A CUSUM Scheme for Autocorrelated Observations

The cumulative sum (CUSUM) chart is a very effective control charting procedure used for the quick detection of small‐sized and moderate‐sized changes. It can detect small process shifts missed by the Shewhart‐type control chart, which is sensitive mainly to large shifts. To further enhance the sensitivity of the CUSUM control chart at detecting very small process disturbances, this article presents CUSUM control charts based on well‐structured sampling procedures, double ranked set sampling, median‐double ranked set sampling, and double‐median ranked set sampling. These sampling techniques significantly improve the overall performance of the CUSUM chart over the entire process mean shift range, without increasing the false alarm rate. The newly developed control schemes do not only dominate most of the existing charts but are also easy to design and implement as illustrated through an application example of real datasets. The control schemes used for comparison in this study include the conventional CUSUM chart, a fast initial response CUSUM chart, a 2‐CUSUM chart, a 3‐CUSUM chart, a runs rules‐based CUSUM chart, the enhanced adaptive CUSUM chart, the CUSUM chart based on ranked set sampling (RSS), and the single CUSUM and combined Shewhart–CUSUM charts based on median RSS. Copyright © 2014 John Wiley & Sons, Ltd.

Control charts applied to simulated sow herd datasets

A neural network-based procedure for the monitoring of exponential mean

Monitoring 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.

Given the low early revision rate after total hip arthroplasty (THA) and total knee arthroplasty (TKA), hospital performance is typically compared using 3 years of data. The purpose of this study was to assess how much earlier worsening hospital performance in 1-year revision rates after THA and TKA can be detected. All 86,468 THA and 73,077 TKA procedures performed from 2014 to 2016 and recorded in the Dutch Arthroplasty Register were included. Negative outlier hospitals were identified by significantly higher O/E (observed divided by expected) 1-year revision rates in a funnel plot. Monthly Shewhart p-charts (with 2 and 3-sigma control limits) and cumulative sum (CUSUM) charts (with 3.5 and 5 control limits) were constructed to detect a doubling of revisions (odds ratio of 2), generating a signal when the control limit was reached. The median number of months until generation of a first signal for negative outliers and the number of false signals for non-negative outliers were calculated. Sensitivity, specificity, and accuracy were calculated for all charts and control limit settings using outlier status in the funnel plot as the gold standard. The funnel plot showed that 13 of 97 hospitals had significantly higher O/E 1-year revision rates and were negative outliers for THA and 7 of 98 hospitals had significantly higher O/E 1-year revision rates and were negative outliers for TKA. The Shewhart p-chart with the 3-sigma control limit generated 68 signals (34 false-positive) for THA and 85 signals (63 false-positive) for TKA. The sensitivity for THA and TKA was 92% and 100%, respectively; the specificity was 69% and 51%, respectively; and the accuracy was 72% and 54%, respectively. The CUSUM chart with a 5 control limit generated 18 signals (1 false-positive) for THA and 7 (1 false-positive) for TKA. The sensitivity was 85% and 71% for THA and TKA, respectively; the specificity was 99% for both; and the accuracy was 97% for both. The Shewhart p-chart with a 3-sigma control limit generated the first signal for negative outliers after a median of 10 months (interquartile range [IQR] = 2 to 18) for THA and 13 months (IQR = 5 to 18) for TKA. The CUSUM chart with a 5 control limit generated the first signal after a median of 18 months (IQR = 7 to 22) for THA and 21 months (IQR = 9 to 25) for TKA. Monthly monitoring using CUSUM charts with a 5 control limit enables earlier detection of worsening 1-year revision rates with accuracy so that initiatives to improve care can start earlier.

In the chemical and biochemical industries, batch processes play an important role in producing high-quality and value-added specialty products. The variables measured in such processes often have periodic trajectories with varying batch duration and fixed operational stages. The measured value of a process variable provides only part of the information about process conditions, while the process monitoring and operation activities mostly rely on the trends and trajectories of these variables. It is necessary to construct a trend-monitoring statistic to detect an abnormality in a batch process variable trajectory to provide operation decision support and guarantee product quality. In this paper, a new trend cumulative sum (CUSUM) chart is introduced as a way to detect the deviation of batch process variable trends, which may be caused by abnormal operations. This method relies on the extraction of variable trend features as a combination of derivatives through a functional description of variable trajectories. This then leads to the construction of an individual trend CUSUM chart and a multiple-trend CUSUM chart. The multiple-trend CUSUM chart, the sum of the trend CUSUM charts of all process variables, aims to improve the monitoring efficiency by combining the trend deviations of all variables in a single chart. The potential of these novel univariate control charts is demonstrated using the batch manufacture of polypropylene (PP). When several simulated and actual batches are studied, the results show that the trend CUSUM chart is capable of capturing trend abnormalities. We also show, by comparison, that alarms can be triggered in the trend CUSUM chart before the measured process variables exceed the control limits of their Shewhart individuals charts, allowing for the corrections to be made at an early stage of an abnormal situation.

