Monitoring therapeutic processes using risk-adjusted multivariate Tukey's CUSUM control chart

Using control charts for monitoring therapeutic processes has become popular lately. As the application of traditional control charts in the therapeutic processes may be misleading due to the inherent differences between patients, a multifactor correlated risk measure is considered in monitoring of these processes. Therefore, using risk-adjusted control charts for monitoring the therapeutic processes is of interest to practitioners. Furthermore, in health care monitoring, statistical models should account for abnormal distributions and outlier data to minimize misinterpretations of monitoring schemes. This study proposes a risk-adjusted multivariate Tukey's cumulative sum (RA-MTCUSUM) control chart. The proposed method is a combination of the accelerated failure time (AFT) regression model, the Tukey's control chart (TCC) featuring robustness against abnormality, and the multivariate cumulative sum (MCUSUM) control chart for monitoring multivariable process. Simulation experiments are performed to evaluate the performance of the proposed control chart using the average run length (ARL) measure. Results show that the RA-MTCUSUM control chart has better performance in comparison with traditional ones for monitoring various distributions (normal and non-normal). Based on the simulation results, outlier data do not disturb the proposed control chart's performance. Moreover, applying the RA-MTCUSUM control chart to a real-world dataset related to sepsis patients of a hospital located in Tehran, Iran indicates that the control chart has more reasonable performance than the traditional control charts in the real applications due to its robustness.     

Variable Selection‐based Multivariate Cumulative Sum Control Chart

High‐dimensional applications pose a significant challenge to the capability of conventional statistical process control techniques in detecting abnormal changes in process parameters. These techniques fail to recognize out‐of‐control signals and locate the root causes of faults especially when small shifts occur in high‐dimensional variables under the sparsity assumption of process mean changes. In this paper, we propose a variable selection‐based multivariate cumulative sum (VS‐MCUSUM) chart for enhancing sensitivity to out‐of‐control conditions in high‐dimensional processes. While other existing charts with variable selection techniques tend to show weak performances in detecting small shifts in process parameters due to the misidentification of the ‘faulty’ parameters, the proposed chart performs well for small process shifts in identifying the parameters. The performance of the VS‐MCUSUM chart under different combinations of design parameters is compared with the conventional MCUSUM and the VS‐multivariate exponentially weighted moving average control charts. Finally, a case study is presented as a real‐life example to illustrate the operational procedures of the proposed chart. Both the simulation and numerical studies show the superior performance of the proposed chart in detecting mean shift in multivariate processes. Copyright © 2016 John Wiley & Sons, Ltd.     

Effect of Autocorrelation on Performance of the MCUSUM Control Chart

Control charts are designed to detect assignable causes of variation that may occur in production processes. When traditional control charts are used there is the implicit assumption that observations are independently and identically distributed over time. It is also assumed that the probability distribution representing the observations has a known functional form and is constant over time. However, in practice, observations generated by continuous as well as discrete production processes are often serially correlated. Autocorrelation not only violates the independence assumption of traditional control charts but also can affect the performance of control charts adversely. This point has received considerable attention in the past few decades. In this paper, we investigate the performance of the multivariate cumulative sum (MCUSUM) control chart in the presence of autocorrelation. Using an average run length (ARL) criterion, it is shown that autocorrelation can deteriorate the performance of MCUSUM control charts. A solution based on a time series model is presented that improves ARL properties of MCUSUM control charts. Copyright © 2005 John Wiley & Sons, Ltd.     

Advanced multivariate cumulative sum control charts based on principal component method with application

Existing multivariate cumulative sum (MCUSUM) control charts involve entire associated variables of a process to monitor variations in the mean vector. In this study, we have offered MCUSUM control charts with principal component method (PCM). The proposed MCUSUM control charts with PCM capture the whole process variations using fewer latent variables (principal components) while preserving as much data variability as possible. To show the significance of proposed MCUSUM control charts with PCM, various performance measures are considered including average run length, extra quadratic loss, relative average run length, and performance comparison index. Furthermore, performance measures are calculated through advanced Monte Carlo simulation method to explore the behavior of proposed MCUSUM control charts and to conduct comparative analysis with existing models. Results revealed that proposed MCUSUM control charts with PCM are efficient to detect variations timely by involving smaller number of principal components instead of considering entire associated variables. Also, proposed MCUSUM control charts have the ability to accommodate the features of existing control charts, which are illustrated as the special cases. Besides, to highlight the implementation mechanism and advantages of proposed MCUSUM control charts with PCM, a real-life example from wind turbine process is included.     

