Generally Weighted Moving Average Research Articles

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.

Comparisons of some memory‐type control chart for monitoring Weibull‐distributed time between events and some new results

AbstractMonitoring Weibull time between event (TBE) processes is essential to avoid deterioration of quality characteristics in various reliability analysis problems. Statistical process monitoring (SPM) charts are widely used to monitor the possible decreasing mean shift. For monitoring such data, many control charts have been proposed in the literature, such as the Weibull exponentially weighted moving average (EWMA) chart, the Weibull cumulative sum (CUSUM) chart, and Mixed EWMA–CUSUM (MEC) chart and Mixed Hybrid EWMA–CUSUM (MHEC) chart. The current investigation revisits their efficacy and compares them with a CUSUM chart based on the generally weighted moving average (GWMA) statistic, labelled as the Mixed GWMA–CUSUM (MGC) chart, to monitor the decreasing mean shift in Weibull TBEs processes. We offer some numerical results based on the Monte‐Carlo simulation, such as average run length (ARL), the standard deviation of the run length (SDRL) and the relative mean index (RMI). Our findings show that the MGC chart often outperforms the existing control charts in detecting small downward shifts. Finally, an illustrative example is given to display the application of the MGC chart for Weibulll data.

Generally weighted moving average control chart for monitoring two-parameter exponential distribution with measurement errors

Control charts for two-parameter exponential distributions are widely used in monitoring quality and lifetime processes, and more useful and flexible than the chart for one parameter exponential distributions. In order to efficiently detect small shifts for two-parameter exponential distributions, a generally weighted moving average (GWMA) control chart is developed. Since measurement errors are inevitable in practice, and they are not considered in the existing control charts, we proposed a GWMA chart with measurement errors (GWMA-M) by two strategies: one is the GWMA chart with adjusted control limits, and the other is the GWMA chart with additive measurement error model to deal with measurement errors. The performance of the proposed GWMA-M scheme is investigated in terms of average run length (ARL) and expected ARL (EARL). Comparisons of the GWMA-M chart with the EWMA-M chart and the GWMA signed-rank (GWMA-SR) chart in the presence of measurement errors are carried out, and results show the advantage and efficiency of the GWMA-M scheme. In addition, the effect of model parameters, measurement errors and sample size on the GWMA-M chart are studied through Monte-Carlo simulation. Finally, two real examples of monitoring flood data and HPM (high-voltage metal oxide semiconductor transistor) data are given to illustrate the implementation of the proposed chart.

The case against generally weighted moving average (GWMA) control charts

We argue against the use of generally weighted moving average (GWMA) control charts. Our primary reasons are the following: (1) There is no recursive formula for the GWMA control chart statistic, so all previous data must be stored and used in the calculation of each chart statistic. (2) The Markovian property does not apply to the GWMA statistics, so computer simulation must be used to determine control limits and the statistical performance. (3) An appropriately designed, and much simpler, exponentially weighted moving average (EWMA) chart provides as good or better statistical performance. (4) In some cases the GWMA chart gives more weight to past data values than to current values.

An extended nonparametric homogeneously weighted moving average sign control chart

AbstractParametric (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 nonparametric triple exponentially weighted moving average sign control chart

AbstractNonparametric 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‐ 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.

Distribution‐free composite Shewhart‐GWMA Mann‐Whitney charts for monitoring the process location

AbstractThe Mann‐Whitney (MW) statistic is one of the most recommended two‐sample statistical tests when the assumption of normality fails to hold due to its robustness and fascinating properties especially when small sample sizes are involved. In order to improve the sensitivity of the generally weighted moving average (GWMA) monitoring scheme toward the detection of large shifts, in this paper, a new distribution‐free phase II composite Shewhart‐GWMA (CSG) scheme is proposed using the MW U statistic. The performance of the proposed scheme is investigated using the average run‐length (ARL) and average extra quadratic loss (AEQL) values through extensive simulations. The performance of this newly proposed monitoring scheme is found to be superior when compared to numerous memory‐type MW schemes (some existing and others introduced in this paper) in many situations. An illustrative example is given to demonstrate the implementation of the CSG MW U scheme using real‐life data.

Generally weighted moving average monitoring schemes: Overview and perspectives

AbstractAn overview of monitoring schemes from a class called generally weighted moving average (GWMA) is provided. A GWMA scheme is an extended version of the exponentially weighted moving average (EWMA) scheme with an additional adjustment parameter that introduces more flexibility in the GWMA model as it adjusts the kurtosis of the weighting function so that the GWMA scheme can be designed such that it has an advantage over the corresponding EWMA scheme in the detection of certain shift values efficiently. The parametric and distribution‐free GWMA schemes to monitor various quality characteristics and its existing enhanced versions (ie, double GWMA, composite Shewhart‐GWMA, mixed GWMA‐CUSUM, and mixed CUSUM‐GWMA) have better performance than their corresponding EWMA counterparts in many situations; hence, all such existing research works discussing GWMA‐related schemes (ie, 66 publications in total) are documented and categorized in such a manner that it is easy to identify research gaps. Finally, a number of possible future research ideas are provided.

