Multivariate CUSUM and EWMA Control Charts for Skewed Populations Using Weighted Standard Deviations

This article proposes a heuristic method of constructing multivariate cumulative sum and exponentially weighted moving average control charts for skewed populations based on the weighted standard deviation method which adjusts the variance–covariance matrix of quality characteristics and approximates the probability density function using several multivariate normal distributions. These control charts, however, reduce to the conventional control charts when the underlying distribution is symmetric. In-control and out-of-control average run lengths of the proposed control charts are compared with those of the conventional control charts for multivariate lognormal and Weibull distributions. Simulation results show that considerable improvements over the standard method can be achieved when the underlying distribution is skewed.     

A mixed GWMA–CUSUM control chart for monitoring the process mean

The memory-type control charts are widely used in the process and service industries for monitoring the production processes. The reason is their sensitivity to quickly react against the small process disturbances. Recently, a new cumulative sum (CUSUM) chart has been proposed that uses the exponentially weighted moving average (EWMA) statistic, called the EWMA–CUSUM chart. Similarly, in order to further enhance the sensitivity of the EWMA–CUSUM chart, we propose a new CUSUM chart using the generally weighted moving average (GWMA) statistic, called the GWMA–CUSUM chart, for efficiently monitoring the process mean. The GWMA–CUSUM chart encompasses the existing CUSUM and EWMA–CUSUM charts. Extensive Monte Carlo simulations are used to explore the run length profiles of the GWMA–CUSUM chart. Based on comprehensive run length comparisons, it turns out that the GWMA–CUSUM chart performs substantially better than the CUSUM, EWMA, GWMA, and EWMA–CUSUM charts when detecting small shifts in the process mean. An illustrative example is also presented to explain the implementation and working of the EWMA–CUSUM and GWMA–CUSUM charts.     

Statistical control charts for monitoring the mean of a stationary process

Recently statistical process control (SPC) methodologies have been developed to accommodate autocorrelated data. A primary method to deal with autocorrelated data is the use of residual charts. Although this methodology has the advantage that it can be applied to any autocorrelated data it needs time series modeling efforts. In addition for a X residual chart the detection capability is sometimes small compared to the X chart and EWMA chart. Zhang (1998) proposed the EWMAST chart which is constructed by charting the EWMA statistic for stationary processes to monitor the process mean. The performance of the EWMAST chart the X chart the X residual chart and other charts were compared in Zhang (1998). In this paper comparisons are made among the EWMAST chart the CUSUM residual chart and EWMA residual chart as well as the X residual chart and X chart via the average run length.     

Statistical Monitoring of Nominal Logistic Profiles in Phase II

Statistical analysis of profile monitoring, a relatively new sub-area of statistical process control due to its applications in different industries, have urged researchers and practitioners to contribute to the developments of new monitoring methods. A statistical profile is a relationship between a quality characteristic (a response) and one or more independent variables to characterize quality of a process or a product. In this article, statistical profiles based on nominal responses are studied, where logistic regression is used to model the responses. Three approaches including likelihood ratio test (LRT), multivariate exponentially weighted moving average (MEWMA), and support vector machines (SVM) approaches are proposed to monitor quality of a process or product in Phase II. Performances of the proposed approaches are evaluated and compared using a case study. Moreover, the effect of two important factors on average run length (ARL) performance, number of levels and number of covariates, has been considered. Results indicate that performance of all approaches depends on the number of covariates and levels. As the number of these factors increases, SVM performance improves while performance of the other approaches deteriorates.     

Phase-II monitoring and diagnosing of multivariate categorical processes using generalized linear test-based control charts

In this paper, two control charts based on the generalized linear test (GLT) and contingency table are proposed for Phase-II monitoring of multivariate categorical processes. The performances of the proposed methods are compared with the exponentially weighted moving average-generalized likelihood ratio test (EWMA-GLRT) control chart proposed in the literature. The results show the better performance of the proposed control charts under moderate and large shifts. Moreover, a new scheme is proposed to identify the parameter responsible for an out-of-control signal. The performance of the proposed diagnosing procedure is evaluated through some simulation experiments.     

A weighted likelihood ratio test-based chart for monitoring process mean and variability

In this paper, a new single exponentially weighted moving average (EWMA) control chart based on the weighted likelihood ratio test, referred to as the WLRT chart, is proposed for the problem of monitoring the mean and variance of a normally distributed process variable. It is easy to design, fast to compute, and quite effective for diverse cases including the detection of the decrease in variability and individual observation case. The optimal parameters that can be used as a design aid in selecting specific parameter values based on the average run length (ARL) and the sample size are provided. The in-control (IC) and out-of-control (OC) performance properties of the new chart are compared with some other existing EWMA-type charts. Our simulation results show that the IC run length distribution of the proposed chart is similar to that of a geometric distribution, and it provides quite a robust and satisfactory overall performance for detecting a wide range of shifts in the process mean and/or variability.     

