A Distribution-Free Multivariate Control Chart

Monitoring multivariate quality variables or data streams remains an important and challenging problem in statistical process control (SPC). Although the multivariate SPC has been extensively studied in the literature, designing distribution-free control schemes are still challenging and yet to be addressed well. This article develops a new nonparametric methodology for monitoring location parameters when only a small reference dataset is available. The key idea is to construct a series of conditionally distribution-free test statistics in the sense that their distributions are free of the underlying distribution given the empirical distribution functions. The conditional probability that the charting statistic exceeds the control limit at present given that there is no alarm before the current time point can be guaranteed to attain a specified false alarm rate. The success of the proposed method lies in the use of data-dependent control limits, which are determined based on the observations online rather than decided before monitoring. Our theoretical and numerical studies show that the proposed control chart is able to deliver satisfactory in-control run-length performance for any distributions with any dimension. It is also very efficient in detecting multivariate process shifts when the process distribution is heavy-tailed or skewed. Supplementary materials for this article are available online.     

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.     

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.     

Control Chart Limits for Monitoring Process Mean Based on Downton's Estimator

Control charts are important tools in statistical process control used to monitor shift in process mean and variance. This paper proposes a control chart for monitoring the process mean using the Downton estimator and provides table of constant factors for computing the control limits for sample size (n ≤ 10). The derived control limits for process mean were compared with control limits based on range statistic. The performance of the proposed control charts was evaluated using the average run length for normal and non‐normal process situations. The obtained results showed that the control chart, using the Downton statistic, performed better than Shewhart chart using range statistic for detection of small shift in the process mean when the process is non‐normal and compares favourably well with Shewhart chart that is normally distributed. Copyright © 2015 John Wiley & Sons, Ltd.     

Multivariate Control Chart for Process Dispersion Based on Individual Observations

In this paper, we will discuss a simple way for monitoring shifts in the covariance matrix of a p-dimensional multivariate normal process distribution, N_p(μ,Σ). An exact method based on the chi-square distribution for constructing multivariate control limits will also be shown. We will illustrate the proposed procedure at work based on an example.     

Comparisons of Shewhart-type rank based control charts for monitoring location parameters of univariate processes

The nonparametric (distribution-free) control charts are robust alternatives to the conventional parametric control charts when the form of underlying process distribution is unknown or complicated. In this paper, we consider two new nonparametric control charts based on the Hogg–Fisher–Randle (HFR) statistic and the Savage rank statistic. These are popular statistics for testing location shifts, especially in right-skewed densities. Nevertheless, the control charts based on these statistics are not studied in quality control literature. In the current context, we study phase-II Shewhart-type charts based on the HFR and Savage statistics. We compare these charts with the Wilcoxon rank-sum chart in terms of false alarm rate, out-of-control average run-length and other run length properties. Implementation procedures and some illustrations of these charts are also provided. Numerical results based on Monte Carlo analysis show that the new charts are superior to the Wilcoxon rank-sum chart for a class of non-normal distributions in detecting location shift. New charts also provide better control over false alarm when reference sample size is small.     

A EWMA Control Chart for Exponential Distributed Quality Based on Moving Average Statistics

We propose an exponentially weighted moving average (EWMA) control chart for monitoring exponential distributed quality characteristics. The proposed control chart first transforms the sample data to approximate normal variables, then calculates the moving average (MA) statistic for each subgroup, and finally constructs the EWMA statistic based on the current and the previous MA statistics. The upper and the lower control limits are derived using the mean and the variance of EWMA statistics. The in‐control and the out‐of‐control average run lengths are derived and tabularized according to process shift parameters and smoothing constants. It is shown that the proposed control chart outperforms the MA control chart for all shift parameters. Copyright © 2015 John Wiley & Sons, Ltd.     

