Nonparametric triple exponentially weighted moving average signed-rank control chart for monitoring shifts in the process location

Nonparametric (or distribution-free) control charts are used for monitoring processes where there is a lack of knowledge about the underlying distribution. In this article, a triple exponentially weighted moving average control chart based on the signed-rank statistic (referred as TEWMA-SR chart) is proposed for monitoring shifts in the location parameter of an unknown, but continuous and symmetric, distribution. The run-length characteristics of the proposed chart are evaluated performing Monte Carlo simulations. A comparison study with other existing nonparametric control charts based on the signed-rank statistic, the TEWMA sign chart, and the parametric TEWMA-$\overline{X}$ chart indicates that the proposed chart is more effective in detecting small shifts, while it is comparable with the other charts for moderate and large shifts. Finally, two illustrative examples are provided to demonstrate the application of the proposed chart.     

A nonparametric triple exponentially weighted moving average sign control chart

Nonparametric control charts are used in process monitoring when there is insufficient information about the form of the underlying distribution. In this article, we propose a triple exponentially weighted moving average (TEWMA) control chart based on the sign statistic for monitoring the location parameter of an unknown continuous distribution. The run-length characteristics of the proposed chart are evaluated performing Monte Carlo simulations. We also compare its statistical performance with existing nonparametric sign charts, such as the cumulative sum (CUSUM), exponentially weighted moving average (EWMA), generally weighted moving average (GWMA), and double exponentially weighted moving average (DEWMA) sign charts as well as the parametric TEWMA-$\overline{X}$ chart. The results show that the TEWMA sign chart is superior to its competitors, especially for small shifts. Moreover, two examples are given to demonstrate the application of the new scheme.     

Exponentially weighted moving average control charts based on the moving average statistic and lnS2 for monitoring a Weibull process with subgroups

In this paper, we propose 2 new exponentially weighted moving average (EWMA) control charts based on the moving average (MA) statistic and lnS2 to monitor the process mean and variability of a Weibull process with subgroups. The inverse error function is used to transform the Weibull‐distributed data to a standard normal distribution. The Markov chain approach is used to derive the average run length (ARL). Subsequently, the performances of the proposed charts with other existing control charts are provided. The comparison shows that the EWMA‐MA outperforms the and EWMA‐ control charts for monitoring the process mean of ARL values. The comparison also shows that the EWMA‐lnS2 outperforms the S2 and S2‐MA control charts for monitoring the process variability of ARL value. Two examples are used to illustrate the application of the proposed control charts.     

Progressive mean control chart is not a special case of an exponentially weighted moving average control chart

The progressive mean (PM) statistic is based on a simple idea of accumulating information of each subgroup by calculating the average progressively. Its weighting structure is based on a subgroup number that changes arithmetically, which makes the PM chart unique and efficient compared with the existing classical memory control charts. In a recent article (see reference 1), it was claimed that the PM chart is a special case of the exponentially weighted moving average (EMWA) chart. In this article, it is shown that even though the PM statistic can be written in the form of an EWMA statistic, the variance of the PM statistic is different from that of the EWMA statistic. Consequently, the limits of the PM chart are different from that of the EWMA chart. Therefore, it is found that the PM chart is not a special case of the EWMA chart; hence, the claim in reference 1 is incorrect. Furthermore, it is pointed out in this paper that no adaptive property in the weighting parameter of the PM statistic exists, further contradicting the claim in reference 1.     

An exponentially weighted moving average chart based on likelihood‐ratio test for monitoring Weibull mean and variance with subgroups

Monitoring changes in the Weibull mean and variance simultaneously is of interest in quality control. The mean and variance of a Weibull process are determined by its shape and scale parameters. Most studies are focused on monitoring the Weibull scale parameter with fixed shape parameter or the Weibull shape parameter with fixed scale parameter. In this paper, we propose an exponentially weighted moving average chart based on the likelihood‐ratio test and an inverse error function called ELR chart to monitor changes in the Weibull mean and variance simultaneously. The simulation approach is used to derive the average run length. We compare our proposed chart with other existing control charts for 3 cases, including scale parameter changes with fixed shape parameter, shape parameter changes with fixed scale parameter, and both parameters changes. The results show that the ELR chart outperforms the other control charts in terms of average run length in most cases. Two numerical examples are used to illustrate the applications of the proposed control chart.     

