The Continuous Run Sum Chart

The run sum chart is an effective two-sided chart that can be used to monitor for process changes. It is known that it is more sensible than the Shewhart chart with runs rules and its performance improves as the number of regions increases. However, as the number of regions increses the resulting chart has more parameters to be defined and its design becomes more involved. In this article, we introduce a one-parameter run sum chart. This chart accumulates scores equal to the subgroup means and signals when the cummulative sum exceeds a limit value. A fast initial response feature is proposed and its run length distribution function is found by a set of recursive relations. We compare this chart with other charts suggested in the literature and find it competitive with the CUSUM, the FIR CUSUM, and the combined Shewhart FIR CUSUM schemes.     

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

The 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.     

Enhanced adaptive CUSUM charts for process mean

A statistical quality control chart is an important tool of the statistical process control, which is widely used to control and monitor a production process. The CUSUM chart is designed to detect a specific shift, provided that the shift size is known in advance. In practice, however, shift sizes are rarely known. It is then customary to use an adaptive CUSUM chart, which can effectively detect a range of shift sizes. In this paper, we enhance the sensitivities of the improved adaptive CUSUM mean charts using an auxiliary-information-based (AIB) mean estimator. The run length performances of the proposed charts are compared with those of the AIB adaptive and non-adaptive CUSUM charts in terms of the average run length (ARL), extra quadratic loss, and integral relative ARL. These run length comparisons reveal that the proposed charts are more sensitive than the existing charts when detecting different kinds of shift in the process mean. An example is given to demonstrate the implementation of existing and proposed charts.     

Conditional design of the CUSUM median chart for the process position when process parameters are unknown

In practice, different practitioners will use different Phase I samples to estimate the process parameters, which will lead to different Phase II control chart's performance. Researches refer to this variability as between-practitioners-variability of control charts. Since between-practitioners-variability is important in the design of the CUSUM median chart with estimated process parameters, the standard deviation of average run length (SDARL) will be used to study its properties. It is shown that the CUSUM median chart requires a larger amount of Phase I samples to sufficiently reduce the variation in the in-control ARL of the CUSUM median chart. Considering the limitation of the amount of the Phase I samples, a bootstrap approach is also used here to adjust the control limits of the CUSUM median chart. Comparisons are made for the CUSUM and Shewhart median charts with estimated parameters when using the adjusted- and unadjusted control limits and some conclusions are made.     

Monitoring parameter shift with Poisson integer-valued GARCH models

This study examines the statistical process control chart used to detect a parameter shift with Poisson integer-valued GARCH (INGARCH) models and zero-inflated Poisson INGARCH models. INGARCH models have a conditional mean structure similar to GARCH models and are well known to be appropriate to analyzing count data that feature overdispersion. Special attention is paid in this study to conditional and general likelihood ratio-based (CLR and GLR) CUSUM charts and the score function-based CUSUM (SFCUSUM) chart. The performance of each of the proposed methods is evaluated through a simulation study, by calculating their average run length. Our findings show that the proposed methods perform adequately, and that the CLR chart outperforms the GLR chart when there is an increased shift of parameters. Moreover, the use of the SFCUSUM chart in particular is found to lead to a lower false alarm rate than the use of the CLR chart.     

A CUSUM control chart for monitoring the variance when parameters are estimated

CUSUM control chart has been widely used for monitoring the process variance. It is usually used assuming that the nominal process variance is known. However, several researchers have shown that the ability of control charts to signal when a process is out of control is seriously affected unless process parameters are estimated from a large in-control Phase I data set. In this paper we derive the run length properties of a CUSUM chart for monitoring dispersion with estimated process variance and we evaluate the performance of this chart by comparing it with the same chart but with assumed known process parameters.     

CUSUM Chart with Transformed Exponential Data

Time Between Events (TBE) charts were proposed to monitor the time between events occur based on exponential distribution, and have been shown to be more effective than monitoring the fraction non conforming directly. In this article, we consider monitoring the TBE data with CUSUM scheme by transformation. The idea behind it is to transform the TBE data to normal, and then apply the CUSUM scheme for the approximate normal data. Several simple transformation methods are examined. The calculation of Average Run Length (ARL) with Markov chain approach is described. Comparative studies on the ARL performance show that the transformed CUSUM is superior to the X-MR (Moving Range) chart with transformation, the Cumulative Quantity Control (CQC) chart, and have comparable performance with exponential CUSUM charts. The design procedures of optimal CUSUM chart are also presented. This study provides another possible alternative for monitoring TBE data with easy design procedures and relatively good performance.     

