On the performance of coefficient of variation control charts in Phase I

Control charts are mainly carried out in 2 interconnected phases: Phase I (retrospective phase) and Phase II (monitoring phase). Phase I uses a stable historical sample to establish control limits that will be used later in Phase II. The preciseness of the control limits obtained from Phase I can greatly affect the performance of control charts in Phase II. Monitoring the coefficient of variation (CV) is an effective approach when the process mean or standard deviation is not constant. Until now, little work has been dedicated on investigating the performance of CV control charts in Phase I. Viewed under this perspective, this study investigates the performance of CV control charts in Phase I in terms of probability to signal. A real‐life example is also provided to illustrate the working of CV charts in Phase I.     

Auxiliary-information-based efficient variability control charts for Phase I of SPC

The inclusion of correlated auxiliary variables into the monitoring scheme of quality characteristic of interest has gained notable attention in recent statistical process control (SPC) literature. Several authors have investigated the use of a correlated auxiliary variable for efficient monitoring of variability in Phase II of SPC. This phase is generally used to detect any shifts in the expected behavior of the process parameters which are often estimated from the historical in-control process in Phase I. However, no study has investigated the performance of auxiliary-based variability charts in Phase I of SPC. Here, we propose auxiliary-based dispersion control charts in Phase I of SPC. The auxiliary information is considered in the forms of regression, ratio, exponential, and power-ratio forms. The performance of the variability charts is evaluated and compared using probability to signal as a performance measure. An illustrative example is also provided to show the application of the charts. This study will provide practitioners with appropriate recommendations on the choice of dispersion charts for Phase I analysis.     

Design and implementation issues for a class of distribution-free Phase II EWMA exceedance control charts

Distribution-free (nonparametric) control charts can play an essential role in process monitoring when there is dearth of information about the underlying distribution. In this paper, we study various aspects related to an efficient design and execution of a class of nonparametric Phase II exponentially weighted moving average (denoted by NPEWMA) charts based on exceedance statistics. The choice of the Phase I (reference) sample order statistic used in the design of the control chart is investigated. We use the exact time-varying control limits and the median run-length as the metric in an in-depth performance study. Based on the performance of the chart, we outline implementation strategies and make recommendations for selecting this order statistic from a practical point of view and provide illustrations with a data-set. We conclude with a summary and some remarks.     

Effect of Phase I Estimation on Phase II Control Chart Performance with Profile Data

This paper illustrates how phase I estimators in statistical process control (SPC) can affect the performance of phase II control charts. The deleterious impact of poor phase I estimators on the performance of phase II control charts is illustrated in the context of profile monitoring. Two types of phase I estimators are discussed. One approach uses functional cluster analysis to initially distinguish between estimated profiles from an in‐control process and those from an out‐of‐control process. The second approach does not use clustering to make the distinction. The phase II control charts are established based on the two resulting types of estimates and compared across varying sizes of sustained shifts in phase II. A simulated example and a Monte Carlo study show that the performance of the phase II control charts can be severely distorted when constructed with poor phase I estimators. The use of clustering leads to much better phase II performance. We also illustrate that the performance of phase II control charts based on the poor phase I estimators not only have more false alarms than expected but can also take much longer than expected to detect potential changes to the process. Copyright © 2014 John Wiley & Sons, Ltd.     

On the design of control charts with guaranteed conditional performance under estimated parameters

When designing control charts the in-control parameters are unknown, so the control limits have to be estimated using a Phase I reference sample. To evaluate the in-control performance of control charts in the monitoring phase (Phase II), two performance indicators are most commonly used: the average run length (ARL) or the false alarm rate (FAR). However, these quantities will vary across practitioners due to the use of different reference samples in Phase I. This variation is small only for very large amounts of Phase I data, even when the actual distribution of the data is known. In practice, we do not know the distribution of the data, and it has to be estimated, along with its parameters. This means that we have to deal with model error when parametric models are used and stochastic error because we have to estimate the parameters. With these issues in mind, choices have to be made in order to control the performance of control charts. In this paper, we discuss some results with respect to the in-control guaranteed conditional performance of control charts with estimated parameters for parametric and nonparametric methods. We focus on Shewhart, exponentially weighted moving average (EWMA), and cumulative sum (CUSUM) control charts for monitoring the mean when parameters are estimated.     