Improved design of robust exponentially weighted moving average control charts for autocorrelated processes

Residual‐based control charts for autocorrelated processes are known to be sensitive to time series modeling errors, which can seriously inflate the false alarm rate. This paper presents a design approach for a residual‐based exponentially weighted moving average (EWMA) chart that mitigates this problem by modifying the control limits based on the level of model uncertainty. Using a Bayesian analysis, we derive the approximate expected variance of the EWMA statistic, where the expectation is with respect to the posterior distribution of the unknown model parameters. The result is a relatively clean expression for the expected variance as a function of the estimated parameters and their covariance matrix. We use control limits proportional to the square root of the expected variance. We compare our approach to two other approaches for designing robust residual‐based EWMA charts and argue that our approach generally results in a more appropriate widening of the control limits. Copyright © 2010 John Wiley & Sons, Ltd.     

A risk-adjusted exponentially weighted moving average control chart for detection of the scale parameter in surgical quality monitoring

Risk-adjusted control charts have been widely used in monitoring surgical quality in detecting risks of surgical performance. Most of the previous approaches focus on shifts in the location parameter as well as the existence of the scale parameter, which cannot get the full measure of the scale parameter under different levels. Ignoring the magnitude of the scale parameter, the monitoring methods cannot detect different variations of surgical mortality that is measured by scale parameter and required to reflect surgical quality improvement. The method of detecting variations in surgical quality is of interest in surgical quality improvement. This paper uses a new weighted h-likelihood method to obtain a weighted score test for the surgical risks from the logistic model. Then an exponentially weighted moving average chart can be constructed to monitor the changes in the variance of risks, which could be of interest in practical surgical monitoring programs. Simulation results indicate that the proposed approach performs more efficiently than existing methods under various magnitudes of shifts in scale parameters on top of different pre-set threshold stability. In addition, the application of the proposed method to real surgical data from the Surgical Outcome Monitoring and Improvement Program in Hong Kong shows the improvement and deterioration in a hospital's outcomes.     

A hybrid homogeneously weighted moving average control chart for process monitoring

Several modifications and enhancements to control charts in increasing the performance of small and moderate process shifts have been introduced in the quality control charting techniques. In this paper, a new hybrid control chart for monitoring process location is proposed by combining two homogeneously weighted moving average (HWMA) control charts. The hybrid homogeneously weighted moving average (HHWMA) statistic is derived using two smoothing constants λ₁ and λ₂ . The average run length (ARL) and the standard deviation of the run length (SDRL) values of the HHWMA control chart are obtained and compared with some existing control charts for monitoring small and moderate shifts in the process location. The results of study show that the HHWMA control chart outperforms the existing control charts in many situations. The application of the HHWMA chart is demonstrated using a simulated data.     

Function-based adaptive exponentially weighted moving average dispersion control chart

In this study, a new process dispersion monitoring control chart is proposed based on a function-based adaptation to select the smoothing constant value, named as a function-based adaptive exponentially weighted moving average (EWMA) dispersion control chart. It is suggested to track shift ranges first expected in process dispersion by opting for smoothing constant computation with the help of a function. The shift magnitude assessment is made by an unbiased estimator that determines the smoothing constant value through the proposed function. The enhanced efficiency of the proposed chart can be assessed in terms of smaller run-length profiles, which are determined through Monte Carlo simulations. The proposed chart is compared with the existing adaptive EWMA dispersion chart, and it turned out to perform substantially efficiently in detecting all kinds of decreasing and increasing process dispersion shift magnitudes. Moreover, a real-life dataset application is explained in the example section to elaborate on the ease of implementation in the real-life scenario.     

