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 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.     

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 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.     

The Effect of Autocorrelation on the Hotelling T2 Control Chart

One of the basic assumptions for traditional univariate and multivariate control charts is that the data are independent in time. For the latter, in many cases, the data are serially dependent (autocorrelated) and cross‐correlated because of, for example, frequent sampling and process dynamics. It is well known that the autocorrelation affects the false alarm rate and the shift‐detection ability of the traditional univariate control charts. However, how the false alarm rate and the shift‐detection ability of the Hotelling T² control chart are affected by various autocorrelation and cross‐correlation structures for different magnitudes of shifts in the process mean is not fully explored in the literature. In this article, the performance of the Hotelling T² control chart for different shift sizes and various autocorrelation and cross‐correlation structures are compared based on the average run length using simulated data. Three different approaches in constructing the Hotelling T² chart are studied for two different estimates of the covariance matrix: (i) ignoring the autocorrelation and using the raw data with theoretical upper control limits; (ii) ignoring the autocorrelation and using the raw data with adjusted control limits calculated through Monte Carlo simulations; and (iii) constructing the control chart for the residuals from a multivariate time series model fitted to the raw data. To limit the complexity, we use a first‐order vector autoregressive process and focus mainly on bivariate data. © 2014 The Authors. Quality and Reliability Engineering International Published by John Wiley & Sons Ltd.     

A robust multivariate Shewhart chart for contaminated normal environments

Lately, the multivariate setup of control charts, especially the memory-less chart has received less attention of researchers as compared to the univariate setup. However, the multivariate setup is of paramount importance in this big-data era. In this research work, we study the multivariate Shewhart chart for monitoring location parameter by examining the robustness of this scheme with the mean estimator. We also explored the scheme with some other robust parametric estimators in different process environments. The multivariate estimators such as median, midrange, tri-mean (TM), and Hodges–Lehmann (HL) estimators were examined under uncontaminated, location contaminated, variance contaminated, and both location–variance contaminated normal environments. Through a synthetic Monte Carlo simulation and application of the schemes on a real-life dataset, the findings suggest that the proposed estimators outperform the default estimator of the multivariate scheme (mean). The performance measures of comparing these estimators through the charts are the average run length, standard deviation run length, extra-quadratic loss, and relative average run length. The charts resulting from applying the schemes on real-life dataset recorded from glass manufacturing process also buttresses the simulation findings.     

Multivariate EWMA control charts using individual observations for process mean and variance monitoring and diagnosis

Most multivariate control charts in the literature are designed to detect either mean or variation shifts rather than both. A simultaneous use of the Hotelling T ² and |S| control charts has been proposed but the Hotelling T ² reacts to mean shifts, dispersion changes, and changes of correlations among responses. The combination of two multivariate control charts into one chart sometimes loses the ability to provide detailed diagnostic information when a process is out-of-control. In this research a new multivariate control chart procedure based on exponentially weighted moving average (EWMA) statistics is proposed to monitor process mean and variance simultaneously to identify proper sources of variations. Two multivariate EWMA control charts using individual observations are proposed to achieve a quick detection of mean or variance shifts or both. Simulation studies show that the proposed charts are capable of identifying appropriate types of shifts in terms of correct detection percentages. A manufacturing example is used to demonstrate how the proposed charts can be properly set-up based on average run length values via simulations. In addition, correct detection rates of the proposed charts are explored.     

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.     

Statistical process control for monitoring non-linear profiles using wavelet filtering and B-Spline approximation

A statistical process control framework is proposed to monitor non-linear profiles. The proposed methodology aims at identifying mean shifts or ‘shape changes’ in a profile. Discrete wavelet transformation (DWT) is applied to separate variation or noise from profile contours. B-splines are adopted to generate critical points to define the shape of a profile. The proposed method is innovative in that users can divide a profile into multiple segments to be monitored simultaneously. The high dimensionality problem that hinders the implementation of multivariate control charts can be solved by this framework. The distance difference statistic for each segment provides diagnostic information when the process of interest is out of control. These proposed statistics form a vector to be fed into any multivariate control chart such as the Hotelling T ² control chart. A decomposition method can also be applied on the T ² statistics when an out-of-control profile is detected. A simulation study applied to a forging process is conducted to demonstrate the property of the proposed method. The proposed method is capable of detecting profile shifts and identifying the exact location of problematic segments.     

A Multivariate T2 Control Chart for Skewed Populations Using Weighted Standard Deviations

This paper proposes a heuristic method of constructing multivariate T² control charts for skewed populations based on ‘weighted standard deviations’, obtained by decomposing the standard deviation into upper and lower deviations according to the direction and degree of skewness. The proposed method adjusts the variance–covariance matrix of quality characteristics and modifies the ellipsoidal probability density function contour of the multivariate normal distribution to a shape similar to that of the skewed distribution. False alarm rates and out‐of‐control average run lengths of the proposed control chart are compared with those of the standard control chart for multivariate lognormal, Weibull and gamma distributions, and the results show that considerable improvement over the standard method can be achieved when the distribution is skewed. Copyright © 2003 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.     

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.     