A Double Generally Weighted Moving Average Chart for Monitoring the COM-Poisson Processes

Generalized exponentially weighted moving average (EWMA) and double EWMA (DEWMA) charts based on the Conway–Maxwell–Poisson (CMP or COM-Poisson) distribution, also known as the GEWMA and CMP-DEWMA charts, are effectively used for monitoring the counts of non-conformities in a process. To further enhance their performance, this study utilizes design and adjustment parameters to develop generally weighted moving average (GWMA) and double GWMA charts, also known as the CMP-GWMA and CMP-DGWMA charts, to monitor COM-Poisson attributes. Numerical simulations indicate that the CMP-DGWMA chart outperforms its prototype CMP-DEWMA and CMP-GWMA charts in detecting small location and dispersion shifts, as well as both shifts together, in terms of average run lengths. Finally, an example is provided to demonstrate the efficiency of the proposed CMP-DGWMA chart and its counterparts.

An Enhanced Auxiliary Information-Based EWMA-t Chart for Monitoring the Process Mean

The exponentially weighted moving average t chart using auxiliary information (AIB-EWMA-t chart) is an effective approach for monitoring small process mean shifts when the process standard deviation is unstable or poorly estimated. To further enhance the sensitivity of the AIB-EWMA-t chart, in this study, we propose an AIB generally weighted moving average (GWMA) t chart (AIB-GWMA-t chart) to monitor the process mean. The existing EWMA-t, GWMA-t, and AIB-EWMA-t charts are special cases of the AIB-GWMA-t chart. Numerical simulation studies indicate that the AIB-GWMA-t chart performs uniformly and substantially better than the EWMA-t and GWMA-t charts in terms of average run length. Moreover, the AIB-GWMA-t chart with large design and adjustment parameters also outperforms the AIB-EWMA-t chart when the correlation coefficients are within a certain range. An illustrative example is provided to highlight the efficiency of the proposed AIB-GWMA-t chart in detecting small process mean shifts.

A generally weighted moving average control chart for zero‐inflated Poisson processes

AbstractZero‐inflated Poisson (ZIP) model is very useful in high‐yield processes where an excessive number of zero observations exist. This model can be viewed as an extension of the standard Poisson distribution. In this paper, a one‐sided generally weighted moving average (GWMA) control chart is proposed for monitoring upward shifts in the two parameters of a ZIP process (regarded as ZIP‐GWMA chart). The design parameters of the proposed chart are provided, and through a simulation study, it is shown that the ZIP‐GWMA performs better than the existing control charts under shifts in both parameters. Moreover, an illustrative example is presented to display the application of the proposed chart on practitioners.

Generally weighted moving average control charts using repetitive sampling

Generally weighted moving average (GWMA) control charts have been validated for effective detection of small process shifts, and perform better than exponentially weighted moving average (EWMA) control charts. These charts are available based on single sampling; however, repetitive sampling charts have received less attention. Here, a GWMA control chart based on repetitive sampling (namely an RS-GWMA chart) is proposed for enhancing detectability of small process shifts. Simulations show that the proposed RS-GWMA chart with large design and small adjustment parameters outperforms existing hybrid EWMA (HEWMA) control charts based on repetitive sampling. An in-silico example is considered for demonstrating the applicability of the proposed RS-GWMA chart.

An enhanced GWMA chart for process mean

Memory-type control charts are frequently used to monitor small-to-moderate disturbances in the process parameter(s). Recently, an auxiliary-information-based (AIB) exponentially weighted moving average (EWMA) chart has been proposed for efficiently monitoring the process mean, called the AIB-EWMA chart, that surpasses the classical EWMA chart. On similar lines, in order to further enhance the sensitivity of the AIB-EWMA chart, we propose an AIB generally weighted moving average (GWMA) chart for monitoring the process mean, named the AIB-GWMA chart. The existing AIB-Shewhart and AIB-EWMA charts are special cases of the AIB-GWMA chart. With the help of Monte Carlo simulation, the run length characteristics of the AIB-GWMA chart are computed. It turns out that the AIB-GWMA chart performs uniformly and substantially better than the classical GWMA chart. Moreover, the AIB-GWMA chart also outperforms the AIB-EWMA chart when detecting mean shifts within certain ranges. A similar trend is observed when the proposed and existing control charts are compared with the fast initial response features. An example is used to explain the working and implementation of the GWMA, AIB-EWMA and AIB-GWMA charts.

New GWMA‐CUSUM control chart for monitoring the process dispersion

AbstractThe 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.

