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.     

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.     

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.     

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.     

An adaptive step-down procedure for fault variable identification

In a process with a large number of process variables (high-dimensional process), identifying which variables cause an out-of-control signal is a challenging issue for quality engineers. In this paper, we propose an adaptive step-down procedure using conditional T² statistic for fault variable identification. While existing procedures focus on selecting variables that have strong evidence of a change, the proposed step-down procedure selects a variable having the weakest evidence of a change at each step based on the variables that are selected in previous steps. The information of selected unchanged variables is effectively utilised in obtaining a powerful conditional T² test statistic for identifying the changed elements of the mean vector. The proposed procedure is designed to utilise the correlation information between fault and non-fault variables for the efficient fault variables identification. Further, the simulation results show that the proposed procedure has the better diagnostic performance compared with existing methods in terms of fault variable identification and computational complexity, especially when the number of the variables is high and the number of fault variables is small.     

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.     

Multivariate multinomial T2 control chart using fuzzy approach

Quality of a product is often measured through various quality characteristics generally correlated. Multivariate control charts are a response to the need for quality control in such situations. If quality characteristics are qualitative, it sometimes happens that the product quality is defined by linguistic variables – where quality levels are represented by some specific words – and product units are classified into several linguistic forms categories, depending on the degree of fulfilment of expectations, creating a situation of fuzzy classifications. This study first reviews the concepts found in the literature on the development of fuzzy multivariate control charts. We propose a method to control these fuzzy quality evaluations, with correlated multiple attributes quality characteristics, through the use of a Hotelling T² control chart.     

Multivariate charting techniques: a review and a line‐column approach

The primary objective of multivariate statistical process control is to monitor the related process quality characteristics over time and identify the assignable causes affecting the process using multivariate control charts. When an out‐of‐control signal is obtained from the chart, it is imperative to be able to detect the component variables that have gone out‐of‐control. In this paper we propose a new charting procedure for T², multivariate exponentially weighted moving average and multivariate cumulative sum control charts. The proposed charts will facilitate in identification of the source of out‐of‐control signal and are simple, economical and easier to implement. Copyright © 2009 John Wiley & Sons, Ltd.     

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

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

Univariate X¯ control charts for individual characteristics in a multinormal model

The early work on multivariate statistical process control was built upon Hotelling's T ² control chart which was developed to simultaneously monitor the means of correlated quality variables. This chart, however, has a drawback, namely, the problem of identifying the responsible variable(s) when an out-of-control signal occurs. One alternative is to use a separate X¯ control chart for each individual characteristic with equal risks, based on Bonferroni inequality. In this study, we show that, from an economic perspective, it may be desirable to have unequal type I risks for the individual charts, because of different inspection and restoration costs associated with each variable. We obtain their risk ratios, which are measures of relative importance of the variables monitored. Then, based on these risk ratios, we develop computer algorithms for finding the exact control limits for individual variables from a multinormal distribution, in the sense that the overall type I risk of the charts is equal to the desired value. Numerical studies show that the proposed methods give optimal or near-optimal results from an economic as well as statistical point of view.     

Economic statistical design of a T2 control chart with double warning lines

Recent studies have shown that enhancing the common T² control chart by using variable sample sizes (VSS) and variable sample intervals (VSI) sampling policies with a double warning line scheme (DWL) yields improvements in shift detection times over either pure VSI or VSS schemes in detecting almost all shifts in the process mean. In this paper, we look at this problem from an economical perspective, certainly at least as an important criterion as shift detection time if one considers what occurs in the industry today. Our method is to first construct a cost model to find the economic statistical design (ESD) of the DWL T² control chart using the general model of Lorenzen and Vance (Technometrics 1986; 28 :3–11). Subsequently, we find the values of the chart parameters which minimize the cost model using a genetic algorithm optimization method. Cost comparisons of Fixed ratio sampling, VSI, VSS, VSIVSS with DWL, and multivariate exponentially weighted moving average (MEWMA) charts are made, which indicate the economic efficacy of using either VSIVSS with DWL or MEWMA charts in practice if cost minimization is of interest to the control chart user. Copyright © 2010 John Wiley & Sons, Ltd.     

A variable‐selection control chart via penalized likelihood and Gaussian mixture model for multimodal and high‐dimensional processes

With the development of the sensor network and manufacturing technology, multivariate processes face a new challenge of high‐dimensional data. However, traditional statistical methods based on small‐ or medium‐sized samples such as T² monitoring statistics may not be suitable because of the “curse of dimensionality” problem. To overcome this shortcoming, some control charts based on the variable‐selection (VS) algorithms using penalized likelihood have been suggested for process monitoring and fault diagnosis. Although there has been much effort to improve VS‐based control charts, there is usually a common distributional assumption that in‐control observations should follow a single multivariate Gaussian distribution. However, in current manufacturing processes, processes can have multimodal properties. To handle the high‐dimensionality and multimodality, in this study, a VS‐based control chart with a Gaussian mixture model (GMM) is proposed. We extend the VS‐based control chart framework to the process with multimodal distributions, so that the high‐dimensionality and multimodal information in the process can be better considered.     

