Model‐based Multivariate Monitoring Charts for Autocorrelated Processes

Autocorrelation or nonstationarity may seriously impact the performance of conventional Hotelling's T² charts. We suggest modeling processes with multivariate autoregressive integrated moving average time series models and propose two model‐based monitoring charts. One monitors the predicted value and provides information about the need for mean adjustments. The other is a Hotelling's T² control chart applied to the residuals. The average run length performance of the residual‐based Hotelling's T² chart is compared with the observed data‐based Hotelling's T² chart for a group of first‐order vector autoregressive models. We show that the new chart in most cases performs well. Copyright © 2013 John Wiley & Sons, Ltd.     

A comparative study of phase II robust multivariate control charts for individual observations

Use of Hotelling's T² charts with high breakdown robust estimates to monitor multivariate individual observations are the recent trend in the control chart methodology. Vargas (J. Qual. Tech. 2003; 35: 367‐376) introduced Hotelling's T² charts based on the minimum volume ellipsoid (MVE) and the minimum covariance determinant (MCD) estimates to identify outliers in Phase I data. Studies carried out by Jensen et al. (Qual. Rel. Eng. Int. 2007; 23: 615‐629) indicated that the performance of these charts heavily depends on the sample size, amount of outliers and the dimensionality of the Phase I data. Chenouri et al. (J. Qual. Tech. 2009; 41: 259‐271) recently proposed robust Hotelling's T² control charts for monitoring Phase II data based on the reweighted MCD (RMCD) estimates of the mean vector and covariance matrix from Phase I. They showed that Phase II RMCD charts have better performance compared with Phase II standard Hotelling's T² charts based on outlier free Phase I data, where the outlier free Phase I data were obtained by applying MCD and MVE T² charts to historical data. Reweighted MVE (RMVE) and S‐estimators are two competitors of the RMCD estimators and it is a natural question whether the performance of Phase II Hotelling's T² charts with RMCD and RMVE estimates exhibits similar pattern observed by Jensen et al. (Qual. Rel. Eng. Int. 2007; 23: 615‐629) in the case of MCD and MVE‐based Phase I Hotelling's T² charts. In this paper, we conduct a comparative study to assess the performance of Hotelling's T² charts with RMCD, RMVE and S‐estimators using large number of Monte Carlo simulations by considering different data scenarios. Our results are generally in favor of the RMCD‐based charts irrespective of sample size, outliers and dimensionality of Phase I data. Copyright © 2010 John Wiley & Sons, Ltd.     

A robust alternative to Hotelling's T2 control chart using trimmed estimators

A control chart is one of the primary techniques used in statistical process control. In phase I, historical observations are analysed in order to construct a control chart with which to determine whether the process has been in control over the period of time in which the data were collected. The presence of multiple outliers may go undetected by the usual control charts, such as Hotelling's T² due to the masking effect. In this paper we propose a robust alternative to Hotelling's T² control chart with estimators defined using trimming. Simulation studies show that the proposed control chart is more effective than T² in detecting outliers. Copyright © 2008 John Wiley & Sons, Ltd.     

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.     

A Robust Bivariate Control Chart Alternative to the Hotelling's T2 Control Chart

In this paper, we proposed a new bivariate control chart denoted by based on the robust estimation as an alternative to the Hotelling's T² control chart. The location vector and the variance‐covariance matrix for the new control chart are obtained using the sample median, the median absolute deviation from the sample median, and the comedian estimator. The performance of the proposed method in detecting outliers is evaluated and compared with the Hotelling's T² method using a Monte‐Carlo simulation study. A numerical example is considered to illustrate the application of the proposed method. Copyright © 2012 John Wiley & Sons, Ltd.     

