The T2 chart with mixed samples to control bivariate autocorrelated processes

In this paper, we propose the use of the T² chart with the mixed sampling strategy (MS) to monitor the mean vector of bivariate processes with observations that fit to a first-order vector autoregressive model. With the MS, rational subgroups of size n are taken from the process and the selected units are regrouped to form the mixed samples. The units of the mixed samples are units selected from the last two rational subgroups. The aim of the proposed sampling strategy is to reduce the negative effect of the autocorrelation on the performance of the T² chart. When the two variables are autocorrelated, the MS always enhances the T² chart performance, however, the mixed samples are not recommended for bivariate processes with only one autocorrelated variable which is rarely affected by the assignable cause.     

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.     

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.     

Simultaneous Univariate X_bar Charts to Control Bivariate Processes with Autocorrelated Data

In this article, we consider the simultaneous univariate X_bar charts (SU X_bar charts) for samples of n bivariate observation vectors that are not only cross‐correlated but also autocorrelated. The cross‐covariance matrix of the sample mean vectors was derived with the assumption that the observations are described by a first‐order vector autoregressive model. The combined effect of the cross‐correlation and autocorrelation on the performance of the SU X_bar charts is investigated. Depending on the autocorrelations and the nature of the disturbance, affecting only one or both variables, the SU _bar charts perform better with samples of size one than with samples of size two; in same cases even better than with samples of size four. When the two variables are affected by the assignable cause, the simultaneous charts tend to perform better than the T² chart as the autocorrelation and cross‐correlation increase. Copyright © 2014 John Wiley & Sons, Ltd.     

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.     

On autoregressive model selection for the exponentially weighted moving average control chart of residuals in monitoring the mean of autocorrelated processes

With the development of automation technologies, data can be collected in a high frequency, easily causing autocorrelation phenomena. Control charts of residuals have been used as a good way to monitor autocorrelated processes. The residuals have been often computed based on autoregressive (AR) models whose building needs much experience. Data have been assumed to be first-order autocorrelated, and first-order autoregressive (AR(1) ) models have been employed to obtain residuals. But for a p th-order autocorrelated process, how the AR(1) model affects the performance of the control chart of residuals remains unknown. In this paper, the control chart of exponentially weighted moving average of residuals (EWMA-R) is used to monitor the p th-order autocorrelated process. Taking the mean and standard deviation of run length as performance indicators, two types of EWMA-R control charts, with their residuals obtained from the p th-order autoregressive AR(p) and AR(1) models, respectively, are compared. The results of the numerical experiment show that for detecting small mean shifts, EWMA-R control charts based on AR(1) models outperform ones based on AR(p) models, whereas for detecting large shifts, they are sometimes slightly worse. A practical application is used to give a recommendation that a large number of samples are necessary for determining an EWMA-R control chart before using it.     

A Bivariate Semiparametric Control Chart Based on Order Statistics

In this article, a new bivariate semiparametric Shewhart‐type control chart is presented. The proposed chart is based on the bivariate statistic (X_(r), Y_(s)), where X_(r) and Y_(s) are the order statistics of the respective X and Y test samples. It is created by considering a straightforward generalization of the well‐known univariate median control chart and can be easily applied because it calls for the computation of two single order statistics. The false alarm rate and the in‐control run length are not affected by the marginal distributions of the monitored characteristics. However, its performance is typically affected by the dependence structure of the bivariate observations under study; therefore, the suggested chart may be characterized as a semiparametric control chart. An explicit expression for the operating characteristic function of the new control chart is obtained. Moreover, exact formulae are provided for the calculation of the alarm rate given that the characteristics under study follow specific bivariate distributions. In addition, tables and graphs are given for the implementation of the chart for some typical average run length values and false alarm rates. The performance of the suggested chart is compared with that of the traditional χ² chart as well as to the nonparametric SN² and SR² charts that are based on the multivariate form of the sign test and the Wilcoxon signed‐rank test, respectively. Finally, in order to demonstrate the applicability of our chart, a case study regarding a real‐world problem related to winery production is presented. Copyright © 2016 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.     

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.     