A Distribution‐free Control Chart for the Joint Monitoring of Location and Scale

Traditional statistical process control for variables data often involves the use of a separate mean and a standard deviation chart. Several proposals have been published recently, where a single (combination) chart that is simpler and may have performance advantages, is used. The assumption of normality is crucial for the validity of these charts. In this article, a single distribution‐free Shewhart‐type chart is proposed for monitoring the location and the scale parameters of a continuous distribution when both of these parameters are unknown. The plotting statistic combines two popular nonparametric test statistics: the Wilcoxon rank sum test for location and the Ansari–Bradley test for scale. Being nonparametric, all in‐control properties of the proposed chart remain the same and known for all continuous distributions. Control limits are tabulated for implementation in practice. The in‐control and the out‐of‐control performance properties of the chart are investigated in simulation studies in terms of the mean, the standard deviation, the median, and some percentiles of the run length distribution. The influence of the reference sample size is examined. A numerical example is given for illustration. Summary and conclusions are offered. Copyright © 2011 John Wiley & Sons, Ltd.     

A Note on the Lower-Sided Synthetic Chart for Exponentials

The synthetic control chart for exponential data is discussed and an expression is derived for its average run length, as well as its design parameters. The synthetic control chart for exponentials is shown analytically to be a two-in-a-row rule. This chart is compared with the Shewhart chart for individuals and with the worst-case, lower-sided exponential EWMA and CUSUM charts. While the synthetic control chart for exponentials outperforms the Shewhart chart for individuals, the EWMA and CUSUM charts are shown to be far superior in detecting decreases in the exponential mean.     

Design of a Control Chart Using a Modified EWMA Statistic

In this paper, the design of a control chart is given using a modified exponentially weighted moving average statistic under the assumption that the quality characteristic of interest follows the normal distribution. The structure of the proposed control chart is developed, and the necessary measures are derived to find the average run length for in‐control and out‐of‐control processes. The efficiency of the proposed chart is compared with two existing control charts in terms of the average run length. The results are explained with the help of industrial example. Copyright © 2016 John Wiley & Sons, Ltd.     

A New Two-Sided Cumulative Sum Quality Control Scheme

A new two-sided cumulative sum quality control scheme is proposed. The new scheme was developed specifically to be generalized to a multivariate cumulative sum quality control scheme. The multivariate version will be examined in a subsequent paper; this article evaluates the univariate version. A comparison of the conventional two-sided cumulative sum scheme and the proposed scheme indicates that the new scheme has slightly better properties (ratio of on-aim to off-aim average run lengths) than the conventional scheme. Steady state average run lengths are discussed. The new scheme and the conventional two-sided cumulative sum scheme have equivalent steady state average run lengths. Methods for implementing the fast initial response feature for the new cumulative sum scheme are given. A comparison of average run lengths for the conventional and proposed schemes with fast initial response features is also favorable to the new scheme. A Markov chain approximation is used to calculate the average run lengths of the new scheme.     

The Generally Weighted Moving Average Control Chart for Detecting Small Shifts in the Process Mean

A generalization of the exponentially weighted moving average (EWMA) control chart is proposed and analyzed. The generalized control chart we have proposed is called the generally weighted moving average (GWMA) control chart. The GWMA control chart, with time-varying control limits to detect start-up shifts more sensitively, performs better in detecting small shifts of the process mean. We use simulation to evaluate the average run length (ARL) properties of the EWMA control chart and GWMA control chart. An extensive comparison reveals that the GWMA control chart is more sensitive than the EWMA control chart for detecting small shifts in the mean of a process. To enhance the detection ability of the GWMA control chart, we submit the composite Shewhart-GWMA scheme to monitor process mean. The composite Shewhart-GWMA control chart with/without runs rules is more sensitive than the GWMA control chart in detecting small shifts of the process mean. The resulting ARLs obtained by the GWMA control chart when the assumption of normality is violated are discussed.     

A Statistical Control Chart for Stationary Process Data

In the statistical process control environment, a primary method to deal with autocorrelated data is the use of a residual chart. Although this methodology has the advantage that it can be applied to any autocorrelated data, it needs some modeling effort in practice. In addition, the detection capability of the residual chart is not always great. This article proposes a statistical control chart for stationary process data. It is simple to implement, and no modeling effort is required. Comparisons are made among the proposed chart, the residual chart, and other charts. When the process autocorrelation is not very strong and the mean changes are not large, the new chart performs better than the residual chart and the other charts.     

