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.     

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.     

Alternative Hotelling's T2 Charts using Winsorized Modified One‐Step M‐estimator

Hotelling's T² chart is a popular tool for monitoring statistical process control. However, this chart is sensitive in the presence of outliers. To alleviate the problem, this paper proposed alternative Hotelling's T² charts for individual observations using robust location and scale matrix instead of the usual mean vector and the covariance matrix, respectively. The usual mean vector in the Hotelling T² chart is replaced by the winsorized modified one‐step M‐estimator (MOM) whereas the usual covariance matrix is replaced by the winsorized covariance matrix. MOM empirically trims the data based on the shape of the data distribution. This study also investigated on the different trimming criteria used in MOM. Two robust scale estimators with highest breakdown point, namely S_n and T_n were selected to suit the criteria. The upper control limits for the proposed robust charts were calculated based on simulated data. The performance of each control chart is based on the false alarm and the probability of outlier's detection. In general, the performance of an alternative robust Hotelling's T² charts is better than the performance of the traditional Hotelling's T² chart. Copyright © 2012 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.     

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.     

Process Control for the Vector Autoregressive Model

Multivariate monitoring techniques for serially correlated observations have been widely used in various applications. This study examines several issues that have arisen in relation to the statistical quality control for the vector autoregressive (VAR) model, using a Monte Carlo approach. Different versions of the Hotelling T² statistic and control limits to monitor the VAR‐type process for both Phase I and Phase II schemes can be specified for different sample sizes and configurations of the model. Our simulation study suggests that the Hotelling's T² statistic can be tested against the χ² critical values during Phase I, but should be tested against scaled F critical values during Phase II. An unbiased covariance estimate of residuals is also recommended during Phase II when sample size is typically small. By reanalyzing some real data examples, the authors offer new conclusions. Copyright © 2012 John Wiley & Sons, Ltd.     

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.     

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.     

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.     

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.     

Monitoring nonlinear profiles using a wavelet-based distribution-free CUSUM chart

WDFTC is a wavelet-based distribution-free CUSUM chart for detecting shifts in the mean of a profile with noisy components. Exploiting a discrete wavelet transform (DWT) of the mean in-control profile, WDFTC selects a reduced-dimension vector of the associated DWT components from which the mean in-control profile can be approximated with minimal weighted relative reconstruction error. Based on randomly sampled Phase I (in-control) profiles, the covariance matrix of the corresponding reduced-dimension DWT vectors is estimated using a matrix-regularisation method; then the DWT vectors are aggregated (batched) so that the non-overlapping batch means of the reduced-dimension DWT vectors have manageable covariances. To monitor shifts in the mean profile during Phase II operation, WDFTC computes a Hotelling's T ²-type statistic from successive non-overlapping batch means and applies a CUSUM procedure to those statistics, where the associated control limits are evaluated analytically from the Phase I data. Experimentation with several normal and non-normal test processes revealed that WDFTC was competitive with existing profile-monitoring schemes.     

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.     

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.     

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.     

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.     

Multivariate SPC for recipe preservation of batch processes

The performances of the Hotelling's T² control chart and the squared prediction error control chart based on the multi‐way principal component analysis are evaluated for monitoring within batch process variation for the purpose of recipe preservation. A nonlinear model for simulated batch process data is provided. The model allows for cross correlation of error terms at a given time period and serial correlation of error terms across time periods. The performance characterizations of the two monitoring schemes are provided for a variety of levels of cross correlation and serial correlation. The impact of the time period at which process shifts occur is also investigated for the monitoring schemes. The T² control chart is recommended for the cases considered. Copyright © 2009 John Wiley & Sons, Ltd.     

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.     

Monitoring type B buyer–supplier relationships

The ability to monitor supplier performance is a critical capability for maintaining strong buyer–supplier relationships. Monitoring type B suppliers is especially challenging as they are not as clearly defined as either type A, with strong strategic partnerships, or type C, with little partnership. This research develops a non-parametric multivariate Hoteling’s T ² control chart to capture the in-control state of a dyadic relationship (Phase I), and show how it would be developed based on survey data of buyer–supplier relationship attributes. Modelling the satisfactory level of dyadic relationship performance is very useful for identifying when the relationship begins to move away from the desired state. We then use the designed control chart to monitor the relationship between dyads over time to determine if any unusual behaviour has occurred (Phase II) and illustrate its implementation through a case study from the auto industry. This research illustrates how supply chain managers can secure and improve their supply chain performance by monitoring and maintaining strong relationships with their partners. The proposed method extends the existing SPC tools to effectively manage Type B buyer–supplier relationships.     

Fault Diagnosis in Multivariate Control Charts Using Artificial Neural Networks

Most multivariate quality control procedures evaluate the in‐control or out‐of‐control condition based upon an overall statistic, like Hotelling's T². Although T² is optimal for finding a general shift in mean vectors, it is not optimal for shifts that occur for some subset of variables. This introduces a persistent problem in multivariate control charts, namely the interpretation of a signal that often discourages practitioners in applying them. In this paper, we propose an artificial neural network based model to diagnose faults in out‐of‐control conditions and to help identify aberrant variables when Shewhart‐type multivariate control charts based on Hotelling's T² are used. The results of the model implementation on two numerical examples and one case of real world data are encouraging. Copyright © 2005 John Wiley & Sons, Ltd.     

The effectiveness of Max-half-Mchart over Max-Mchart in simultaneously monitoring process mean and variability of individual observations

Control charts are widely used in industrial environments for the simultaneous or separate monitoring of the process mean and process variability. The Max-Mchart is a multivariate Shewhart-type simultaneous control chart that is used when monitoring subgroups. While this sampling design allows the computation of the generalized variance (GV) used to calculate the process variability, a GV chart cannot be plotted for individual observations. Hence, we cannot compute the single statistic in the Max-Mchart. This study aims to overcome the aforementioned issue. To this end, first, we develop a new Max-Mchart for individual observations by utilizing the statistic in the dispersion control chart. Second, instead of the standard normal distribution, we propose a new transformation using a half-normal distribution to calculate the statistic for the process mean and process variability. Thus, the proposed chart is called the Max-Half-Mchart, whose control limit is calculated using the bootstrap approach. An evaluation based on the average run length values shows the robustness of the Max-Half-Mchart for the simultaneous monitoring of the process mean and process variability. The single statistic in the Max-Half-Mchart is more consistent with the statistic in Hotelling's T² and the dispersion chart than that of the Max-Mchart.