M‐ATTRIVAR: An attribute‐variable chart to monitor multivariate process means

In this paper, an attribute‐variable control chart, namely, M‐ATTRIVAR, is introduced to monitor possible shifts in a vector of means. The monitoring starts using an attribute chart (classifying the units as approved or not using a gauge) and continues in such a way until a warning signal is given, shifting the control to a variable chart for the next sampling. If the variable chart does not confirm the warning, the monitoring returns to an attribute control. Otherwise, the monitoring remains with the variable chart. Whenever any of the charts (attribute or variable) signals an alarm, the control scheme triggers an alarm. The main advantage of this new proposal is the possibility of judging the state of the process only by the attribute chart most of the time (normally more economical and faster). The performance of the M‐ATTRIVAR control chart is compared versus the main competitor (T² control chart) in terms of performance detection (out‐of‐control average run length) but also economically (average sampling cost). The M‐ATTRIVAR is always cheaper than T², and in many scenarios, it detects quicker process shifts than the T² control chart. A numerical example illustrates a practical situation.     

A New Synthetic Exponentially Weighted Moving Average Control Chart for Monitoring Process Dispersion

Exponentially weighted moving average (EWMA) control charts have been widely recognized as a potentially powerful process monitoring tool of the statistical process control because of their excellent speed in detecting small to moderate shifts in the process parameters. Recently, new EWMA and synthetic control charts have been proposed based on the best linear unbiased estimator of the scale parameter using ordered ranked set sampling (ORSS) scheme, named EWMA‐ORSS and synthetic‐ORSS charts, respectively. In this paper, we extend the work and propose a new synthetic EWMA (SynEWMA) control chart for monitoring the process dispersion using ORSS, named SynEWMA‐ORSS chart. The SynEWMA‐ORSS chart is an integration of the EWMA‐ORSS chart and the conforming run length chart. Extensive Monte Carlo simulations are used to estimate the run length performances of the proposed control chart. A comprehensive comparison of the run length performances of the proposed and the existing powerful control charts reveals that the SynEWMA‐ORSS chart outperforms the synthetic‐R, synthetic‐S, synthetic‐D, synthetic‐ORSS, CUSUM‐R, CUSUM‐S, CUSUM‐ln S², EWMA‐ln S² and EWMA‐ORSS charts when detecting small shifts in the process dispersion. A similar trend is observed when the proposed control chart is constructed under imperfect rankings. An application to a real data is also provided to demonstrate the implementation and application of the proposed control chart. Copyright © 2014 John Wiley & Sons, Ltd.     

One‐sided runs‐rules schemes to monitor autocorrelated time series data using a first‐order autoregressive model with skip sampling strategies

In some industrial or health‐related processes, it makes more practical sense to monitor either an increase only or a decrease only in the quality characteristic of interest. Consequently, in this paper, we propose four one‐sided Shewhart charts supplemented with runs‐rules to monitor the mean of autocorrelated normally distributed samples using a stationary first‐order autoregressive model. To counteract the negative effect of autocorrelation, we implement a sampling strategy which involves sampling of non‐neighboring observations to form rational subgroups. The Markov chain technique is used to derive zero‐state and steady‐state closed‐form expressions of the specific shift performance metric, ie, average run‐length (ARL). Moreover, we compute the expected ARL metric which evaluates each monitoring scheme based on all specified range of possible values of the shift parameter, or more specifically, from a global point of view and thus gives a different perspective from the specific shift ARL metric. We observed that the steady‐state improved w‐of‐w and the improved 2‐of‐(H + 1) schemes yield a better overall performance than their corresponding basic counterparts for all different levels of autocorrelation. A real‐life example is provided to illustrate the implementation of the monitoring schemes proposed here.     

Using FIR to Improve CUSUM Charts for Monitoring Process Dispersion

Statistical process control deals with monitoring process to detect disturbances in the process. These disturbances may be from the process mean or variance. In this study, we propose some charts that are efficient for detecting early shifts in dispersion parameter, by applying the Fast Initial Response feature. Performance measures such as average run length, standard deviation of the run length, extra quadratic loss, relative average run length, and performance comparison index are used to compare the proposed charts with their existing counterparts, including the Shewhart R chart and the Shewhart S chart with and without warning lines. Others include the CUSUM R chart, the CUSUM S chart, the EWMA of ln S², the CUSUM of ln S², the P_σ CUSUM, the χ CUSUM, and the Change Point (CP) CUSUM charts. The proposed charts do not only detect early shifts in the process dispersion faster, but also have better overall performance than their existing counterparts. Copyright © 2016 John Wiley & Sons, Ltd.     