An assessment of the kernel‐distance‐based multivariate control chart through an industrial application

Traditional multivariate quality control charts assume that quality characteristics follow a multivariate normal distribution. However, in many industrial applications the process distribution is not known, implying the need to construct a flexible control chart appropriate for real applications. A promising approach is to use support vector machines in statistical process control. This paper focuses on the application of the ‘kernel‐distance‐based multivariate control chart’, also known as the ‘k‐chart’, to a real industrial process, and its assessment by comparing it to Hotelling's T² control chart, based on the number of out‐of‐control observations and on the Average Run Length. The industrial application showed that the k‐chart is sensitive to small shifts in mean vector and outperforms the T² control chart in terms of Average Run Length. Copyright © 2010 John Wiley & Sons, Ltd.     

Robust T2 control chart using median-based estimators

One of the most widely used multivariate control charts is the Hotelling T². In order to construct a Hotelling T² control chart, the mean vector (μ) and the variance–covariance matrix (Σ) must be first estimated. The classical estimators of μ and Σ are usually used to design Hotelling T² control chart. The classical estimators are sensitive to the presence of outliers. One way to deal with outliers is to use robust estimators. In this study, a robust T² control chart is proposed. The mean vector is obtained from the sample median. The median absolute deviation and the comedian are used as the estimates of the elements of the variance–covariance matrix. The proposed robust estimators of the mean vector and the variance–covariance matrix are compared with the sample mean vector and the sample variance–covariance matrix, and the M estimator of these parameters, through efficiency and robustness measures. The performances of the proposed robust T² control chart and the classical and the M estimators are also compared by means of average run length. Simulation results reveal that the proposed robust T² control chart has much better performance than the traditional Hotelling T² and similar performance to the M estimator in detecting shifts in process mean vector. Use of other robust estimators to estimate the process parameters is an area for further research.     

Hotelling's T2 control chart with adaptive sample sizes

Some quality control schemes have been developed when several related quality characteristics are to be monitored: simultaneous X¯ charts, Hotelling's T² chart, multivariate CUSUM and multivariate EWMA. Hotelling's T² control chart has the advantage of its simplicity but it is slow in detecting small process shifts. The latest developments in variable sample sizes for univariate control charts are applied in this paper to define an adaptive sample sizes T² control chart. As occurs in the univariate case the ARL improvements are very important particularly for small process shifts. An example is given to illustrate the use of the proposed scheme.     

The Run Sum Hotelling's χ2 Control Chart with Variable Sampling Intervals

Control charts are widely used for process monitoring and quality control in manufacturing industries. Implementing variable sampling interval (VSI) control schemes on control charts rather than traditional fixed sampling interval procedure can significantly improve the control chart's efficiency. In this paper, the VSI run sum (RS) Hotelling's χ² chart is proposed. The optimal scores and parameters of the proposed chart are determined using an optimization technique to minimize the following: (i) out‐of‐control average time to signal (ATS); (ii) adjusted ATS (AATS), when the exact shift size can be specified; (iii) expected ATS; or (iv) expected AATS, when the exact shift size cannot be specified. The Markov chain method is used to evaluate the zero‐state ATS and expected ATS, and steady‐state AATS and expected AATS of the proposed chart. The results show that the VSI RS Hotelling's χ² chart significantly outperforms the standard RS Hotelling's χ² chart and the former also performs well compared with other competing charts. By adding more scoring regions, the efficiency of the VSI RS Hotelling's χ² chart can be further enhanced. An illustrative example using data from a manufacturing process is presented to demonstrate the application of the VSI RS Hotelling's χ² chart. The application of the proposed chart in a quality improvement program can be extended to management and service industries. Copyright © 2016 John Wiley & Sons, Ltd.     

Phase I Analysis of Individual Observations with Missing Data

The effect of the methods for handling missing values on the performance of Phase I multivariate control charts has not been investigated. In this paper, we discuss the effect of four imputation methods: mean substitution, regression, stochastic regression and the expectation maximization algorithm. Estimates of mean vector and variance covariance matrix from the treated data set are used to estimate the unknown parameters in the Hotelling's T² chart statistic. Based on a Monte Carlo simulation study, the performance of each of the four methods is investigated in terms of its ability to obtain the nominal in‐control and out‐of‐control overall probability of a signal. We consider three sample sizes, five levels of the percentage of missing values and three types of variable numbers. Our simulation results show that the stochastic regression method has the best overall performance among all the competing methods. Copyright © 2013 John Wiley & Sons, Ltd.     