Deviance residual-based control charts for monitoring the beta-distributed processes

Beta-distributed process outputs are common in manufacturing industry because they range from 0 to 1 based on inputs like yield. Under the normality assumption, Shewarts control charts and Hotelling's control charts based on the deviance residual have been applied to monitor the process mean of the beta-distributed process outputs. The normality assumption can be violated according to the shape of the beta distribution. Therefore, without the normality assumption, we propose antirank control charts, exponentially weighted moving average (EWMA) control charts and cumulative sum (CUSUM) control charts. The proposed control charts outperform the existing control charts in the experimental results. The previous research has been focused on monitoring the process mean only. For the first time, in order to monitor the process variance of the beta-distributed process outputs, we propose the multivariate exponentially weighted mean squared deviation (MEWMS) chart, the first norm distance of the MEWMS deviation from its expected value (MEWMSL₁) chart, the chart based on MEWMS deviation with the approximated distribution of trace (MEWMS_AT), the multivariate trace sum squared deviation (MTSSD) chart and the multivariate matrix sum squared deviation (MMSSD) chart based on the deviance residual. The proposed control charts are compared and recommended in terms of the experimental results. This research can be a guideline for practitioners who monitor the deviance residual.     

A Multivariate Exponentially Weighted Moving Average Control Chart

A multivariate extension of the exponentially weighted moving average (EWMA) control chart is presented, and guidelines given for designing this easy-to-implement multivariate procedure. A comparison shows that the average run length (ARL) performance of this chart is similar to that of multivariate cumulative sum (CUSUM) control charts in detecting a shift in the mean vector of a multivariate normal distribution. As with the Hotelling's χ² and multivariate CUSUM charts, the ARL performance of the multivariate EWMA chart depends on the underlying mean vector and covariance matrix only through the value of the noncentrality parameter. Worst-case scenarios show that Hotelling's χ² charts should always be used in conjunction with multivariate CUSUM and EWMA charts to avoid potential inertia problems. Examples are given to illustrate the use of the proposed procedure.     

Optimal designs of the multivariate synthetic chart for monitoring the process mean vector based on median run length

The average run length (ARL) is usually used as a sole measure of performance of a multivariate control chart. The Hotelling's T², multivariate exponentially weighted moving average (MEWMA) and multivariate cumulative sum (MCUSUM) charts are commonly optimally designed based on the ARL. Similar to the case of univariate quality control, in multivariate quality control, the shape of the run length distribution changes in accordance to the magnitude of the shift in the mean vector, from highly skewed when the process is in‐control to nearly symmetric for large shifts. Because the shape of the run length distribution changes with the magnitude of the shift in the mean vector, the median run length (MRL) provides additional and more meaningful information about the in‐control and out‐of‐control performances of multivariate charts, not given by the ARL. This paper provides a procedure for optimal designs of the multivariate synthetic T² chart for the process mean, based on MRL, for both the zero and steady‐state modes. Two Mathematica programs, each for the zero state and steady‐state modes are given for a quick computation of the optimal parameters of the synthetic T² chart, designed based on MRL. These optimal parameters are provided in the paper, for the bivariate case with sample sizes, nin{4, 7, 10}. The MRL performances of the synthetic T², MEWMA and Hotelling's T² charts are also compared. Copyright © 2011 John Wiley & Sons, Ltd.     

