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.     

A Multivariate Adaptive Exponentially Weighted Moving Average Control Chart

A multivariate extension of the adaptive exponentially weighted moving average (AEWMA) control chart is proposed. The new multivariate scheme can detect small and large shifts in the process mean vector effectively. The proposed scheme can be viewed as a smooth combination of a multivariate exponentially weighted moving average (MEWMA) chart and a Shewhart χ²-chart. The optimal design of the proposed chart is given according to a pre-specified in-control average run length and two shift sizes; a small and large shift each measured in terms of the non centrality parameter. The signal resistance of the newly proposed multivariate chart is also given. Comparisons among the new chart, the MEWMA chart, and the combined Shewhart-MEWMA (S-MEWMA) chart in terms of the standard and worst-case average run length profiles are presented. In addition, the three charts are compared with respect to their worst-case signal resistance values. The proposed chart gives somewhat better worst-case ARL and signal resistance values than the competing charts. It also gives better standard ARL performance especially for moderate and large shifts. The effectiveness of our proposed chart is illustrated through an example with simulated data set.     

Multivariate EWMA Control Chart with Adaptive Sample Sizes

Standard multivariate control charts usually employ fixed sample sizes at equal sampling intervals to monitor a process. In this study, a multivariate exponential weighted moving average (MEWMA) chart with adaptive sample sizes is investigated. Performance measure of the adaptive-sample-size MEWMA chart is obtained through a Markov chain approach. The performance of the adaptive-sample-size MEWMA chart is compared with the fixed-sample-size control chart in terms of steady-state average run length for different magnitude of shifts in the process mean. It is shown that the adaptive-sample-size chart is more efficient than the fixed-sample-size MEWMA control chart in detecting shifts in the process mean.     

Mixed multivariate EWMA-CUSUM control charts for an improved process monitoring

Multivariate exponential weighted moving average and cumulative sum charts are the most common memory type multivariate control charts. They make use of the present and past information to detect small shifts in the process parameter(s). In this article, we propose two new multivariate control charts using a mixed version of their design setups. The plotting statistics of the proposed charts are based on the cumulative sum of the multivariate exponentially weighted moving averages. The performances of these schemes are evaluated in terms of average run length. The proposals are compared with their existing counterparts, including HotellingT², MCUSUM, MEWMA, and MC1 charts. An application example is also presented for practical considerations using a real dataset.     

Using the predictive distribution to determine control limits for the Bayesian MEWMA chart

Bayesian control charts have been proposed for monitoring multivariate processes with the multivariate exponentially weighted moving average (MEWMA) statistic. It has been suggested that we use limits based on the predictive distribution of the MEWMA statistic. This analysis, however is based on the erroneous result that the average run length (ARL) is a function of the exceedance probability, that is, the probability that the first point exceeds the control limit. We show how this result can be corrected and we discuss how the Bayesian MEWMA chart with limits based on the predictive distribution compares with other multivariate control chart procedures.     

The economic design of multivariate binomial EWMA VSSI control charts

Since multi-attribute control charts have received little attention compared with multivariate variable control charts, this research is concerned with developing a new methodology to employ the multivariate exponentially weighted moving average (MEWMA) charts for m-attribute binomial processes; the attributes being the number of nonconforming items. Moreover, since the variable sample size and sampling interval (VSSI) MEWMA charts detect small process mean shifts faster than the traditional MEWMA, an economic design of the VSSI MEWMA chart is proposed to obtain the optimum design parameters of the chart. The sample size, the sampling interval, and the warning/action limit coefficients are obtained using a genetic algorithm such that the expected total cost per hour is minimized. At the end, a sensitivity analysis has been carried out to investigate the effects of the cost and the model parameters on the solution of the economic design of the VSSI MEWMA chart.     

On-line monitoring data quality of high-dimensional data streams

ABSTRACT

In recent years, effective monitoring of data quality has increasingly attracted attention of researchers in the area of statistical process control. Among the relevant research on this topic, none used multivariate methods to control the multidimensional data quality process, but instead relied on multiple univariate control charts. Based on a novel one-sided multivariate exponentially weighted moving average (MEWMA) chart, we propose a conditional false discovery rate-adjusted scheme to on-line monitor the data quality of high-dimensional data streams. With thousands of input data streams, the average run length loses its usefulness because one will likely have out-of-control signals at each time period. Hence, we first control the percentage of signals that are false alarms. Then, we compare the power of the proposed MEWMA scheme with that of two alternative methods. Compared with two competitors, numerical results show that the proposed MEWMA scheme has higher average power.     

