Statistical process monitoring with principal components

Most industrial processes are characterized by a system of several variables, all of which are subject to drifts, disturbances, and assignable causes of variation. In the chemical and process industries, there are often inertial forces arising from raw material streams, reactors and tanks that introduce serial correlation over time into these variables. This autocorrelation can have a profound impact on the effectiveness of the statistical monitoring methods used for such processes. This paper reviews some of the available methodology for multivariate process monitoring and shows the effectiveness of principal components in this context. An application of the principal components approach with correlated observation vectors is presented. The effectiveness of this procedure to indicate process upsets is discussed.     

A new adaptive CUSUM control chart for detecting the multivariate process mean

We propose a new multivariate CUSUM control chart, which is based on self adaption of its reference value according to the information from current process readings, to quickly detect the multivariate process mean shifts. By specifying the minimum magnitude of the process mean shift in terms of its non‐centrality parameter, our proposed control chart can achieve an overall performance for detecting a particular range of shifts. This adaptive feature of our method is based on two EWMA operators to estimate the current process mean level and make the detection at each step be approximately optimal. Moreover, we compare our chart with the conventional multivariate CUSUM chart. The advantages of our control chart detection for range shifts over the existing charts are greatly improved. The Markovian chain method, through which the average run length can be computed, is also presented. Copyright © 2010 John Wiley & Sons, Ltd.     

Optimum variable-dimension EWMA chart for multivariate statistical process control

The variable-dimension T² control chart (VDT² chart) was recently proposed for monitoring the mean of multivariate processes in which some of the quality variables are easy and inexpensive to measure while other variables require substantially more effort or expense. The chart requires most of the times that only the inexpensive variables be sampled, switching to sampling all the variables only when a warning is triggered. It has good ARL performance compared with the standard T² chart, while significantly reducing the sampling cost. However, like the T² chart, it has limited sensitivity to small and moderate mean shifts. To detect such shifts faster, we developed an exponentially weighted moving average (EWMA) version of the VDT² chart, along with Markov chain models for ARL calculation, and software (made available) for optimizing the chart design. The optimization software, which is based on genetic algorithms, runs in Windows^© and has a friendly user interface. The performance analysis shows the great gain in performance achieved by the incorporation of the EWMA procedure.     

A multivariate CUSUM control chart for monitoring Gumbel's bivariate exponential data

Exponentially distributed data are commonly encountered in high-quality processes. Control charts dedicated to the univariate exponential distribution have been extensively studied by many researchers. In this paper, we investigate a multivariate cumulative sum (MCUSUM) control chart for monitoring Gumbel's bivariate exponential (GBE) data. Some tables are provided to determine the optimal design parameters of the proposed MCUSUM GBE chart. Furthermore, both zero-state and steady-state properties of the proposed MCUSUM GBE chart for the raw and the transformed GBE data are compared with the multivariate exponentially weighted moving average (MEWMA) chart and the paired individual cumulative sum (CUSUM) chart. The results show that the proposed MCUSUM GBE chart outperforms the other two types of control charts for most shift domains. In addition, an extension to Gumbel's multivariate exponential (GME) distribution is also investigated. Finally, an illustrative example is provided in order to explain how the proposed MCUSUM GBE chart can be implemented in practice.     

Monitoring therapeutic processes using risk-adjusted multivariate Tukey's CUSUM control chart

Using control charts for monitoring therapeutic processes has become popular lately. As the application of traditional control charts in the therapeutic processes may be misleading due to the inherent differences between patients, a multifactor correlated risk measure is considered in monitoring of these processes. Therefore, using risk-adjusted control charts for monitoring the therapeutic processes is of interest to practitioners. Furthermore, in health care monitoring, statistical models should account for abnormal distributions and outlier data to minimize misinterpretations of monitoring schemes. This study proposes a risk-adjusted multivariate Tukey's cumulative sum (RA-MTCUSUM) control chart. The proposed method is a combination of the accelerated failure time (AFT) regression model, the Tukey's control chart (TCC) featuring robustness against abnormality, and the multivariate cumulative sum (MCUSUM) control chart for monitoring multivariable process. Simulation experiments are performed to evaluate the performance of the proposed control chart using the average run length (ARL) measure. Results show that the RA-MTCUSUM control chart has better performance in comparison with traditional ones for monitoring various distributions (normal and non-normal). Based on the simulation results, outlier data do not disturb the proposed control chart's performance. Moreover, applying the RA-MTCUSUM control chart to a real-world dataset related to sepsis patients of a hospital located in Tehran, Iran indicates that the control chart has more reasonable performance than the traditional control charts in the real applications due to its robustness.     