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.     

Diagnosing the variance shifts signal in multivariate process control using ensemble classifiers

Statistical process control charts have been successfully used to monitor process stability in various industries. The need to simultaneously monitor two or more quality characteristics has led to the prevalent adoption of multivariate control charts. However, out-of-control signals in multivariate control charts may be caused by one or more variables, or a set of variables. Therefore, effective quality control requires not only the rapid detection of process fluctuations, but also the correct identification of the variable(s) responsible for those changes. This study approaches the diagnosis of out-of-control signals as a classification task and proposes a support vector machine (SVM)-based ensemble classification model focused on variance shifts in multivariate processes. We address the issues of data diversity and ensemble method in constructing an ensemble model. Simulation results demonstrate the effectiveness of the proposed ensemble classification model in identifying the source of variance change. The proposed method clearly outperforms single classifiers as well as other comparable models including bagging and boosting. The results also reveal that the use of extracted features as input vectors for SVM provides better classification performance than the use of raw data. The proposed SVM-based ensemble classification system provides a reliable tool for the interpretation of out-of-control signals in multivariate process control.     

Adaptive multivariate double sampling and variable sampling interval Hotelling's T2 charts

In this article, two adaptive multivariate charts, which combine the double sampling (DS) and variable sampling interval (VSI) features, called the adaptive multivariate double sampling variable sampling interval T² (AMDSVSI T²) and the adaptive multivariate double sampling variable sampling interval combined T² (AMDSVSIC T²) charts, are proposed. The real purpose of using the proposed charts is to provide flexibility by enabling the sampling interval length of the DS T² chart to be varied so that the chart's sensitivity can be enhanced. The fundamental difference between the two proposed charts is that when a second sample is taken, the AMDSVSI T² chart uses the information of the combined sample mean vectors while the AMDSVSIC T² chart uses the information of the combined T² statistics, in deciding about the process status. This research is motivated by existing combined DS and VSI charts in the literature, which show convincing performance improvement over the standard DS chart. Consequently, it is believed that adopting this existing approach in the multivariate case will enable superior multivariate DS charts to be proposed. Numerical results show that the proposed charts outperform the existing standard T² and other adaptive multivariate charts, in detecting shifts in the mean vector, for the zero‐state and steady‐state cases. The performances of both charts when the shift sizes in the mean vector are unknown are also measured. The application of the AMDSVSI T² chart is illustrated with an example.     

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 variable sample size variable dimension T2 control chart

In this paper, we describe the development of the variable dimension and variable sample size T² control chart (VSSVDT²), which is an enhancement of the variable dimension T² chart (VDT²). In the VDT² control chart, the number of variables that are measured to compute the T² statistic is made variable. Some of the variables are easy or inexpensive to measure and are always monitored. The variables that are more difficult or expensive to measure are measured only when the T² value from the previous sample exceeds a specified threshold. The VDT² control chart performs well for moderate and large shifts in the mean vector. However, its performance for small shifts is poor. To improve the chart’s performance in detecting such shifts, we propose the application of the variable sample size technique to the VDT² control chart, resulting in the VSSVDT² control chart. To promote the use of the VSSVDT² chart, a user-friendly software has been developed, which the final user can use to find the best parameters of the VSSVDT² chart for a particular process.     

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.     

A Phase II depth-based variable dimension EWMA control chart for monitoring process mean

Statistical process control consists of tools and techniques that are useful for improving a process or ensuring that a process is in a stable and satisfactory state. In many modern industrial applications, it is critically important to simultaneously monitor two or more correlated process quality variables, thus necessitating the development of multivariate statistical process control (MSPC) as an important area of research for the new century. Nevertheless, the existing MSPC research is mostly based on the assumption that the process data follow a multinormal distribution or a known distribution. However, it is well recognized that in many applications the underlying process distribution is unknown. In practice, among a set of correlated variables to be monitored, there is oftentimes a subset of variables that are easy and/or inexpensive to measure, whereas the remaining variables are difficult and/or expensive to measure but contain information that may help more quickly detect a shift in the process mean. We are motivated to develop a Phase II control chart to monitor variable dimension (VD) mean vector for unknown multivariate processes. The proposed chart is based on the exponentially weighted moving average (EWMA) of a depth-based statistic. The proposed chart is shown to lead to faster detection of mean shifts than the existing VDT² and VD EWMAT² charts studied in Aparisi et al. and Epprecht et al., respectively.     

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.