A Double Moving Average Control Chart

The double exponentially weighted moving average (DEWMA) technique has been investigated in recent years for detecting shifts in the process mean and has been shown to be more efficient than the corresponding exponentially weighted moving average (EWMA) technique. In this article, we extend the DEWMA technique of performing exponential smoothing twice to the double moving average (DMA) technique by computing the moving average twice. Using simulation, we show that our proposed DMA chart improves upon the ARL performance of the moving average (MA) chart in detecting mean shifts of small to moderate magnitudes. It is also shown through simulation that, generally, the DMA charts with spans, w = 10 and 15 provide comparable average run length (ARL) performances to the EWMA and cumulative sum (CUSUM) charts, designed for detecting small shifts.     

Economic-statistical design of synthetic double sampling T2 chart

Abstract

An economic-statistical design of the synthetic double sampling (synDS) T² chart is presented in this study. The cost function is minimized to obtain the optimal design parameters of the synDS T² chart by incorporating the statistical constraints or the constraints on the average number of samples. An example is provided and a sensitivity analysis is conducted to study the effect of model parameters on the optimal solution of the design. The numerical comparison shows that the synDS T² chart performs better than the synthetic T² chart and the multivariate exponentially weighted moving average chart, in terms of the cost.     

Accounting for phase I sampling variability in the performance of the MEWMA control chart with estimated parameters

In this article, we assess the performance of the multivariate exponentially weighted moving average (MEWMA) control chart with estimated parameters while considering the practitioner-to-practitioner variability. We evaluate the chart performance in terms of the in-control average run length (ARL) distributional properties; mainly the average (AARL), the standard deviation (SDARL), and some percentiles. We show through simulations that using estimates in place of the in-control parameters may result in an in-control ARL distribution that almost completely lies below the desired value. We also show that even with the use of larger amounts of historical data, there is still a problem with the excessive false alarm rates. We recommend the use of a recently proposed bootstrap-based design technique for adjusting the control limits. The technique is quite effective in controlling the percentage of short in-control ARLs resulting from the estimation error.     

Parameter optimization for a modified MEWMA control chart based on a PSO algorithm

The standard multivariate exponentially weighted moving average (MEWMA) control chart with a constant smoothing parameter or diagonal matrix is based on the assumption that the samples obey standard normal distribution. With improvements in manufacturing quality and product complexity, there is always correlativity among quality characteristics, and samples will not always obey standard normal distribution. Considering the correlativity among quality characteristics, a new modified general MEWMA (GEWMA) control chart is proposed, and its performance is analyzed. Based on the Particle Swarm Optimization (PSO) algorithm, a smoothing matrix optimized under certain conditions is selected and applied to a sample analysis. As a result of the parameter combination chosen by PSO, the statistic function of the GEWMA control chart is better than that of the full matrix MEWMA (FEWMA) control chart.     

The Effect of Methods for Handling Missing Values on the Performance of the MEWMA Control Chart

This article is concerned with the effect of the methods for handling missing values in multivariate control charts. We discuss the complete case, mean substitution, regression, stochastic regression, and the expectation–maximization algorithm methods for handling missing values. Estimates of mean vector and variance–covariance matrix from the treated data set are used to build the multivariate exponentially weighted moving average (MEWMA) control chart. Based on a Monte Carlo simulation study, the performance of each of the five methods is investigated in terms of its ability to obtain the nominal in-control and out-of-control average run length (ARL). We consider three sample sizes, five levels of the percentage of missing values, and three types of variable numbers. Our simulation results show that imputation methods produce better performance than case deletion methods. The regression-based imputation methods have the best overall performance among all the competing methods.     

Maximum Chi-Square Generally Weighted Moving Average Control Chart for Monitoring Process Mean and Variability

In this article, we propose a new control chart called the maximum chi-square generally weighted moving average (MCSGWMA) control chart. This control chart can effectively combine two generally weighted moving average (GWMA) control charts into a single one and can detect both increases as well as decreases in the process mean and/or variability simultaneously. The average run length (ARL) characteristics of the MCSGWMA and maximum exponentially weighted moving average (MaxEWMA) charts are evaluated by performing computer simulations. The comparison of the ARLs shows that the MCSGWMA control chart performs better than the MaxEWMA control chart.     

A Sum of Squares Generally Weighted Moving Average Control Chart

In this article, we propose a new control chart called the sum of squares generally weighted moving average (SS-GWMA) control chart to simultaneously detect both the increase and decrease in the mean and/or variability. This new scheme is compared with the sum of squares exponentially weighted moving average (SS-EWMA) control chart. A simulation study is conducted to show that SS-GWMA control charts outperform SS-EWMA charts, in terms of the average run length (ARL), standard deviation of run length (SDRL), and diagnostic abilities. The design of SS-GWMA control charts is also discussed.     