On the multivariate progressive control chart for effective monitoring of covariance matrix

With the development of modern acquisition techniques, data with several correlated quality characteristics are increasingly accessible. Thus, multivariate control charts can be employed to detect changes in the process. This study proposes two multivariate control charts for monitoring process variability (MPVC) using a progressive approach. First, when the process parameters are known, the performance of the MPVC charts is compared with some multivariate dispersion schemes. The results showed that the proposed MPVC charts outperform their counterparts irrespective of the shifts in the process dispersion. The effects of the Phase I estimated covariance matrix on the efficiency of the MPVC charts were also evaluated. The performances of the proposed methods and their counterparts are evaluated by calculating some useful run length properties. An application of the proposed chart is also considered for the monitoring of a carbon fiber tubing process.     

Self-starting control chart for simultaneously monitoring process mean and variance

A self-starting control chart, based on the likelihood ratio test and the exponentially weighted moving average procedure, is proposed for monitoring the process mean and variance simultaneously when the process parameters are unknown. A table is presented to assist in the design of the control chart with different parameters. Its in-control average run length can be evaluated by a two-dimensional Markov chain model. Moreover, the diagnostic aids of the proposed chart are given. Monte Carlo simulation results compared with some competing methods in the literature show that the proposed approach has quite satisfactory charting performance across a range of possible shifts when the process parameters are unknown, even including the detection of a decrease in variability. A real data example from industrial manufacturing is used to demonstrate its implementation.     

Statistically constrained economic design of the multivariate exponentially weighted moving average control chart

This paper shows that economic statistical design can provide for better statistical properties without significantly increasing optimal total costs. Cost comparisons between optimal economic statistical designs and optimal economic designs show no significant cost increases. The average run length (ARL) constraints added by economic statistical design significantly improve the statistical properties of the control chart scheme. False alarm frequency is limited while keeping good shift detection characteristics. In addition, the Multivariate Exponentially Weighted Moving Average (MEWMA) control schemes performed better from the cost standpoint than the benchmark pure statistical design—the Hotelling T² control chart. This improvement held for unconstrained and constrained designs. Finally, cost comparisons at small values of n showed significant advantage for the MEWMA schemes. Copyright © 2001 John Wiley & Sons, Ltd.     

Statistical monitoring and control of a low defect process

Advanced technologies today are such that it is possible to keep the occurrence of defects in manufactured products at very low levels. The use of the conventional c-chart for statistical control of defects in such products would encounter serious practical difficulties because the low defect counts would render invalid the theoretical assumptions used in the construction of the chart. Based on reasoning with fundamental probability distributions, this paper offers a simple and reliable solution that is particularly suited to on-line inspection and testing operations such as those found in an automated manufacturing environment.     

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 variable parameters multivariate control chart for simultaneous monitoring of the process mean and variability with measurement errors

Evaluating the effect of measurement errors on either adaptive or simultaneous control charts has been a topic of interest for the researchers in the recent years. Nevertheless, the effect of measurement errors on both adaptive and simultaneous monitoring control charts has not been considered yet. In this paper, through extensive numerical studies, we evaluate the effect of measurement errors on an adaptive (variable parameters) simultaneous multivariate control chart for the mean vector and the variance-covariance matrix of p quality characteristics assumed to follow a multivariate normal distribution. In order to do so, (a) we use eight performance measures computed using a Markov chain model, (b) we consider the effects of multiple measurements as well as the error model's parameters, and (c) we also consider the overall performance of this adaptive simultaneous chart including the chart parameters values optimization, which have never been considered so far for this scheme. At last, a real case is presented in order to illustrate the proposed scheme.     

A control chart for monitoring the lognormal process variation using repetitive sampling

Control charts are developed to make the specific quality measures for a successful production process and follow normal distribution behaviors. But some real-life practices do not match such practices and exhibit some positively skewed behavior like lognormal distribution. The present study has considered this situation and proposed a monitoring control chart based on lognormal process variation using a repetitive sampling scheme. This concept proved better for detecting shifts as quickly as possible, and compared with the existing concept, results are elaborated through extensive tables. The average run lengths and standard deviations of the run lengths are being used as a performance evaluation measures and computed by using Monte Carlo simulations performed in R language. A real-life situation has been discussed in the example section to strengthen the proposed control chart concept in a real-life situation.     

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.     

An EWMA chart for monitoring the covariance matrix of a multivariate process based on dissimilarity index

In this article, we propose an exponentially weighted moving average (EWMA) control chart for monitoring the covariance matrix of a multivariate process based on the dissimilarity index of 2 matrices. The proposed control chart essentially monitors the covariance matrix by comparing the individual eigenvalues of the estimated EWMA covariance matrix with those of the estimated covariance matrix from the in‐control (IC) phase I data. It is different from the conventional EWMA charts for monitoring the covariance matrix, which are either based on comparing the sum or product or both of the eigenvalues of the estimated EWMA covariance matrix with those of the IC covariance matrix. We compare the performance of the proposed chart with that of the best existing chart under the multivariate normal process. Furthermore, to prevent the control limit of the proposed EWMA chart developed using the limited IC phase I data from having extensively excessive false alarms, we use a bootstrap resampling method to adjust the control limit to guarantee that the proposed chart has the actual IC ARL(average run length) not less than the nominal level with a certain probability. Finally, we use an example to demonstrate the applicability and implementation of the proposed EWMA chart.     