A nonparametric double generally weighted moving average signed-rank control chart for monitoring process location

Most control charts have been developed based on the actual distribution of the quality characteristic of interest. However, in many applications, there is a lack of knowledge about the process distribution. Therefore, in recent years, nonparametric (or distribution-free) control charts have been introduced for monitoring the process location or scale parameter. In this article, a nonparametric double generally weighted moving average control chart based on the signed-rank statistic (referred as DGWMA-SR chart) is proposed for monitoring the location parameter. We provide the exact approach to compute the run-length distribution, and through an extensive simulation study, we compare the performance of the proposed chart with existing nonparametric charts, such as the exponentially weighted moving average signed-rank (EWMA-SR), the generally weighted moving average signed-rank (GWMA-SR), the double exponentially weighted moving average signed-rank (DEWMA-SR), and the double generally weighted moving average sign (DGWMA-SN) charts, as well as the parametric DGWMA- $\overline{X}$ chart for subgroup averages. The simulation results show that the DGWMA-SR chart (with suitable parameters) is more sensitive than the other competing charts for small shifts in the location parameter and performs as well as the other nonparametric charts for larger shifts. Finally, two examples are given to illustrate the application of the proposed chart.     

A new nonparametric double exponentially weighted moving average control chart

There are many practical situations where the underlying distribution of the quality characteristic either deviates from normality or it is unknown. In such cases, practitioners often make use of the nonparametric control charts. In this paper, a new nonparametric double exponentially weighted moving average control chart on the basis of the signed-rank statistic is proposed for monitoring the process location. Monte Carlo simulations are carried out to obtain the run length characteristics of the proposed chart. The performance comparison of the proposed chart with the existing parametric and nonparametric control charts is made by using various performance metrics of the run length distribution. The comparison showed the superiority of the suggested chart over its existing parametric and nonparametric counterparts. An illustrative example for the practical implementation of the proposed chart is also provided by using the industrial data set.     

A moving average S control chart for monitoring process variability

An efficient alternative to the S control chart for detecting shifts of small magnitude in the process variability using a moving average based on the sample standard deviation s statistic is proposed. Control limit factors are derived for the chart for different values of sample size and span w. The performance of the moving average S chart is compared to the S chart in terms of average run length. The result shows that the performance of moving average S chart for varying values of w outweigh those of the S chart for small and moderate shifts in process variability.     

An extended nonparametric homogeneously weighted moving average sign control chart

Parametric (or traditional) control charts are based on the assumption that the quality characteristic of interest follows a specific distribution. However, in many applications, there is a lack of knowledge about the underlying distribution. To this end, nonparametric (or distribution-free) control charts have been developed in recent years. In this article, a nonparametric double homogeneously weighted moving average (DHWMA) control chart based on the sign statistic is proposed for monitoring the location parameter of an unknown and continuous distribution. The performance of the proposed chart is measured through the run-length distribution and its associated characteristics by performing Monte Carlo simulations. The DHWMA sign chart is compared with other nonparametric sign charts, such as the homogeneously weighted moving average, generally weighted moving average (GWMA), double GWMA, and triple exponentially weighted moving average sign charts, as well as the traditional DHWMA chart. The results indicate that the proposed chart performs just as well as and in some cases better than its competitors, especially for small shifts. Finally, two examples are provided to show the application and implementation of the proposed chart.     

A side-sensitive synthetic coefficient of variation chart

Control charts for monitoring the coefficient of variation (γ) are useful for processes with an inconsistent mean (μ) and a standard deviation (σ) which changes with μ, by monitoring the consistency in the ratio σ over μ. The synthetic-γ chart is one of the charts proposed to monitor γ, and its attractiveness lie in waiting until a second point to fall outside the control limits before a decision is made. However, existing synthetic-γ charts do not differentiate between the points falling outside the upper control limit (UCL) and lower control limit (LCL). Hence, this paper proposes a side-sensitive synthetic-γ chart, where successive nonconforming samples must either fall above the UCL or below the LCL. Formulae to compute the average run length (ARL), the standard deviation of the run length (SDRL) and expected average run length (EARL) are derived using the Markov chain approach, and the algorithms to obtain the optimal charting parameters are proposed. Subsequently, the optimal charting parameters, ARL, SDRL and EARL values for various numerical examples are shown. Comparisons show that the side-sensitive synthetic-γ chart consistently outperforms the existing synthetic-γ chart, especially for small shifts. The proposed chart also consistently outperforms the Shewhart-γ chart, while showing comparable or better performance than the Exponentially Weighted Moving Average (EWMA) chart for most shift sizes, except for very small shifts. Finally, this paper shows the implementation of the proposed chart on an industrial example.     