A Multivariate Exponentially Weighted Moving Average Control Chart

A multivariate extension of the exponentially weighted moving average (EWMA) control chart is presented, and guidelines given for designing this easy-to-implement multivariate procedure. A comparison shows that the average run length (ARL) performance of this chart is similar to that of multivariate cumulative sum (CUSUM) control charts in detecting a shift in the mean vector of a multivariate normal distribution. As with the Hotelling's χ² and multivariate CUSUM charts, the ARL performance of the multivariate EWMA chart depends on the underlying mean vector and covariance matrix only through the value of the noncentrality parameter. Worst-case scenarios show that Hotelling's χ² charts should always be used in conjunction with multivariate CUSUM and EWMA charts to avoid potential inertia problems. Examples are given to illustrate the use of the proposed procedure.     

Increasing the Sensitivity of Multivariate EWMA Control Chart

A multivariate exponentially weighted moving average (MEWMA) control chart is used for fast detection of small shifts in multivariate statistical quality control. However, for ease of computation, the MEWMA control chart statistics are computed based on the asymptotic form of their covariance matrix in most cases. Another reason that justifies the design of the MEWMA control chart using the asymptotic covariance matrix is that the chart will be insensitive at start-up since processes are more likely to be away from the target value when the control scheme is initiated due to start-up problems. However, if initial out-of-control conditions are deemed important for quick detection, then the MEWMA statistics should be computed based on the exact covariance matrix, as it leads to a natural fast initial response for the MEWMA chart. It will also be shown in this paper the importance of computing the MEWMA statistics based on the exact form of their covariance matrix to further enhance the MEWMA control chart's sensitivity for detecting small shifts. The MEWMA statistics based on the asymptotic and the exact form of their covariance matrix will be referred to as the asymptotic and the exact MEWMA statistics, respectively. Plots and factors that simplify the design of the exact MEWMA control chart are also given.     

A new nonparametric double exponentially weighted moving average control chart

There are many practical situations where the underlying distribution of the quality characteristic either deviates from normality or it is unknown. In such cases, practitioners often make use of the nonparametric control charts. In this paper, a new nonparametric double exponentially weighted moving average control chart on the basis of the signed-rank statistic is proposed for monitoring the process location. Monte Carlo simulations are carried out to obtain the run length characteristics of the proposed chart. The performance comparison of the proposed chart with the existing parametric and nonparametric control charts is made by using various performance metrics of the run length distribution. The comparison showed the superiority of the suggested chart over its existing parametric and nonparametric counterparts. An illustrative example for the practical implementation of the proposed chart is also provided by using the industrial data set.     

Alternatives to the Multivariate Control Chart for Process Dispersion

In this article, we compare the performances of six new multivariate control chart schemes for process dispersion to the standard multivariate process dispersion control chart. The six new schemes are designed by transforming the standard multivariate control chart statistic for process dispersion into a standard scale so that runs rules can be incorporated into these schemes. This article discusses a simple extension for using runs rules in a multivariate control chart for process dispersion. The extension is deemed important since the use of runs rules is always confined to univariate control charts only. The performances of the six control chart schemes together with the standard control chart are based on the computed average run length (ARL) profiles. Five of the six schemes have shown better ARL performances than the standard multivariate process dispersion control chart.     

A New Hybrid Exponentially Weighted Moving Average Control Chart for Monitoring Process Mean: Discussion

A new hybrid exponentially weighted moving average (HEWMA) control chart has been proposed in the literature for efficiently monitoring the process mean. In that paper, the computed variance of the HEWMA statistic was, unfortunately, not correct! In this discussion, the correct variance of the HEWMA statistic is given, and the run length characteristics of the HEWMA control chart are studied and explored. It is noticed that not only the superiority of the HEWMA control chart remains over the existing (considered before) charts but also the new results based on the corrected control limits are more profound and reflective. Copyright © 2016 John Wiley & Sons, Ltd.     

A Neural Network Method for Modelling the Parameters of a CUSUM Chart

The cumulative sum (CUSUM) chart is widely employed in quality control to monitor a process or to evaluate historic data. CUSUM charts are designed to exhibit acceptable average run lengths both when the process is in and out of control. This paper introduces a functional technique for generating the parameters h and k for such a chart that will have specified average run lengths. It employs the method of artificial neural networks to derive the appropriate coefficients. An EXCEL spreadsheet to assist computing the parameters is presented.