Monitoring for Increases in Process Variance

Process variation is made up of a wandering mean (signal) and the variation of the process around this wandering mean (noise). The paper advocates a new process monitoring method that uses various forms of moving variances to signal changes in process noise or variance. The monitoring plans are compared to the current approaches in terms of their average run length properties.     

Likelihood ratio test-based chart for monitoring the process variability

This article proposes a new chart with the generalized likelihood ratio (GLR) test statistics for monitoring the process variance of a normally distributed process. The new chart can be easily designed and constructed and the computation results show that it provides quite a satisfactory performance, including the detection of the decrease in the variance and the individual observation at the sampling point which are very important in many practical applications. Average run length (ARL) comparisons between other procedures and the new chart are presented. The optimal parameters that can be used as a design aid in selecting specific parameter values based on the ARL are described. The application of our proposed method is illustrated by a real data example from chemical process control.     

One-sided EWMA control charts for monitoring Poisson processes with varying sample sizes

ABSTRACT

Recently considerable research has been devoted to monitoring increases of incidence rate of adverse rare events. This paper extends some one-sided upper exponentially weighted moving average (EWMA) control charts from monitoring normal means to monitoring Poisson rate when sample sizes are varying over time. The approximated average run length bounds are derived for these EWMA-type charts and compared with the EWMA chart previously studied. Extensive simulations have been conducted to compare the performance of these EWMA-type charts. An illustrative example is given.     

Phase I monitoring of social network with baseline periods using poisson regression

Social network analysis is an important analytic tool to forecast social trends by modeling and monitoring the interactions between network members. This paper proposes an extension of a statistical process control method to monitor social networks by determining the baseline periods when the reference network set is collected. We consider probability density profile (PDP) to identify baseline periods using Poisson regression to model the communications between members. Also, Hotelling T² and likelihood ratio test (LRT) statistics are developed to monitor the network in Phase I. The results based on signal probability indicate a satisfactory performance for the proposed method.     

Monitoring the mean of autocorrelated observations with one generally weighted moving average control chart

Traditionally, using a control chart to monitor a process assumes that process observations are normally and independently distributed. In fact, for many processes, products are either connected or autocorrelated and, consequently, obtained observations are autocorrelative rather than independent. In this scenario, applying an independence assumption instead of autocorrelation for process monitoring is unsuitable. This study examines a generally weighted moving average (GWMA) with a time-varying control chart for monitoring the mean of a process based on autocorrelated observations from a first-order autoregressive process (AR(1)) with random error. Simulation is utilized to evaluate the average run length (ARL) of exponentially weighted moving average (EWMA) and GWMA control charts. Numerous comparisons of ARLs indicate that the GWMA control chart requires less time to detect various shifts at low levels of autocorrelation than those at high levels of autocorrelation. The GWMA control chart is more sensitive than the EWMA control chart for detecting small shifts in a process mean.     

The monitoring of simple linear regression profiles with two observations per sample

We evaluate and compare the performance of Phase II simple linear regression profile approaches when only two observations are used to establish each profile. We propose an EWMA control chart based on average squared deviations from the in-control line, to be used in conjunction with two EWMA control charts based on the slope and Y-intercept estimators, to monitor changes in the three regression model parameters, i.e., the slope, intercept and variance. Simulations establish that the performance of the proposed technique is generally better than that of other approaches in detecting parameter shifts.     

Effect of measurement error on exponentially weighted moving average control charts under ranked set sampling schemes

Control charts are a powerful statistical process monitoring tool often used to monitor the stability of manufacturing processes. In quality control applications, measurement errors adversely affect the performance of control charts. In this paper, we study the effect of measurement error on the detection abilities of the exponentially weighted moving average (EWMA) control charts for monitoring process mean based on ranked set sampling (RSS), median RSS (MRSS), imperfect RSS (IRSS) and imperfect MRSS (IMRSS) schemes. We also study the effect of multiple measurements and non-constant error variance on the performances of the EWMA control charts. The EWMA control chart based on simple random sampling is compared with the EWMA control charts based on RSS, MRSS, IRSS and IMRSS schemes. The performances of the EWMA control charts are evaluated in terms of out-of-control average run length and standard deviation of run lengths. It turns out that the EWMA control charts based on MRSS and IMRSS schemes are better than their counterparts for all measurement error cases considered here.     

The Design of the Multivariate Synthetic Exponentially Weighted Moving Average Control Chart

A multivariate synthetic exponentially weighted moving average (MSEWMA) control chart is presented in this study. The MSEWMA control chart consists of a multivariate exponentially weighted moving average (MEWMA) control chart and a conforming run length control chart. The average run length of the MSEWMA control chart is obtained using a Markov chain approach. From the numerical comparisons, it is shown that the MSEWMA control chart is more efficient than the multivariate synthetic T ² control chart and the MEWMA control chart for detecting shifts in the process mean vector.     