Memory‐type multivariate control charts with auxiliary information for process mean

In statistical process control, it is a common practice to increase the sensitivity of a control chart with the help of an efficient estimator of the underlying process parameter. In this paper, we consider an efficient estimator that requires information on several study variables along with one or more auxiliary variables when estimating the mean of a multivariate normally distributed process. Using this auxiliary‐information‐based (AIB) process mean estimator, we propose new multivariate EWMA (MEWMA), double MEWMA (DMEWMA), and multivariate CUSUM (MCUSUM) charts for monitoring the process mean, denoted by the AIB‐MEWMA, AIB‐DMEWMA, and AIB‐MCUSUM charts, respectively. The run length characteristics of the proposed multivariate charts are computed using Monte Carlo simulations. The proposed charts are compared with their existing counterparts in terms of the run length characteristics. It turns out that the AIB‐MEWMA, AIB‐DMEWMA, and AIB‐MCUSUM charts are uniformly and substantially better than the MEWMA, DMEWMA, and MCUSUM charts, respectively, when detecting different shifts in the process mean. A real dataset is considered to explain the implementation of the proposed and existing multivariate control charts.     

A Copula Markov CUSUM Chart for Monitoring the Bivariate Auto‐correlated Binary Observations

Although in the statistical process control (SPC) literature, there are considerable number of researches related to the multivariates variables control charting (focusing on the variable quality characteristics), fewer investigations could be found regarding the multivariate attributes control charts (relying on the attribute quality characteristics). More specifically considering the multivariate attributes control charting, it would be more interesting to monitor the auto‐correlated data, since the real‐world processes usually include the data based on an auto‐correlation structure. Ignoring the auto‐correlation structure in developing a multivariate control chart increases the type I and type II errors simultaneously and consequently reduces the performance of the chart. The most important difficulty with developing multivariate attributes control charts is the absence of the joint distribution for the quality characteristics. This deficiency can be dispelled through the use of the copula approach for developing the joint distribution. In this paper, we use the Markov approach for modeling the auto‐correlated data. Then, the copula approach is used to make the joint distribution of two auto‐correlated binary data series. Finally, based on this joint distribution, we develop a cumulative sum (CUSUM) chart. Hence, the proposed chart is entitled the copula Markov CUSUM chart. The proposed control chart is compared with the most recent and effective existing one in the literature. Based on the average number of observations to signal (ANOS) measure, it is considered that the developed control chart performs better than the other one. In addition, a real case study related to two correlated diseases such as the Type 2 Diabetes Mellitus and the Obesity, in which each has an auto‐correlated structure, is investigated to verify the applicability of the control chart. Copyright © 2012 John Wiley & Sons, Ltd.     

Sum of Squares Control Charts for Monitoring of Multivariate Multiple Linear Regression Profiles in Phase II

In this paper, we propose four control charts for simultaneous monitoring of mean vector and covariance matrix in multivariate multiple linear regression profiles in Phase II. The proposed control charts include sum of squares exponential weighted moving average (SS‐EWMA) and sum of squares cumulative sum (SS‐CUSUM) for monitoring regression parameters and corresponding covariance matrix and SS‐EWMARe and SS‐CUSUMRe control charts for monitoring mean vector and covariance matrix of residual. Proposed methods are able to identify the out‐of‐control parameter responsible for shift. The performance of the proposed control charts is compared with existing method through Monte‐Carlo simulations. Moreover, the diagnostic performance of the proposed control charts is evaluated through simulation studies. The results show better performance of the proposed control charts rather than competing control chart. Finally, the applicability of the proposed control charts is illustrated using a real case of calibration application in the automotive industry. Copyright © 2016 John Wiley & Sons, Ltd.     

Dual multivariate CUSUM charts with auxiliary information for process mean

It is customary to increase the sensitivity of a control chart using an efficient estimator of the underlying process parameter which is being monitored. In this paper, using an auxiliary information-based (AIB) mean estimator, we propose dual multivariate CUSUM (DMCUSUM) and mixed DMCUSUM (MDMCUSUM) charts, called the AIB-DMCUSUM and AIB-MDMCUSUM charts, with and without fast initial response features for monitoring the mean vector of a multivariate normally distributed process. The DMCUSUM chart combines two similar-type multivariate CUSUM (MCUSUM) charts while the MDMCUSUM chart combines two different-type MCUSUM charts, into a single chart. The objective of two multivariate subcharts in the DMCUSUM/MDMCUSUM chart is to simultaneously detect small-to-moderate and moderate-to-large shifts in the process mean vector. Monte Carlo simulations are used to compute the run length characteristics, including the average run length (ARL), extra quadratic loss, and integral of the relative ARL. Based on detailed run length comparisons, it turns out that the AIB-DMCUSUM and AIB-MDMCUSUM charts uniformly and substantially outperform the DMCUSUM and MDMCUSUM charts when detecting different sizes of shift in the process mean vector. A real dataset is used to explain the implementation of proposed AIB multivariate charts.     