A Synthetic Control Chart for Monitoring Process Variability

The control chart based on Downton's estimator (D chart) has recently been introduced in the literature for monitoring the process variability. The D chart is found to be equally efficient to the S chart in terms of detecting shifts in process variability. In this paper, salient features of D chart and the conforming run length chart are combined to produce synthetic D chart. The average run length performance of the synthetic D chart is investigated using simulation study and is compared with the originally proposed D chart and some other procedures proposed in the literature. It is found that it has an improved performance in comparison with the traditional control charts for process variability. Copyright © 2013 John Wiley & Sons, Ltd.     

The Design of the ARL‐Unbiased S2 Chart When the In‐Control Variance Is Estimated

The S² chart has been known as a powerful tool to monitor the variability of the normal process. When the variance of the process is unknown, it needs to be estimated by Phase I samples. It is well known that there are serious effects of parameter estimation on the performance of the S² chart based on known parameter assumption. If the effects of parameter estimation are not considered, it can lead to an increase in the number of false alarms and a reduction in the ability of the chart to detect process changes except for very small shifts in the variance. Based on the criterion of average run length (ARL) unbiased, a S² control chart is developed when the in‐control variance is estimated. The performance of the proposed control chart is also evaluated in terms of the ARL and standard deviation of the run length. Finally, an example is used to illustrate the proposed control chart. Copyright © 2013 John Wiley & Sons, Ltd.     

Toward identifying the source of mean shifts in multivariate SPC: a neural network approach

To identify the source(s) of process shifts under a multivariate setting is a challenging problem. Though some statistical techniques have been proposed, they are limited or restricted in their level of success and ease of use. In this paper, we propose a neural-network based identifier (NNI) to detect process mean shifts as well as indicate the variable(s) responsible for the shifts in a process where variables are correlated. Various network configurations and training strategies were investigated to develop an effective network. This research demonstrates how the NNI with a simple network structure, i.e. without any hidden layers, can perform superiorly to the Hotelling T ² chart and comparably to the MEWMA chart in detecting small to moderate shifts for bivariate processes. The run length analysis also indicates that the NNI performs much more stably than the Hotelling T ² chart and the MEWMA chart. One of the great advantages of this approach is that the proposed identifier, aided with the NNI output chart, can indicate the source(s) of the shift(s), i.e. the variable(s) responsible for the shift(s). The NNI output chart allows this monitoring scheme to easily interpret the underlying structures of the process variables.     

Nonparametric Synthetic Control Charts for Process Variation

In this article, we propose nonparametric synthetic and side‐sensitive synthetic control charts for controlling fraction nonconforming due to increase in the process variation. Synthetic control chart is a combination of sign and conforming run length control charts. We compare performance of the proposed control charts with the Shewhart sign and S² charts. Our performance study shows that the proposed control charts have a higher power of detecting out‐of‐control signal. We also study the steady‐state behavior of a nonparametric synthetic control chart. We present a Markov chain model to evaluate the steady‐state average run length of the synthetic and side‐sensitive synthetic control charts. Copyright © 2011 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.     

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.     

Efficient bivariate EWMA charts for monitoring process dispersion

To ensure high quality standards of a process, the application of control charts to monitor process performance has become a regular routine. Multivariate charts are a preferred choice in the presence of more than one process variable. In this article, we proposed a set of bivariate exponentially weighted moving average (EWMA) charts for monitoring the process dispersion. These charts are formulated based on a variety of dispersion statistics considering normal and non-normal bivariate parent distributions. The performance of the different bivariate EWMA dispersion charts is evaluated and compared using the average run length and extra quadratic loss criteria. For the bivariate normal process, the comparisons revealed that the EWMA chart based on the maximum standard deviation (SMAX_E) was the most efficient chart when the shift occurred in one quality variable. It also performed well when the sample size is small and the shift occurred in both quality variables. The EWMA chart based on the maximum average absolute deviation from median (MDMAX_E) performed better than the other charts in most situations when the shift occurred in the covariance matrix for the bivariate non-normal processes. An illustrative example is also presented to show the working of the charts.