Optimisation of a set of or principal components control charts using genetic algorithms

When a multivariate process is to be monitored, there are the options of employing a set of univariate control charts or a single multivariate chart. This paper shows how to effectively design a multivariate control scheme consisting of two or three X charts, using genetic algorithms to optimise the charts parameters. The procedure is implemented using software tools, which we designed. A complete performance comparison of the scheme with the Hotelling's T ² control chart can be made in order to help the user in choosing the most adequate option for the process under consideration. Also, if the user prefers to employ charts based on principal components rather than on the original variables, the software can be used in the same way to optimise a set of two or three control charts based on these components, and to compare their performance with the performance of the T ² chart on the principal components.     

To shrink or not to shrink: Hotelling's T2 chart based on shrunken covariance estimates

Several authors have studied the effect of parameter estimation on the performance of Phase II control charts and shown that large in‐control reference samples are necessary for the Phase II control charts to perform as desired. For higher dimensional data, even larger reference samples are required to achieve stable estimation of the in‐control parameters. Shrinkage estimation has been widely studied as a method to achieve stable estimation of the covariance matrix for high‐dimensional data. We investigate the average run length (ARL) distribution of the Hotelling T² chart when using a shrunken covariance matrix. Specifically, we explore the following questions: (1) Does the use of a shrinkage estimator of the covariance matrix result in reduced variability in the ARL performance of the T² chart? (2) Does the use of a shrinkage estimator of the covariance matrix result in a reduced occurrence of “strictly multivariate” false alarms on the T²chart? (3) How does shrinkage of the covariance matrix affect the out‐of‐control performance of the T² chart? We use a simulation study to investigate the use of shrinkage estimation with the Hotelling T² chart in Phase II. Our results indicate that, while shrinkage estimation affects the ARL performance of the T² chart, the benefits are small and occur in fairly specific circumstances. The benefits of shrinking may not justify the use of more advanced techniques.     

A boosting approach for understanding out-of-control signals in multivariate control charts

The most widely used tools in statistical quality control are control charts. However, the main problem of multivariate control charts, including Hotelling's T ² control chart, lies in that they indicate that a change in the process has happened, but do not show which variable or variables are the source of this shift. Although a number of methods have been proposed in the literature for tackling this problem, the most usual approach consists of decomposing the T ² statistic. In this paper, we propose an alternative method interpreting this task as a classification problem and solving it through the application of boosting with classification trees. The classifier is then used to determine which variable or variables caused the change in the process. The results prove this method to be a powerful tool for interpreting multivariate control charts.     

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.     

Monitoring Highly Correlated Multivariate Processes Using Hotelling's T2 Statistic: Problems and Possible Solutions

Hotelling's T² statistic is the default control statistic for continuous multivariate data, but there are dangers in applying this statistic without the appropriate level of checks and balances. This paper discusses the potential issues with using the Hotelling's T² statistic when the quality variable measures are highly correlated and provides some solutions that will help mitigate the risks with applying the Hotelling's T² control charts in such practical examples. Copyright © 2014 John Wiley & Sons, Ltd.     

Stochastic Processes

In quality control, a proper Phase I analysis is essential to the success of Phase II monitoring. A literature review reveals no distribution-free Phase I multivariate techniques in existence. This research develops a Phase I location control chart for multivariate elliptical processes. The resulting in-control reference sample can then be used to estimate the parameters for Phase II monitoring. Using Monte Carlo simulation, the proposed method is compared with the Hotelling's T² Phase I chart. Although Hotelling's T² chart is preferred when the data are multivariate normal, the proposed method is shown to perform significantly better under nonnormality. This article has supplementary material online.     