A wavelet‐based nonparametric CUSUM control chart for autocorrelated processes with applications to network surveillance

Statistical process control (SPC) has natural applications in data network surveillance. However, network data are commonly autocorrelated, which presents challenges to the basic SPC methods. Most existing SPC methods for correlated data assume parametric models to account for the correlation structure within the data. Those model assumptions can be difficult to justify in practice. In this paper, we propose a nonparametric cumulative sum (CUSUM) control chart for autocorrelated processes. In our proposed approach, we incorporate a wavelet decomposition and a nonparametric multivariate CUSUM control chart to obtain a robust procedure for autocorrelated processes without distribution assumptions. Extensive simulations show that the procedure appropriately controls the in‐control average run length and also has good sensitivity for detecting location shifts.     

A comparison study of control charts for Weibull distributed time between events

Monitoring decreases in the mean of Weibull time between events data to address process quality deteriorations is an important task in reliability analysis. Two new control charts such as Weibull exponentially weighted moving average and mixed cumulative sum‐exponentially weighted moving average by transforming the Weibull data to the exponential data are proposed and compared with 2 existing control charts such as Weibull cumulative sum and mixed exponentially weighted moving average‐cumulative sum. The performance comparison provides a way to select a specific control chart in a given situation. The average run length and the standard deviation of the run length are used as performance measures. The relative mean index is also utilized to measure the overall performance. The smaller the value of the relative mean index, the better the performance of the control chart and vice versa. Two illustrative examples are provided to show the applications of the proposed control charts.     

On developing an exponentially weighted moving average chart under progressive setup: An efficient approach to manufacturing processes

The exponentially weighted moving average (EWMA) control chart is a memory chart that is widely used in process monitoring to spot small and persistent disturbances in the process parameter(s). This chart requires normality of the quality characteristic(s) of interest and a smaller choice of smoothing parameter. Any deviations from these conditions affect its performance in terms of efficiency and robustness. For the said two concerns, this study develops a new mixed EWMA chart under progressive setup (mixed EWMA–progressive mean [MEP] chart). The proposed MEP chart combines the advantages of robustness (under nonnormal scenarios) and high sensitivity to small and persistent shifts in the process mean. The performance of the proposed MEP control chart is evaluated in terms of average run length and some other characteristics of run length distribution. The assessment of the proposed chart is made under standard normal, student's t, gamma, Laplace, logistic, exponential, contaminated normal and lognormal distributions. The performance of the proposed MEP chart is also compared with some existing competitors including the classical EWMA, the classical cumulative sum (CUSUM), the homogenously weighted moving average, the mixed EWMA–CUSUM, the mixed CUSUM–EWMA and the double EWMA charts. The analysis reveals that the proposal of this study offers a superior design structure relative to its competing counterparts. An application from substrates manufacturing process (in which flow width of the resist is the key quality characteristic) is also provided in the study.     

The effects of constructed bivariate copulas on multivariate control charts effectiveness

The average control chart monitors the shifts in the process. The familiar multivariate control charts are used to detect the mean vector of the process such as multivariate cumulative sum (MCUSUM) and Hotelling's T² control charts. In this paper, the effects of constructing bivariate copulas on multivariate control charts, that is, MCUSUM and Hotelling's T² control charts are intensively investigated when observations are drawn from the exponential distribution. Moreover, the dependence levels of observations are classified to be weak, moderate, and strong in both positive and negative values by Kendall's tau. The numerical results were obtained by Monte Carlo simulation to explore the average run length (ARL). The simulation results show that the performance of Hotelling's T² control chart is superior to the MCUSUM control chart for all shifts in the mean vector of process. Furthermore, from applying the presented control chart to two sets of real data, data set of the strength of 1.5 cm glass fibers measured at the National Physical Laboratory, England and data set of the strength of glass of the aircraft window, it was found that for a small shift ($\delta \le 0.1$), the MCUSUM control chart is better than Hotelling's T² control chart.     