QUALITY CONTROL WITH MULTIVARIATE DATA

The paper examines statistical process control of bivariate and multivariate data, using in particular the multivariate equivalents of the univariate Shewhart chart, CUSUM chart and the Exponentially Weighted Moving Average chart. This illustrates the usefulness of Principal Component methods in statistical process control with multivariate data.     

Statistical control charts for monitoring the mean of a stationary process

Recently statistical process control (SPC) methodologies have been developed to accommodate autocorrelated data. A primary method to deal with autocorrelated data is the use of residual charts. Although this methodology has the advantage that it can be applied to any autocorrelated data it needs time series modeling efforts. In addition for a X residual chart the detection capability is sometimes small compared to the X chart and EWMA chart. Zhang (1998) proposed the EWMAST chart which is constructed by charting the EWMA statistic for stationary processes to monitor the process mean. The performance of the EWMAST chart the X chart the X residual chart and other charts were compared in Zhang (1998). In this paper comparisons are made among the EWMAST chart the CUSUM residual chart and EWMA residual chart as well as the X residual chart and X chart via the average run length.     

Measurement error effect on the CUSUM control chart

The performance of the cumulative sum (CUSUM) control chart for the mean when measurement error exists is investigated. It is shown that the CUSUM chart is greatly affected by the measurement error. A similar result holds for the case of the CUSUM chart for the mean with linearly increasing variance. In this paper, we consider multiple measurements to reduce the effect of measurement error on the charts performance. Finally, a comparison of the CUSUM and EWMA charts is presented and certain recommendations are given.     

The random intrinsic fast initial response of one-sided CUSUM charts

This article analyses the performance of a one-sided cumulative sum (CUSUM) chart that is initialized using a random starting point following the natural or intrinsic probability distribution of the CUSUM statistic. By definition, this probability distribution remains stable as the chart is used. The probability that the chart starts at zero according to this intrinsic distribution is always smaller than one, which confers on the chart a fast initial response feature. The article provides a fast and accurate algorithm to compute the in-control and out-of-control average run lengths and run-length probability distributions for one-sided CUSUM charts initialized using this random intrinsic fast initial response (RIFIR) scheme. The algorithm also computes the intrinsic distribution of the CUSUM statistic and random samples extracted from this distribution. Most importantly, no matter how the chart was initialized, if no level shifts and no alarms have occurred before time τ?>?0, the distribution of the run length remaining after τ is provided by this algorithm very accurately, provided that τ is not too small.     

On increasing the sensitivity of mixed EWMA–CUSUM control charts for location parameter

Control chart is an important statistical technique that is used to monitor the quality of a process. Shewhart control charts are used to detect larger disturbances in the process parameters, whereas cumulative sum (CUSUM) and exponential weighted moving average (EWMA) are meant for smaller and moderate changes. In this study, we enhanced mixed EWMA–CUSUM control charts with varying fast initial response (FIR) features and also with a runs rule of two out of three successive points that fall above the upper control limit. We investigate their run-length properties. The proposed control charting schemes are compared with the existing counterparts including classical CUSUM, classical EWMA, FIR CUSUM, FIR EWMA, mixed EWMA–CUSUM, 2/3 modified EWMA, and 2/3 CUSUM control charting schemes. A case study is presented for practical considerations using a real data set.     

Optimal CUSUM and adaptive CUSUM charts with auxiliary information for process mean

The CUSUM chart is good enough to detect small-to-moderate shifts in the process parameter(s) as it can be optimally designed to detect a particular shift size. The adaptive CUSUM (ACUSUM) chart provides good detection over a range of shift sizes because of its ability to update the reference parameter using the estimated process shift. In this paper, we propose auxiliary-information-based (AIB) optimal CUSUM (OCUSUM) and ACUSUM charts, named AIB-OCUSUM and AIB-ACUSUM charts, using a difference estimator of the process mean. The performance comparisons between existing and proposed charts are made in terms of the average run length (ARL), extra quadratic loss and integral relative ARL measures. It is found that the AIB-OCUSUM and AIB-ACUSUM charts are more sensitive than the AIB-CUSUM and ACUSUM charts, respectively. Moreover, the AIB-ACUSUM chart surpasses the AIB-OCUSUM chart when detecting a range of mean shift sizes. Illustrative examples are given to support the theory.     