This study aimed to design an exponentially weighted moving average (EWMA) chart for the quality review of nuchal translucency (NT) and to assess its performance compared with the methods currently in use: retrospective distribution-based methods and the cumulative sum (CUSUM) chart. The EWMA model was designed for NT quality review using simulation. The NT measurements obtained during routine first-trimester screening in our centre over a two-and-a-half-year period were retrieved from the database. The NT distribution parameters, EWMA and CUSUM chart were established, and the methods were compared. On the basis of the results from the simulation, the optimal EWMA settings were established. A set of 9338 NT measurements obtained from nine sonographers was used to construct the EWMA and CUSUM charts and to calculate the distribution parameters. Distribution-based methods were unable to reveal the temporal periods of poor performance. The EWMA model agreed closely with the CUSUM but had the advantage promptly indicating when the process returned to an in-control state, thus extending its use to long-term prospective and retrospective quality assessments. The EWMA provides a universal, easy and efficient tool for NT quality review when the prompt and effective detection of suboptimal performance is desired.

Monitoring the quality of new or ongoing surgical activities is a necessity. Several Statistical Process Control (SPC) tools are available to professionals. Among them, Shewhart charts and cumulative sum charts (CUSUM charts) are useful methods to provide visual feedback before significant quality issues arise. In this paper, we discuss both methods based on our current approach. On Shewhart charts, one variable value is plotted on a time-series line. This method provides information about every single determination. Random variations of the values appear and by adjusting the adequate control limits it is possible to know whether those variations are random or out-of-control. Although large variations are easily detected, small but relevant changes are not. On the contrary, CUSUM charts have the capability of detecting small changes quickly. CUSUM is defined as a statistical tool that graphically represents the sequential monitoring of cumulative performance of any dichotomized or continuous variable under assessment. It emphasizes failures penalizing them against the correct performance when individual risk is adjusted. This makes CUSUM especially sensitive to negative changes. CUSUM can be created without the need of a specific sample size and grow with every new case included. Besides the variable under control (with specific definitions of acceptable and unacceptable outcomes), the type I and II errors for the defined parameter and the individual risk of acceptable or unacceptable outcomes must be included in the chart. Graphical representation of these three parameters is easy and intuitive to read making CUSUM graphs a reliable tool to understand the trending of the parameter under control. If performance is considered inadequate: analysis, discussion and implementation of agreed measures should be taken. Despite its limitations, CUSUM analysis is considered the best tool for quality control in health care domain.

This paper considers the problem of monitoring the mean of a normally distributed process variable when the objective is to effectively detect both small and large shifts in this mean. The performance of a generalized likelihood ratio (GLR) control chart is evaluated, where the likelihood ratio is based on a moving window of past observations. The performance of the GLR chart is compared with the performance of other options, such as combinations of Shewhart and cumulative sum (CUSUM) charts and an adaptive CUSUM chart, that have been proposed for detecting a wide range of shift sizes. Performance is evaluated for sustained shifts, transient shifts, and drifts in the mean. It is shown that the overall performance of the GLR chart is at least as good as these other options. These other options have multiple control-chart parameters that allow for the charts to be tuned to be more sensitive to certain shifts that may be of interest. However, the GLR chart does not require users to specify the values of any control-chart parameters other than the size of the window and the control limit. We recommend a specific window size and provide a table of control limits corresponding to specified values of the false-alarm rate, so it is very easy to design the GLR chart for use in applications. Simulating the performance of the GLR chart is time consuming, but approximating the GLR chart with a set of CUSUM charts provides a much faster way of evaluating the performance of the GLR chart for research purposes.

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 (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.

As technology advances, the need for methods to monitor processes that produce readily-available inspection data becomes essential. In this paper, a multinomial cumulative sum (CUSUM) chart is proposed to monitor in situations where items can be classified into more than two categories, the items are not put into subgroups, and the direction of the out-of-control shift in the parameter vector can be specified. It is shown through examples that the multinomial CUSUM chart can detect shifts in category probabilities at least as quickly and, in most cases, faster than using multiple Bernoulli CUSUM charts. The properties of the multinomial CUSUM chart are determined through a Markov chain representation. If the direction of the out-of-control shift in the parameter vector cannot be specified, we recommend the use of multiple Bernoulli CUSUM charts.

Cumulative Sum Curves and Their Prediction Limits

Companies routinely perform life tests on their products. Each of these life tests typically involves testing several units simultaneously with interest in the times to failure. Two aspects often associated with lifetime data that make the development of a control-charting procedure more demanding are that the data tend to be nonnormally distributed and censored. In this paper, one-sided lower and upper likelihood-ratio—based cumulative sum (CUSUM) control charting procedures are developed for Type I right-censored Weibull lifetime data to monitor changes in the scale parameter, also known as the characteristic life, for a fixed value of the Weibull shape parameter. Because a decrease in the characteristic life indicates a decrease in the mean lifetime of a product, a one-sided lower CUSUM chart is the main focus. We illustrate the development and implementation of the chart and evaluate its properties through a simulation study. The proposed CUSUM chart is compared with an exponentially weighted moving-average (EWMA) chart using the steady-state average run length (ARL) performance. The CUSUM chart is shown to perform better than the EWMA chart in detecting shifts for which it is designed.

The use of the cumulative sum (CUSUM) charting procedure in process monitoring is first described. The relationship between the CUSUM charting procedure and Wald's sequential probability ratio test is then explained. This is followed by three main approaches of determining the average run length. The strengths and weaknesses of the CUSUM charting procedure, with respect to the Shewhart charting procedure, are illustrated with simulated data sets. Various other aspects of the CUSUM chart are also discussed. These include estimation of process mean using the CUSUM, weighting of samples for signaling, fast initial response for a newly start‐up process, simultaneous CUSUM schemes, design of CUSUM chart, multivariate CUSUM schemes, and risk‐adjusted CUSUM charts.