A generally weighted moving average exceedance chart

ABSTRACTDistribution-free control charts gained momentum in recent years as they are more efficient in detecting a shift when there is a lack of information regarding the underlying process distribution. However, a distribution-free control chart for monitoring the process location often requires information on the in-control process median. This is somewhat challenging because, in practice, any information on the location parameter might not be known in advance and estimation of the parameter is therefore required. In view of this, a time-weighted control chart, labelled as the Generally Weighted Moving Average (GWMA) exceedance (EX) chart (in short GWMA-EX chart), is proposed for detection of a shift in the unknown process location; this chart is based on exceedance statistic when there is no information available on the process distribution. An extensive performance analysis shows that the proposed GWMA-EX control chart is, in many cases, better than its contenders.

A mixed GWMA–CUSUM control chart for monitoring the process mean

ABSTRACTThe memory-type control charts are widely used in the process and service industries for monitoring the production processes. The reason is their sensitivity to quickly react against the small process disturbances. Recently, a new cumulative sum (CUSUM) chart has been proposed that uses the exponentially weighted moving average (EWMA) statistic, called the EWMA–CUSUM chart. Similarly, in order to further enhance the sensitivity of the EWMA–CUSUM chart, we propose a new CUSUM chart using the generally weighted moving average (GWMA) statistic, called the GWMA–CUSUM chart, for efficiently monitoring the process mean. The GWMA–CUSUM chart encompasses the existing CUSUM and EWMA–CUSUM charts. Extensive Monte Carlo simulations are used to explore the run length profiles of the GWMA–CUSUM chart. Based on comprehensive run length comparisons, it turns out that the GWMA–CUSUM chart performs substantially better than the CUSUM, EWMA, GWMA, and EWMA–CUSUM charts when detecting small shifts in the process mean. An illustrative example is also presented to explain the implementation and working of the EWMA–CUSUM and GWMA–CUSUM charts.

Non parametric double generally weighted moving average sign charts based on process proportion

ABSTRACTNon parametric control charts have received increasing attention in the field of statistical process control. This paper presents a non parametric double generally weighted moving average (DGWMA) sign chart for monitoring small deviations when the quality characteristics of a process are unknown. The statistical performance of the non parametric DGWMA sign chart is evaluated and compared with those of other charts, including the exponentially weighted moving average (EWMA), generally weighted moving average (GWMA), and double EWMA (DEWMA) sign charts. Simulation studies indicate that the non parametric DGWMA sign chart with a large design and median adjustment parameters is always more sensitive than other charts in detecting small changes.

Novel design of composite generally weighted moving average and cumulative sum charts

AbstractThe cumulative sum (CUSUM) and exponentially weighted moving average (EWMA) charts are popular statistical tools to improve the performance of the Shewhart chart in detecting small process shifts. In this study, we propose the mixed generally weighted moving average (GWMA)‐CUSUM chart and its reverse‐order CUSUM‐GWMA chart to enhance detection ability compared with existing counterparts. The simulation revealed that the mixed GWMA‐CUSUM and mixed CUSUM‐GWMA charts have the sensitivity to detect small process shifts and efficient structures compared with the mixed EWMA‐CUSUM and mixed CUSUM‐EWMA charts, respectively. Moreover, the mixed GWMA‐CUSUM chart with a large design parameter has robust performance, regardless of the high tail t distribution or right skewness gamma distribution.

Applying fast initial response features on GWMA control charts for monitoring autocorrelation data

ABSTRACTA generally weighted moving average (GWMA) control chart with fast initial response (FIR) features is addressed to monitor an autoregressive process mean shift. Numerical simulations based on average run length (ARL) show that the GWMA control chart with additional FIR feature requires less time to detect small or moderate shifts than GWMA control chart at low level of autocorrelation; whereas these two control charts perform similarly at high level of autocorrelation. Regardless of any level of autocorrelation, GWMA control charts provided with additional FIR feature have a good performance in detecting large shifts during the initial stage.

Power curve monitoring using weighted moving average control charts

A method for the monitoring of a wind turbine generator is proposed, based on its power curve and using control charts. Exponentially Weighted Moving Average (EWMA) and Generally Weighted Moving Average (GWMA) control charts are used to detect underperformances such as blade surface erosion. These variations in production amount to a few percent per year. The reference power curve is modeled using the bin method. A validation bench using simulated shifts on data from an MW-class wind turbine generator is used to assess the performance of the proposed method. Results show great potential, with both the EWMA and GWMA control charts able to detect a 1% per year underperformance inside 300 days of operation, based on simulated data. A short example is also given of an application using data involving a real case of underperformance: this example illustrates both the applicability and potential of this method. In this case, a shift of 3.4% in annual energy production over a period of five years could have been detected in time to plan proper maintenance. The rate of false alarms observed is one for every 667 points, which demonstrate the method's robustness.