A multivariate EWMA control chart for monitoring process variability with individual observations

A control chart is proposed to effectively monitor changes in the population variance-covariance matrix of a multivariate normal process when individual observations are collected. The proposed control chart is constructed based on first taking the exponentially weighted moving average of the product of each observation and its transpose. Appropriate statistics which are based on square distances between estimators and true parameters are then developed to detect changes in the variances and covariances of the variance-covariance matrix. The simulation studies show that the proposed control chart outperforms existing procedures in cases where either the variances or correlations increase or both increase. The improvement in performance of the proposed control chart is particularly notable when variables are strongly positively correlated. The proposed control chart is applied to a real-life example taken from the semiconductor industry.     

Cycle-based signal monitoring using a directionally variant multivariate control chart system

Cycle-based signals are generally obtained through the automatic sensing of critical process variables during each repetitive operation cycle of a manufacturing process, and they thus contain a significant amount of information about the process condition. Increasing attention has been paid recently to the problem of effectively monitoring these signals as an aid to the detection of process changes. In general, either based on process engineering knowledge or on historical data analysis, it is possible to obtain process faults and the corresponding signal patterns (the direction and magnitude of a mean shift). In order to fully utilize such fault pattern information in process monitoring, this paper proposes a directionally variant control chart obtained through the effective combination of a multivariate χ² chart and a univariate projection chart. It is shown that the addition of the univariate projection chart can improve the detection power for pre-known process faults, however, this may be at the cost of a deterioration in the detection power for unknown faults. A detailed quantitative analysis is provided to justify the application conditions of the proposed chart. A case study of cycle-based tonnage monitoring of a forging process is presented to illustrate the design procedures and the effectiveness of the proposed control chart system.     

A proposed variable parameter control chart for monitoring the multivariate coefficient of variation

An efficient process monitoring system is important for achieving sustainable manufacturing. The control charting technique is one of the most effective techniques to monitor process quality. In certain processes where the process mean and variance are not independent of one another, the coefficient of variation (CV), which measures the ratio of the standard deviation to the mean, should be monitored. In line with industrial settings, where at least two or more variables are monitored simultaneously in most processes, this paper proposes a variable parameter (VP) chart to monitor the multivariate CV (MCV). Formulae and algorithms to optimize the various performance measures are discussed. The proposed VP MCV chart is designed based on a Markov chain approach. The performance comparison shows that the proposed VP MCV chart prevails over the existing MCV charts, in terms of the average time to signal (ATS), standard deviation of the time of signal (SDTS), and expected average time to signal (EATS) criteria. An example is presented to illustrate the implementation of the proposed VP MCV chart.     

Statistical monitoring of over-dispersed multivariate count data using approximate likelihood ratio tests

In this paper, we develop a statistical monitoring scheme for multivariate count data. In many applications involving multivariate count data, individual variables are not only correlated to each other, but also over-dispersed. Traditional statistical monitoring methods for multivariate count data that assume simple statistical models fail to fit the data collected when the underlying process is under normal working state, also referred to as the in-control state. Therefore, we propose a monitoring scheme which is based on the Poisson–multivariate Gaussian mixed model. Although such models are quite flexible, efficient statistical monitoring schemes for such models have not been developed. In this paper, we develop likelihood ratio test-based monitoring schemes that are shown to be superior to standard multivariate statistical monitoring schemes. The key challenge in developing likelihood ratio test for the Poisson–multivariate Gaussian mixed models is that the likelihood function can only be calculated by multidimensional numerical integration. We tackle this issue using an approximation of this complex likelihood function.     

A new Bayesian multivariate exponentially weighted moving average control chart for phase II monitoring of multivariate multiple linear profiles

Recently, statistical profile monitoring methods have become efficient tools for monitoring the quality of a product (or a production process) using control charts. The key idea is to describe the relationship between a response variable and a set of explanatory variables in the form of a statistical regression model, which called profile. Traditionally, those control charts are constructed with standard “frequentistic” regression models. Recently, it has been proposed to apply Bayesian regression models instead, and it has been empirically demonstrated that Bayesian regression models have the potential to perform significantly better. In this paper, we introduce a novel Bayesian multivariate exponentially weighted moving average control chart for monitoring multivariate multiple linear profiles in phase II. The key idea is to use the data from historical data sets to generate informative prior distributions for the regression models in phase II. The results of our empirical simulation studies show that the Bayesian multivariate multiple linear regression model is superior to its classical “frequentistic” counterpart in terms of the average run length. Our empirical findings are in agreement with findings reported in recently published articles. To shed more light onto the merit of the proposed Bayesian method, we carry out a sensitivity analysis, in which we investigate how the amount of phase I data influences the results. We also demonstrate the applicability and superiority of the proposed Bayesian method by a real‐world application.