Monitoring bivariate and trivariate mean vectors with a Shewhart chart

In this article, we propose the use of the mean chart to control multivariate processes. The basic idea is to control the mean vector of bivariate (X, Y) and trivariate (X, Y, Z) processes by alternating the charting statistic of the Shewhart chart. If the mean of X observations was the charting statistic to obtain the current sample point, then the mean of Y observations will be the charting statistic to obtain the next sample point (for the trivariate case, the mean of Z observations will be the charting statistic to obtain the sample point subsequent to the next one). As a Shewhart chart, the signal is given anytime a sample point is plotted beyond the control limits, independent of the charting statistic in use. A fair comparison between the proposed chart and the Hotelling chart is based on an equal number of measurements per sample. The Shewhart chart with alternated charting statistic (ACS) always outperforms the Hotelling chart, except for specific types of disturbances in quality characteristics highly correlated (ρ = 0.7). The ACS chart is substantially easier to operate and faster than the Hotelling chart in signaling changes in the mean vector of bivariate and trivariate processes. Even with fewer measurements per sample, the trivariate ACS chart outperforms the Hotelling chart.     

High breakdown estimation methods for Phase I multivariate control charts

A goal of Phase I analysis of multivariate data is to identify multivariate outliers and step changes so that the Phase II estimated control limits are sufficiently accurate. High breakdown estimation methods based on the minimum volume ellipsoid (MVE) or the minimum covariance determinant (MCD) are well suited for detecting multivariate outliers in data. As a result of the inherent difficulties in their computation, many algorithms have been proposed to detect multivariate outliers. Due to their availability in standard software packages, we consider the subsampling algorithm to obtain the MVE estimators and the FAST‐MCD algorithm to obtain the MCD estimators. Previous studies have not clearly determined which of these two available estimation methods is best for control chart applications. The comprehensive simulation study presented in this paper gives guidance for the correct use of each estimator. Control limits are provided. High breakdown estimation methods based on the MCD and MVE approaches can be applied to a wide variety of multivariate quality control data. Copyright © 2006 John Wiley & Sons, Ltd.     

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.     

A clustering algorithm-based control chart for inhomogeneously distributed TFT-LCD processes

Statistical process control techniques have been widely used to improve processes by reducing variations and defects. In the present paper, we propose a multivariate control chart technique based on a clustering algorithm that can effectively handle a situation in which the distribution of in-control observations is inhomogeneous. A simulation study was conducted to examine the characteristics of the proposed control chart and to compare them with Hotelling’s T ² multivariate control charts that are widely used in real-world processes. Moreover, an experiment with real data from the thin film transistor liquid crystal display (TFT-LCD) manufacturing process demonstrated the effectiveness and accuracy of the proposed control chart.     

Robust inversion method for jointly estimating parameters and variance components from heterogeneous monitoring data

In this paper, a novel inversion method is proposed for jointly robust estimation of parameters and variance components from disjunctive groups of observations affected by outliers. This method, named robust non-negative variance component estimation (RVCE), is a coupling of variance component estimation (VCE) technique with a robust estimation method, developed to cope with outliers and to avoid negative variance, leading thus, to an estimation reliable enough. The principle of RVCE method is based on the refinement of the stochastic model via an equivalent weight matrix established from the original measurement weight matrix and an adapted full weight matrix with hard rejection to outliers. This last one is derived from the robust standardized residuals, using a highly robust estimator, as an initial solution of the inverse problem, and a cut-off value adapted to sort out the good observations from the bad ones. Furthermore, because the original weight matrix is partly known, the integration of the VCE technique plays a key role to reach an optimal solution and to provide valuable information on the precision of the estimates. The performance of the proposed method is demonstrated by considering a rockfill dam as an example, where the material parameters and variance components are jointly estimated from geotechnical and geodetic measurements. The results of comparison study between RVCE method with other methods such as the classical NN-VCE, RIMCO, least squares and the combined Huber’s M-estimator with VCE (HVCE) for various configuration options have highlighted the pertinence of the proposed method.     

Local polynomial regression smoothers with AR-error structure

Consider the fixed regression model with random observation error that follows an AR(1) correlation structure. In this paper, we study the nonparametric estimation of the regression function and its derivatives using a modified version of estimators obtained by weighted local polynomial fitting. The asymptotic properties of the proposed estimators are studied: expressions for the bias and the variance/covariance matrix of the estimators are obtained and the joint asymptotic normality is established. In a simulation study, a better behavior of the Mean Integrated Squared Error of the proposed regression estimator with respect to that of the classical local polynomial estimator is observed when the correlation of the observations is large. This work has been partially supported by grants PB98-0182-C02-01, PGIDT01PXI10505PR and MCyT Grant BFM2002-00265 (European FEDER support included).     

Online evaluation of the process noise covariance matrix for event-based state estimators

In state estimation, adjusting the process noise covariance matrix is an important and often difficult task. Well-known methods use the innovation vector to perform an adaptive adjustment, but when using event-based sensors, the innovation vector is not available for the estimator. We propose an online method for adjusting the process noise covariance matrix using the expected and observed event rates, which is based on the golden section search optimization algorithm. Simulation results confirm the suitability and efficiency of our proposed method. The process noise covariance parameter converges to the actual covariance iteratively, reducing the sensor transmission rate and the estimation error.