New Approaches in Monitoring Multivariate Categorical Processes based on Contingency Tables in Phase II

In some statistical process control (SPC) applications, quality of a process or product is characterized by contingency table. Contingency tables describe the relation between two or more categorical quality characteristics. In this paper, two new control charts based on the WALD and Stuart score test statistics are designed for monitoring of contingency table‐based processes in Phase‐II. The performances of the proposed control charts are compared with the generalized linear test (GLT) control chart proposed in the literature. The results show the better performance of the proposed control charts under small and moderate shifts. Moreover, new schemes are proposed to diagnose which cell corresponding to different levels of categorical variables is responsible for out‐of‐control signal. In addition, we propose EWMA–WALD and EWMA–Stuart score test control charts to improve the performance of Shewhart‐based control charts in detecting small and moderate shifts in contingency table parameters. Meanwhile, we compare the performances of two proposed EWMA‐based control charts with the ones of three existing control charts called EWMA–GLT, EWMA–GLRT and an EWMA‐type control chart for multivariate binomial/multinomial processes along with the ones of the corresponding Shewhart‐based control charts. A numerical example is given to show the efficiency of the proposed methods. Finally, the effect of parameter estimation in Phase I based on m historical contingency table on the performance of the Shewhart‐based control charts is studied. Copyright © 2016 John Wiley & Sons, Ltd.     

Design and implementation of CUSUM exceedance control charts for unknown location

Nonparametric control charts provide a robust alternative in practice when the form of the underlying distribution is unknown. Nonparametric CUSUM (NPCUSUM) charts blend the advantages of a CUSUM with that of a nonparametric chart in detecting small to moderate shifts. In this paper, we examine efficient design and implementation of Phase II NPCUSUM charts based on exceedance (EX) statistics, called the NPCUSUM-EX chart. We investigate the choice of the order statistic from the reference (Phase I) sample that defines the exceedance statistic. We see that choices other than the median, such as the 75th percentile, can yield improved performance of the chart in certain situations. Furthermore, observing certain shortcomings of the average run-length, we use the median run-length as the performance metric. The NPCUSUM-EX chart is compared with the NPCUSUM-Rank chart based on the popular Wilcoxon rank-sum statistic. We also study the choice of the reference value, k, of the CUSUM charts. An illustration with real data is provided.     

Control Charts for Monitoring Linear Profiles with Within‐Profile Correlation Using Gaussian Process Models

Profile monitoring is the utilization of control charts for checking the stability of the quality of a product over time when the product quality is characterized by a function at each time point. Most existing control charts for monitoring profiles are based on the assumption that the observations within each profile are independent of each other, which is often invalid in practice. Successive measurements within profiles often exhibit spatial or serial correlation. This paper focuses on Phase II linear profile monitoring when within‐profile data are correlated. A Gaussian process model is used to describe the within‐profile correlation (WPC). Two Shewhart‐type multivariate control charts are proposed to monitor the linear trend term and the WPC separately in Phase II. Our proposed approaches are compared with alternative methods through numerical simulations in which different in‐control WPCs are considered. Simulation studies show that the proposed control charts are sensitive to changes in the linear trend term when the correlation is strong and effective in detecting large shifts in the WPC. Finally, an example is given to illustrate the implementation of our proposed control charts. Copyright © 2013 John Wiley & Sons, Ltd.     

Phase II control charts for monitoring dispersion when parameters are estimated

Shewhart control charts are among the most popular control charts used to monitor process dispersion. To base these control charts on the assumption of known in-control process parameters is often unrealistic. In practice, estimates are used to construct the control charts and this has substantial consequences for the in-control and out-of-control chart performance. The effects are especially severe when the number of Phase I subgroups used to estimate the unknown process dispersion is small. Typically, recommendations are to use around 30 subgroups of size 5 each.