New control charts for simultaneous monitoring of the mean vector and covariance matrix of multivariate multiple linear profiles

Simultaneous monitoring of the mean vector and covariance matrix in multivariate processes allows practitioners to avoid the inflated false alarm rate that results from using two independent control charts. In this paper, we extend exponentially weighted moving average semicircle and generally weighted moving average semicircle control charts to monitor the mean vector and covariance matrix of multivariate multiple linear regression profiles in Phase II simultaneously. These new control charts are compared with the existing control charts in the literature in terms of the average run length criterion. Finally, a case is considered to show the application of the proposed charts.     

Surveillance of the covariance matrix of multivariate nonlinear time series

In this paper, sequential procedures for the surveillance of the covariance matrices of multivariate nonlinear time series are introduced. Two different types of control charts are proposed. The first type is based on the exponential smoothing of each component of a local measure for the covariances. The control statistic is equal to the Mahalanobis distance of this quantity with its in-control mean. In our second approach, the Mahalanobis distance is first determined and after that it is exponentially smoothed. We discuss three examples of local measures.

Several properties of the proposed schemes are discussed assuming the target process to be generated by a multivariate GARCH(1, 1) model. The generalization to the family of spherical distributions allows the modelling of frequently observed fat tails in financial data. Some results of an extensive Monte Carlo simulation study are provided in order to judge the performance of the presented control schemes. As a performance measure we use the average run length. An empirical example illustrates the importance of the fast detection of the changes in the covariance structure of the returns of financial assets.     

Grouped data exponentially weighted moving average control charts

In the manufacture of metal fasteners in a progressive die operation, and other industrial situations, important quality dimensions cannot be measured on a continuous scale, and manufactured parts are classified into groups by using a step gauge. This paper proposes a version of exponentially weighted moving average (EWMA) control charts that are applicable to monitoring the grouped data for process shifts. The run length properties of this new grouped data EWMA chart are compared with similar results previously obtained for EWMA charts for variables data and with those for cumulative sum (CUSUM) schemes based on grouped data. Grouped data EWMA charts are shown to be nearly as efficient as variables-based EWMA charts and are thus an attractive alternative when the collection of variables data is not feasible. In addition, grouped data EWMA charts are less affected by the discreteness that is inherent in grouped data than are grouped data CUSUM charts. In the metal fasteners application, grouped data EWMA charts were simple to implement and allowed the rapid detection of undesirable process shifts.     

Robustness to non-normality and autocorrelation of individuals control charts

This paper studies the effects of non-normality and autocorrelation on the performances of various individuals control charts for monitoring the process mean and/or variance. The traditional Shewhart X chart and moving range (MR) chart are investigated as well as several types of exponentially weighted moving average (EWMA) charts and combinations of control charts involving these EWMA charts. It is shown that the combination of the X and MR charts will not detect small and moderate parameter shifts as fast as combinations involving the EWMA charts, and that the performana of the X and MR charts is very sensitive to the normality assumption. It is also shown that certain combinations of EWMA charts can be designed to be robust to non-normality and very effective at detecting small and moderate shifts in the process mean and/or variance. Although autocorrelation can have a significant effect on the in-control performances of these combinations of EWMA charts, their relative out-of-control performances under independence are generally maintained for low to moderate levels of autocorrelation.     

Combined Shewhart–EWMA control charts with estimated parameters

Shewhart and EWMA control charts can be suitably combined to obtain a simple monitoring scheme sensitive to both large and small shifts in the process mean. So far, the performance of the combined Shewhart–EWMA (CSEWMA) has been investigated under the assumption that the process parameters are known. However, parameters are often estimated from reference Phase I samples. Since chart performances may be even largely affected by estimation errors, we study the behaviour of the CSEWMA with estimated parameters in both in- and out-of-control situations. Comparisons with standard Shewhart and EWMA charts are presented. Recommendations are given for Phase I sample size requirements necessary to achieve desired in-control performance.     

Optimal Statistical Design of a Multivariate EWMA Chart Based on ARL and MRL

Statistical design is applied to a multivariate exponentially weighted moving average (MEWMA) control chart. The chart parameters are control limit H and smoothing constant r. The choices of the parameters depend on the number of variables p and the size of the process mean shift δ. The MEWMA statistic is modeled as a Markov chain and the Markov chain approach is used to determine the properties of the chart. Although average run length has become a traditional measure of the performance of control schemes, some authors have suggested other measures, such as median and other percentiles of the run length distribution to explain run length properties of a control scheme. This will allow a thorough study of the performance of the control scheme. Consequently, conclusions based on these measures would provide a better and comprehensive understanding of a scheme. In this article, we present the performance of the MEWMA control chart as measured by the average run length and median run length. Graphs are given so that the chart parameters of an optimal MEWMA chart can be determined easily.