A nonparametric CUSUM chart for process dispersion

In the present article, we propose a nonparametric cumulative sum control chart for process dispersion based on the sign statistic using in‐control deciles. The chart can be viewed as modified control chart due to Amin et al, 6 which is based on in‐control quartiles. An average run length performance of the proposed chart is studied using Markov chain approach. An effect of non‐normality on cumulative sum S² chart is studied. The study reveals that the proposed cumulative sum control chart is a better alternative to parametric cumulative sum S² chart, when the process distribution is non‐normal. We provide an illustration of the proposed cumulative sum control chart.     

Nonparametric Progressive Mean Control Chart for Monitoring Process Target

Nonparametric control charts are widely used when the parametric distribution of the quality characteristic of interest is questionable. In this study, we proposed a nonparametric progressive mean control chart, namely the nonparametric progressive mean chart, for efficient detection of disturbances in process location or target. The proposed chart is compared with the recently proposed nonparametric exponentially weighted moving average and nonparametric cumulative sum charts using different run length characteristics such as the average run length, standard deviation of the run length, and the percentile points of the run length distribution. The comparisons revealed that the proposed chart outperformed recent nonparametric exponentially weighted moving average and nonparametric cumulative sum charts, in terms of detecting the shifts in process target. A real life example concerning the fill heights of soft drink beverage bottles is also provided to illustrate the application of the proposed nonparametric control chart. Copyright © 2012 John Wiley & Sons, Ltd.     

Multivariate statistical process control charts based on the approximate sequential χ2 test

Similar to the univariate CUSUM chart, a multivariate CUSUM (MCUSUM) chart can be designed to detect a particular size of the mean shift optimally based on the scheme of a sequential likelihood ratio test for the noncentrality parameter. However, in multivariate case, the probability ratio of a sequential test is intractable mathematically and the test statistic based on the ratio does not have a closed form expression which makes it impractical for real application. We drive an approximate log-likelihood ratio and propose a multivariate statistical process control chart based on a sequential χ² test to detect a change in the noncentrality parameter. The statistical properties of the proposed test statistic are explored. The average runs length (ARL) performance of the proposed charts is compared with other MCUSUM charts for process mean monitoring. The experimental results reveal that the proposed charts achieve superior, both zero-state and steady-state, ARL performance over a wide range of mean shifts, especially when the dimension of measurements is large.     

Directionally sensitive MCUSUM mean charts

In the context of a disease outbreak detection, a prime interest is to only detect increases in the process mean. It is thus desirable to have a directionally sensitive multivariate chart that can effectively detect either increases or decreases in the process mean vector. In this paper, with a suitable transformation that truncates multivariate observations either above or below the process mean vector, we propose one-sided and two one-sided MCUSUM charts for monitoring the mean of a multivariate normal process. Among the proposed charts, the one-sided MCUSUM charts are directionally sensitive, while the two one-sided MCUSUM charts are directionally insensitive. In addition, the fast initial response feature is also incorporated into the proposed charts to enhance their sensitivities against initial process shifts. The run length characteristics of these control charts are computed with the Monte Carlo simulation. Based on the run length comparisons, it is found that the proposed charts are more sensitive than the existing charts when detecting moderate-to-large shifts in the process mean. The proposed charts are also applied on real datasets to support the theory.     

Weighted adaptive multivariate CUSUM control charts

An adaptive multivariate cumulative sum (AMCUSUM) control chart has received considerable attention because of its ability to dynamically adjust the reference parameter whereby achieving a better performance over a range of mean shifts than the conventional multivariate cumulative sum (CUSUM) charts. In this paper, we introduce a progressive mean–based estimator of the process mean shift and then use it to devise new weighted AMCUSUM control charts for efficiently monitoring the process mean. These control charts are easy to design and implement in a computerized environment compared with their existing counterparts. Monte Carlo simulations are used to estimate the run‐length characteristics of the proposed control charts. The run‐length comparison results show that the weighted AMCUSUM charts perform substantially and uniformly better than the classical multivariate CUSUM and AMCUSUM charts in detecting a range of mean shifts. An example is used to illustrate the working of existing and proposed multivariate CUSUM control charts.