?We derive and tabulate new corrected charting constants that should be used to construct the estimated probability limits of the Phase II Shewhart dispersion (e.g., range and standard deviation) control charts for a given number of Phase I subgroups, subgroup size and nominal in-control average run-length (ICARL). These control limits account for the effects of parameter estimation. Two approaches are used to find the new charting constants, a numerical and an analytic approach, which give similar results. It is seen that the corrected probability limits based charts achieve the desired nominal ICARL performance, but the out-of-control average run-length performance deteriorate when both the size of the shift and the number of Phase I subgroups are small. This is the price one must pay while accounting for the effects of parameter estimation so that the in-control performance is as advertised. An illustration using real-life data is provided along with a summary and recommendations.     

A simulation comparison of some distance-based EWMA control charts for monitoring the covariance matrix with individual observations

It is common in modern manufacturing to simultaneously monitor more than one process quality characteristic. In such a multivariate scenario, the monitoring of the covariance matrix, along with the mean vector, plays an important role in assessing whether a process stays in control or not. However, monitoring the covariance matrix is technically more difficult, especially when there is only one observation available in each subgroup, disabling the usual sample covariance matrix as an effective estimator. To monitor the covariance matrix with individual observations in Phase II stage, several exponentially weighted moving average (EWMA) control charts have been constructed based on the distance between the estimated process covariance matrix and its target value. In this paper, two new control charts are devised using the sum of the square roots of the absolute deviations and its combination with the sum of squared deviations. These distance-based control charts are compared via the simulation experiments on different simulated out-of-control covariance matrices with respect to the number of quality characteristics being monitored, the shift pattern, and the shift magnitude. The simulation results identify the control charts that perform relatively robust and show that these various control charts may have their respective merits on different out-of-control scenarios.     

Phase II process monitoring for the balanced one-way random effects model

Statistical process control (SPC) and monitoring techniques are useful in a variety of applications. In this paper, we consider prospective (Phase II) process monitoring for the balanced random effects (variance components) model with Shewhart-type charts when parameters are estimated from a Phase I study. Such a model is a nonstandard application of control charts and arises in a number of situations in practice. Effects of parameter estimation need to be accounted for or there maybe too many false alarms that will disrupt the monitoring regime. To this end, two types of corrected (adjusted) control limits are proposed, based on two perspectives, namely the unconditional and the conditional, as recommended in the recent literature. Results and derivations are provided along with tabulations and illustrations with real data. Robustness of the charts is examined and a summary and recommendations are given. An accompanying R package is provided for deploying the methodology. Other Phase II charts such as the EWMA and the CUSUM can be considered along similar lines and will be presented elsewhere.     

Evaluation of Phase I analysis scenarios on Phase II performance of control charts for autocorrelated observations

Phase I analysis of a control chart implementation comprises parameter estimation, chart design, and outlier filtering, which are performed iteratively until reliable control limits are obtained. These control limits are then used in Phase II for online monitoring and prospective analyses of the process to detect out-of-control states. Although a Phase I study is required only when the true values of the parameters of a process are unknown, this is the case in many practical applications. In the literature, research on the effects of parameter estimation (a component of Phase I analysis) on the control chart performance has gained importance recently. However, these studies consider availability of complete and clean data sets, without outliers and missing observations, for estimation. In this article, we consider AutoRegressive models of order 1 and study the effects of two extreme cases for Phase I analysis; the case where all outliers are filtered from the data set (parameter estimation from incomplete but clean data) and the case where all outliers remain in the data set during estimation. Performance of the maximum likelihood and conditional sum of squares estimators are evaluated and effects on the Phase II use are investigated. Results indicate that the effect of not detecting outliers in Phase I can be severe on the Phase II application of a control chart. A real-world example is provided to illustrate the importance of an appropriate Phase I analysis.     