A Multivariate Synthetic Control Chart for Monitoring the Process Mean Vector of Skewed Populations Using Weighted Standard Deviations

This article proposes a multivariate synthetic control chart for skewed populations based on the weighted standard deviation method. The proposed chart incorporates the weighted standard deviation method into the standard multivariate synthetic control chart. The standard multivariate synthetic chart consists of the Hotelling's T ² chart and the conforming run length chart. The weighted standard deviation method adjusts the variance–covariance matrix of the quality characteristics and approximates the probability density function using several multivariate normal distributions. The proposed chart reduces to the standard multivariate synthetic chart when the underlying distribution is symmetric. In general, the simulation results show that the proposed chart performs better than the existing multivariate charts for skewed populations and the standard T ² chart, in terms of false alarm rates as well as moderate and large mean shift detection rates based on the various degrees of skewnesses.     

A new synthetic control chart for monitoring process mean using auxiliary information

ABSTRACT

Quality control charts have been widely recognized as a potentially powerful statistical process monitoring tool in statistical process control because of their superior ability in detecting shifts in the process parameters. Recently, auxiliary-information-based control charts have been proposed and shown to have excellent speed in detecting process shifts than those based without it. In this paper, we design a new synthetic control chart that is based on a statistic that utilizes information from both the study and auxiliary variables. The proposed synthetic chart encompasses the classical synthetic chart. The construction, optimal design, run length profiles, and the performance evaluation of the new chart are discussed in detail. It turns out that the proposed synthetic chart performs uniformly better than the classical synthetic chart when detecting different kinds of shifts in the process mean under both zero-state and steady-state run length performances. Moreover, with reasonable assumptions, the proposed chart also surpasses the exponentially weighted moving average control chart. An application with a simulated data set is also presented to explain the implementation of the proposed control chart.     

A gradient approach to efficient design and analysis of multivariate EWMA control charts

Compared to the grid search approach to optimal design of control charts, the gradient-based approach is more computationally efficient as the gradient information indicates the direction to search the optimal design parameters. However, the optimal parameters of multivariate exponentially weighted moving average (MEWMA) control charts are often obtained by using grid search in the existing literature. Note that the average run length (ARL) performance of the MEWMA chart can be calculated based on a Markov chain model, making it feasible to estimate the ARL gradient from it. Motivated by this, this paper develops an ARL gradient-based approach for the optimal design and sensitivity analysis of MEWMA control charts. It is shown that the proposed method is able to provide a fast, accurate, and easy-to-implement algorithm for the design and analysis of MEWMA charts, as compared to the conventional design approach based on grid search.     

The effect of parameter estimation on phase II monitoring of poisson regression profiles

ABSTRACT

The effect of parameters estimation on profile monitoring methods has only been studied by a few researchers and only the assumption of a normal response variable has been tackled. However, in some practical situation, the normality assumption is violated and the response variable follows a discrete distribution such as Poisson. In this paper, we evaluate the effect of parameters estimation on the Phase II monitoring of Poisson regression profiles by considering two control charts, namely the Hotelling’s T² and the multivariate exponentially weighted moving average (MEWMA) charts. Simulation studies in terms of the average run length (ARL) and the standard deviation of the run length (SDRL) are carried out to assess the effect of estimated parameters on the performance of Phase II monitoring approaches. The results reveal that both in-control and out-of-control performances of these charts are adversely affected when the regression parameters are estimated.     

A weibull process failure mechanism for the economic design of mewma control charts

The design parameters of the multivariate exponentially weighted moving average (MEWMA) control chart may be chosen according to economic and/or statistical considerations. The economic model proposed for the design of the'MEWMA chart assumes a Markovian process failure mechanism following an exponential distribution. We'assess the sensitivity of the resulting economic design for the MEWMA to deviations from this assumption. In particular, the generalization, from an exponential to a Weibull distribution of process failure, is used to study the selection of MEWMA chart parameters given process cost and time information. We conclude that the quality of the resulting design (in terms of expected cost) is not substantially affected by mis-specification of the distribution of process failure.     

A New GWMA Control Chart for Monitoring Process Mean and Variability

In this article, we extend a single exponentially weighted moving average semicircle (EWMA-SC) chart to a single generally weighted moving average (GWMA) chart. This new control chart can effectively combine the features of the SC chart with GWMA techniques, and can easily indicate the source and direction of a change. We perform simulations to evaluate the average run length, standard deviation of the run length, and diagnostic abilities of the GWMA-SC and EWMA-SC charts. An extensive comparison shows that the GWMA-SC control chart is more sensitive than the EWMA-SC chart for detecting small shifts in the process mean and/or variability.