) and cumulative sum (CUSUM) charts have been regularly used to monitor small process mean shifts. More recently, a mixture of EWMA and CUSUM charts known as mixed EWMA-CUSUM (MEC) control chart has been introduced for better small shift detection. However, like its predecessor, the MEC chart requires the normality assumption to ensure optimal performances. In the presence of outliers, which is the cause of non-normality, the parameters of the chart may be overestimated, leading to an unreliable monitoring process. To mitigate this problem, this paper employed median-based estimators namely, the median and modified one-step M-estimator (MOM), to control the location parameter via the MEC control chart. In this study, the performance of robust MEC charts for Phase II monitoring of location was compared with the standard MEC chart that is based on the sample mean. The performance of the robust MEC charts in terms of the average run length (ARL) on various g-and-h distributions clearly shows that a robust MEC chart based on the MOM estimator performs well regardless of the distributional shapes.

A Weibull process monitoring with AEWMA control chart: an application to breaking strength of the fibrous composite

In recent times, there has been a growing focus among researchers on memory-based control charts. The Exponentially Weighted Moving Average (EWMA) and Cumulative Sum (CUSUM) charts and the adaptive control charting approaches got the attention. Control charts are commonly employed to oversee processes, assuming the monitored variable follows a normal distribution. However, it's worth noting that this assumption does not hold true in many real-world situations. The use of the algebraic expression for normalization, which can be used for all kinds of skewed distributions with a closed-form distribution function, using the proposed continuous function to adapt a smoothing constant, motivates this study. In the present manuscript, we design an EWMA statistic-based adaptive control chart to monitor the irregular variations in the mean of two parametric Weibull distribution and use Hasting approximation for normalization. The adaptive control charts are used to update the smoothing constant according to the estimated shift. Here we use the proposed continuous function to adapt the smoothing constant. The average run length and standard deviation of run length are calculated under different parameter settings. The effectiveness of the proposed chart is argued in terms of ARLs over the considered EWMA chart through Monte-Carlo (MC) simulation method. The proposed chart is examined, followed by a real data set to demonstrate the design and application procedures.

ON EXPONENTIALLY WEIGHTED MOVING AVERAGE CONTROL CHART FOR TIME TRUNCATED LIFE TEST USING INVERSE WEIBULL DISTRIBUTION

Statistical control charts are widely used in industry for process and measurement control to monitor production processes in order to discover any problem or issues that may arise during the production process and to help in finding solutions for these issues. This research proposes the exponentially weighted moving average (EWMA) control chart, comparing it’s performance to NP and MA control chart for monitoring the number of nonconforming with the truncated lifetime of a product when the lifetimes of products follows inverse Weibull (IW) distribution. The number of failures during the life test is used as an indicator of the quality of the product. EWMA control chart was proposed for this specific situation, thus extending the applicability of control charts methodology to situations involving truncated life tests. The performance of these control charts was measured with the average run length (ARL). The simulation shows that the results obtained from the EWMA chart using IW distribution is more sensitive in detecting small shift and performs much better to monitoring shifts as when compared to other control charts considered in the study.

Effect of estimation error on the conditional false alarm rate design of the EWMA chart

AbstractIn this article, we study the effect of estimation error on the conditional false alarm rate (CFAR) of the exponentially weighted moving average (EWMA) chart based on the estimated dynamic probability control limits (EDPCLs). It is found that the effect is large, but with an increase in the Phase‐I sample size, the effect of estimation error may be reduced. In addition, an alternative approach is suggested for determining the EDPCLs for the EWMA chart that may help reduce the effect of estimation error on the CFAR profiles.