Shewhart-type Phase II control charts for monitoring times to an event with a guaranteed in-control and good out-of-control performance

Monitoring time to event (failure) data is important in many applications. Proper monitoring and control can make the production process more efficient and provide economic advantages. In this paper, we consider the efficacy of a class of Shewhart-type control charts for monitoring time to event data following an exponential distribution with an unknown mean, which is estimated from a class of estimators. An estimator is chosen within this class, so that the in-control performance is maximized with respect to a number of popular criteria in the recent literature, and the proposed optimal charts are compared on the basis of their in-control and out-of-control performance. The comparisons include the traditional Phase II exponential Shewhart chart using the maximum likelihood estimator. Improved in-control and out-of-control performances of these charts can enhance the quality and productivity of manufacturing processes. Since no chart is best under all the criteria, a ranking system is used to choose a chart to use in practice with a good overall performance. Two illustrative examples using real data are given; summary and conclusions are offered.     

Phase II monitoring of variability using Cusum and EWMA charts with individual observations

When monitoring a process mean in Phase II, it is well known that time-weighted control charts (such as the Cusum or EWMA) of individual observations are more sensitive for detecting small mean changes than are the traditional Shewhart control charts for individuals. Further, by collecting one observation every 12?minutes, rather than a subgroup of five every hour, the time-weighted charts of individual values result in a shorter ATS (average time to signal) than would be possible using Shewhart charts of subgrouped data. This article explores a similar strategy of monitoring process variability using time-weighted control charts and individual observations. The average time to signal a change in variability using these charts is studied when there are targets or known values for the in-control process mean and standard deviation. The results show that the ATS of both the Cusum and EWMA are substantially shorter than the ATS for the standard R charts or the more efficient S² chart using subgroups of 5. The article also describes how the control limits for the EWMA chart to monitor process variability should be modified if the in-control process mean and standard deviations are unknown and must be estimated from a Phase I study. Computer functions that are available in R packages for creating Cusum-EWMA charts and computing their ARL (average run length) are demonstrated in this study and are included in the appendix.     

On the Performance of Auxiliary‐based Control Charting under Normality and Nonnormality with Estimation Effects

Process monitoring through control charts is a quite popular practice in statistical process control. From a statistical point of view, a superior control chart is the one which has an efficient design structure, for the case of both known and unknown parameters. There are auxiliary information–based location charts for an improved monitoring of process mean level. These charting structures have some limitations like assuming normality, the parameters to be known and focusing mainly on phase I monitoring. In many practical situations, nonnormal process behaviors are more frequent. Information about process parameters is not available, and we have to rely on the limited data available from the process to establish the limits in phase I and then use them in phase II monitoring. To have a compromise between the statistical and the practical purposes, a natural desire is to have a control chart that can serve both the concerns efficiently. This study is planned for the same objective focusing the auxiliary‐based Shewhart's control charts for location parameter. We have investigated the properties of the design structures of different location charts based on some already used and some new estimators with known and unknown parameters for normal and nonnormally distributed processes. By evaluating the performance of different charting structures in terms of power and run length properties in phase I and phase II, we have identified those more capable of making a good compromise between the abovementioned purposes in terms of statistical efficiency and practical desires. Copyright © 2012 John Wiley & Sons, Ltd.     

On the multivariate progressive control chart for effective monitoring of covariance matrix

With the development of modern acquisition techniques, data with several correlated quality characteristics are increasingly accessible. Thus, multivariate control charts can be employed to detect changes in the process. This study proposes two multivariate control charts for monitoring process variability (MPVC) using a progressive approach. First, when the process parameters are known, the performance of the MPVC charts is compared with some multivariate dispersion schemes. The results showed that the proposed MPVC charts outperform their counterparts irrespective of the shifts in the process dispersion. The effects of the Phase I estimated covariance matrix on the efficiency of the MPVC charts were also evaluated. The performances of the proposed methods and their counterparts are evaluated by calculating some useful run length properties. An application of the proposed chart is also considered for the monitoring of a carbon fiber tubing process.     