A HEWMA-CUSUM control chart for the Weibull distribution

The Weibull distribution is one of the most popular distributions for lifetime modeling. However, there has not been much research on control charts for a Weibull distribution. Shewhart control is known to be inefficient to detect a small shift in the process, while exponentially weighted moving average (EWMA) and cumulative sum control chart (CUSUM) charts have the ability to detect small changes in the process. To enhance the performance of a control chart for a Weibull distribution, we introduce a new control chart based on hybrid EWMA and CUSUM statistic, called the HEWMA-CUSUM chart. The performance of the proposed chart is compared with the existing chart in terms of the average run length (ARL). The proposed chart is found to be more sensitive than the existing chart in ARL. A simulation study is provided for illustration purposes. A real data is also applied to the proposed chart for practical use.     

Robust dual-CUSUM control charts for contaminated processes

ABSTRACT

In this article, we improve the efficiency of the Dual CUSUM chart (which combines the designs of two CUSUM structures to detect a range of shift) by focusing on its robustness, ability to resist some disturbances in the process environment and violation of basic assumptions. We do that, by proposing some robust estimators for constructing the chart for both contaminated and uncontaminated environments. The average run length is used as the performance evaluation measure of the charts. After comparing the performances of the proposed charts based on the estimators, it is noticed that the tri-mean estimator out-performs others in all ramifications. Next to it in performance is the Hodges-Lehmann and midrange estimators. We substantiated the simulation results of the study by applying the scheme on a real-life data set.     

An optimal design of CUSUM control charts for binomial counts

The Shewhart p-chart or np-chart is commonly used for monitoring the counts of non-conforming items which are usually well modelled by a binomial distribution with parameters n and p where n is the number of items inspected each time and p is the process fraction of non-conforming items produced. It is well known that the Shewhart chart is not sensitive to small shifts in p. The cumulative sum (CUSUM) chart is a far more powerful charting procedure for detecting small shifts in p and only marginally less powerful in detecting large shifts in p. The choice of chart parameters of a Shewhart chart is well documented in the quality control literature. On the other hand, very little has been done for the more powerful CUSUM chart, possibly due to the fact that the run length distribution of a CUSUM chart is much harder to compute. An optimal design strategy is given here which allows the chart parameters of an optimal CUSUM chart to be determined easily. Optimal choice of n and the relationship between the CUSUM chart and the sequential probability ratio test are also investigated.     

Bayesian X̄ control limits for exponentially distributed measurements

This article deals with the construction of an X? control chart using the Bayesian perspective. We obtain new control limits for the X? chart for exponentially distributed data-generating processes through the sequential use of Bayes’ theorem and credible intervals. Construction of the control chart is illustrated using a simulated data example. The performance of the proposed, standard, tolerance interval, exponential cumulative sum (CUSUM) and exponential exponentially weighted moving average (EWMA) control limits are examined and compared via a Monte Carlo simulation study. The proposed Bayesian control limits are found to perform better than standard, tolerance interval, exponential EWMA and exponential CUSUM control limits for exponentially distributed processes.     

Comparison of different control charts for a Weibull process with type-I censoring

Some control charts have been proposed to monitor the mean of a Weibull process with type-I censoring. One type of control charts is to monitor changes in the scale parameter because it indicates changes in the mean. With this approach, we compare different control charts such as Shewhart-type and exponentially weighted moving average (EWMA) charts based on conditional expected value (CEV) and cumulative sum (CUSUM) chart based on likelihood-ratio. A simulation approach is employed to compute control limits and average run lengths. The results show that the CUSUM chart has the best performance. However, the EWMA-CEV chart is recommendable for practitioners with its competitive performance and ease of use advantage. An illustrative example is also provided.     

Detection of changes in a random financial sequence with a stable distribution

Quick detection of unanticipated changes in a financial sequence is a critical problem for practitioners in the finance industry. Based on refined logarithmic moment estimators for the four parameters of a stable distribution, this article presents a stable-distribution-based multi-CUSUM chart that consists of several CUSUM charts and detects changes in the four parameters in an independent and identically distributed random sequence with the stable distribution. Numerical results of the average run lengths show that the multi-CUSUM chart is superior (robust and quick) on the whole to a single CUSUM chart in detecting the shift change of the four parameters. A real example that monitors changes in IBM's stock returns is used to demonstrate the performance of the proposed method.