New GWMA‐CUSUM control chart for monitoring the process dispersion

The cumulative sum (CUSUM) and exponentially weighted moving average (EWMA) control charts have been widely accepted because of their fantastic speed in identifying small‐to‐moderate unusual variations in the process parameter(s). Recently, a new CUSUM chart has been proposed that uses the EWMA statistic, called the CS‐EWMA chart, for monitoring the process variability. On similar lines, in order to further improve the detection ability of the CS‐EWMA chart, we propose a CUSUM chart using the generally weighted moving average (GWMA) statistic, named the GWMA‐CUSUM chart, for monitoring the process dispersion. Monte Carlo simulations are used to compute the run length profiles of the GWMA‐CUSUM chart. On the basis of the run length comparisons, it turns out that the GWMA‐CUSUM chart outperforms the CUSUM and CS‐EWMA charts when identifying small variations in the process variability. A simulated dataset is also used to explain the working and implementation of the CS‐EWMA and GWMA‐CUSUM charts.     

An improved design of exponentially weighted moving average scheme for monitoring attributes

The Exponentially Weighted Moving Average (EWMA) schemes are a potent tool for monitoring small to moderate variations in the quality characteristics in production lines of manufacturing industries. Practitioners in various sectors widely use the EWMA schemes for scrutinising both the variables and attributes. In the present article, we investigate a modified EWMA scheme based on the power of the difference between the actual number of nonconforming items and its technical specification in an in-control (IC) situation. We abbreviate it as a wEWMA scheme and show that the traditional EWMA scheme is a particular case of the proposed scheme when the power is unity. We establish that the powers lower than unity are more effective for detecting smaller shifts, while for detecting substantial variations in process parameter, one should prefer higher powers greater than unity. Noting that possible magnitude of a shift is often unknown, we propose the optimal design procedure of the scheme, including the determination of its charting parameters to ensure the best overall performance. The results reveal that the optimal wEWMA schemes can be beneficial in detecting a shift very quickly when the sample size is small, particularly for high-precision production processes.     

Variable sampling interval exponentially weighted moving average median chart with estimated process parameters

The variable sampling interval exponentially weighted moving average median chart with estimated process parameters is proposed. The charting statistic, optimal design, performance evaluation, and implementation of the proposed chart are discussed. The average of the average time to signal (AATS) criterion is adopted to evaluate the performance of the proposed chart. The estimated process parameter‐based VSI EWMA median (VSI EWMA median‐e) chart is compared with the estimated process parameter‐based Shewhart median (SH median‐e), EWMA median (EWMA median‐e), and variable sampling interval run sum median (VSI RS median‐e) charts, in terms of the AATS criterion, where the VSI EWMA median‐e chart is shown to be superior. When process parameters are estimated, the standard deviation of the average time to signal (SDATS) criterion is used to evaluate the AATS performance of the VSI EWMA median‐e chart. Based on the SDATS criterion, the minimum number of phase‐I samples required by the VSI EWMA median‐e chart so that its performance is close to the known process parameters VSI EWMA median chart is recommended.     

Another objection to the homogeneously weighted moving average control chart

It was recently demonstrated that homogeneously weighted moving average (HWMA) control charts are improper devices for an online detection of changes within a series of random variables. However, another attempt was made to bail out the concept underlying these charts. Here we want to re-emphasize that HWMA charts are dispensable by rebutting the arguments of the latter contribution.     

Auxiliary information-based maximum generally weighted moving average chart for simultaneously monitoring process mean and variability

An auxiliary information-based (AIB) maximum exponentially weighted moving average (MaxEWMA) chart has been proposed to simultaneously monitor both increases and decreases in the process mean and/or variability, called the AIB-MaxEWMA chart, which is superior to the existing MaxEWMA chart. In this paper, we propose the AIB maximum generally weighted moving average chart, called the AIB-MaxGWMA chart, to further enhance the sensitivity of the AIB-MaxEWMA chart. Numerical simulation studies indicate that the AIB-MaxGWMA chart is sensitive to small shifts in the process mean and/or variability. The performance of the AIB-MaxGWMA chart based on average run lengths (ARLs) also outperforms than its counterparts including AIB-MaxEWMA, MaxGWMA and MaxEWMA charts. An example is used to illustrate the efficiency of the proposed AIB-MaxGWMA chart in detecting small process shifts.