Revisiting reweighted robust standard deviation estimators for univariate Shewhart S‐charts

Phase I outliers, unless screened during process parameter estimation, are known to deteriorate Phase II performance of process control charts. Reweighting estimators, ie, trimming outlier subgroups and individual observations, were suggested in the literature to improve both the robustness and efficiency of the resulting parameter estimates. In the current study, effects of various reweighted estimators at different trimming levels on the Phase II performance of S‐charts are elucidated using computer simulations including isolated and mixtures of contamination models. Outlier magnitudes in the simulations are held at a moderately low level to mimic industrial practice. Subtleties, such as varying Type I error rate among different trimming levels with respect to quantiles of dispersion estimates, prevent a single method to be revealed as the best performing one under all circumstances, and choice of estimators and trimming levels should depend on the number of subgroups in Phase I and the specifics of the process. Nevertheless, S‐chart using scale M‐estimator with logistic ρ and location M‐estimator at 2% trimming generally stands out in terms of Phase II performance, and high trimming levels are particularly recommended for high number of Phase I subgroups.     

Use of median-based estimator to construct Phase II exponential chart

The Shewhart-type exponential control chart is a popular and extensively used among all time-between-events control charts for its simplicity. When the parameter is unknown, Phase II control limits are constructed, and the success of its implementation depends to an extent on the estimated value of the parameter, obtained from Phase I dataset. However, when the Phase I data are contaminated with spurious observations/outliers, the performance of the chart is suspected to deviate from what is normally expected. Traditionally, maximum likelihood estimator (MLE) and minimum variance unbiased estimator (MVUE) are used to estimate the unknown process parameter. Both of estimators are the functions of sample mean. In this paper, the median-based estimator (MBE) that is a function of sample median is used to construct Phase II control limits. Moreover, performance of the proposed chart is examined when Phase I sample consists of contaminated observations/outliers. It is found that the proposed chart outperforms the existing charts whether the sample is contaminated or not.     

A Kullback-Leibler information control chart for linear profiles monitoring

Profile monitoring is referred to as monitoring the functional relationship between the response and explanatory variables. Traditionally, process control charts monitor the mean and/or the variance of a univariate quality characteristic. For several correlated quality characteristics, multivariate process control charts monitor the mean vector and/or the covariance matrix. However, monitoring the functional relationship between variables is sometimes more beneficial. One example is the power output of a Diesel engine and the amount of fuel injected should be related. In this paper, we propose a Kullback-Leibler information (KLI) control chart for linear profiles monitoring in Phase II. The plotted statistics of the KLI chart are calculated based on a backward procedure. The functional relationship is described by linear regression. The performance of the proposed KLI control chart is compared with those of other existing control charts, especially the Generalized Likelihood Ratio (GLR) chart for both KLI and GLR charts do not require design parameters. The results show that (1) the KLI control chart is better than the GLR control chart in detecting the shift of variance in terms of Average Time to Signal, and (2) if the shift of the regression coefficient is small, the GLR chart outperforms the KLI chart, but if the size of shift is large, the KLI chart outperforms the GLR chart. The plotted statistics of KLI are compared to that of GLR. Their similarity is discussed.     

The Phase I Dispersion Charts for Bivariate Process Monitoring

Multivariate control charts are usually implemented in statistical process control to monitor several correlated quality characteristics. Process dispersion charts are used to determine the stability of process variation (which is typically done before monitoring the process location/mean). A Phase‐I study is generally used when population parameters are unknown. This article develops Phase‐I |S| and |G| control charts, to monitor the dispersion of a bivariate normal process. The charting constants are determined to achieve the required nominal false alarm probability (FAP₀). The performance of the proposed charts is evaluated in terms of (i) the attained false rate and (ii) the probability of signaling for out‐of‐control situations. The analysis shows that the proposed Phase‐I bivariate charts correctly control the FAP (the false alarm probability) and detect a shift occurring in the bivariate dispersion matrix with adequate probability. An example is given to explain the practical implementation of these charts. Copyright © 2015 John Wiley & Sons, Ltd.