New sampling strategies to reduce the effect of autocorrelation on the synthetic T2 chart to monitor bivariate process

In this paper, synthetic T² chart is developed to monitor bivariate process with correlated variables and autocorrelated observations. The proposed chart is a combination of the Hotelling's T² chart and the conforming run length chart. The operation and design of the chart are described when observations are autocorrelated and cross correlated. The first‐order vector autoregressive process VAR (1) is used to model the bivariate data from an autocorrelated process of interest. Using an average run length as performance measure criterion in the VAR (1) model, it is observed that autocorrelation seriously impact the performance of the synthetic T² chart. To reduce the effect of autocorrelation on the performance of the synthetic T² chart, the skip and mixed sampling strategies are implemented to form rational subgroups in the construction of synthetic T² chart. The average run length performance of the synthetic T² chart implementing these strategies is compared with that of the standard strategy of formation of rational subgroups. It is observed that implementing skip and mixed sampling strategies within rational subgroup improves the performance of the synthetic T² chart.     

Hotelling's T2 control charts with fixed and variable sample sizes for monitoring short production runs

Short production runs are common in enterprises that require a high degree of flexibility and variety in manufacturing processes. To date, past research on short production runs has little focus on the multivariate control charts. In view of this, fixed sample size (FSS) and variable sample size (VSS) Hotelling's T² charts are designed to monitor the process mean when the production horizon is finite. Optimal parameters to minimize the out‐of‐control (1) truncated average run length (TARL) and (2) expected TARL (ETARL) are provided such that the in‐control TARL is equal to the number of inspections (say I). The numerical study considers the run length performances of the FSS and VSS T² short‐run charts for both known and unknown shift sizes. The VSS T² short‐run chart performs well in swiftly detecting various mean shifts in comparison with the FSS T² short‐run chart. Additionally, the VSS T² short‐run chart is superior to the FSS T² short‐run chart, in terms of the truncated standard deviation of the run length, expected truncated standard deviation of the run length, probability that the chart signals an alarm within the I inspections, ie, P(I) and expected P(I). A case study on the impurity profile of a crystalline drug substance illustrates the implementation of the VSS T² short‐run chart.     

Variable sampling interval hotelling's T2 charts with runs rules for switching between sampling interval lengths

The use of runs rules is proposed for switching between the sampling interval lengths of variable sampling interval Hotelling's T² charts. The purpose of applying these rules is to reduce the frequency of the switches which causes inconvenience in the administration of the charts. The expressions for the performance measures for the charts with these rules are derived. The effects of different runs rules on the performances are evaluated through numerical comparisons. The runs rules substantially reduce the frequency of switches during the in‐control period and during the out‐of‐control periods due to the small to moderate shifts in the process mean vector. They also fairly improve the statistical performances of the charts in detecting the small shifts and do not affect that in detecting the large shifts. However, some runs rules slightly worsen the statistical performances in detecting the moderate shifts. Copyright © 2011 John Wiley & Sons, Ltd.     

Memory‐Type Control Charts for Monitoring the Process Dispersion

Control charts have been broadly used for monitoring the process mean and dispersion. Cumulative sum (CUSUM) and exponentially weighted moving average (EWMA) control charts are memory control charts as they utilize the past information in setting up the control structure. This makes CUSUM and EWMA‐type charts good at detecting small disturbances in the process. This article proposes two new memory control charts for monitoring process dispersion, named as floating T ? S² and floating U ? S² control charts, respectively. The average run length (ARL) performance of the proposed charts is evaluated through a simulation study and is also compared with the CUSUM and EWMA charts for process dispersion. It is found that the proposed charts are better in detecting both positive as well as negative shifts. An additional comparison shows that the floating U ? S² chart has slightly smaller ARLs for larger shifts, while for smaller shifts, the floating T ? S² chart has better performance. An example is also provided which shows the application of the proposed charts on simulated datasets. Copyright © 2013 John Wiley & Sons, Ltd.