Marien Alet Graham

Phase I Statistical Process Control Charts: An Overview and Some Results

ABSTRACT In practice, Phase I analysis constitutes an integral part of an overall SPC program in which control charts play a crucial role. An overview of the literature on Phase I parametric control charts for univariate variables data is presented. Since the Phase I signaling events are dependent and multiple signaling events are to be dealt with simultaneously in making an in-control or out-of-control decision, the joint distribution of the charting statistics is used to control the false alarm probability, which is defined as the probability of at least one false alarm, while designing the charts. An example is given. Concluding remarks include suggestions regarding future research problems.

Distribution-free exponentially weighted moving average control charts for monitoring unknown location

Distribution-free (nonparametric) control charts provide a robust alternative to a data analyst when there is lack of knowledge about the underlying distribution. A two-sided nonparametric Phase II exponentially weighted moving average (EWMA) control chart, based on the exceedance statistics (EWMA-EX), is proposed for detecting a shift in the location parameter of a continuous distribution. The nonparametric EWMA chart combines the advantages of a nonparametric control chart (known and robust in-control performance) with the better shift detection properties of an EWMA chart. Guidance and recommendations are provided for practical implementation of the chart along with illustrative examples. A performance comparison is made with the traditional (normal theory) EWMA chart for subgroup averages and a recently proposed nonparametric EWMA chart based on the Wilcoxon-Mann-Whitney statistics. A summary and some concluding remarks are given.

Distribution-Free Exceedance CUSUM Control Charts for Location

Distribution-free (nonparametric) control charts can be useful to the quality practitioner when the underlying distribution is not known. A Phase II nonparametric cumulative sum (CUSUM) chart based on the exceedance statistics, called the exceedance CUSUM chart, is proposed here for detecting a shift in the unknown location parameter of a continuous distribution. The exceedance statistics can be more efficient than rank-based methods when the underlying distribution is heavy-tailed and/or right-skewed, which may be the case in some applications, particularly with certain lifetime data. Moreover, exceedance statistics can save testing time and resources as they can be applied as soon as a certain order statistic of the reference sample is available. Guidelines and recommendations are provided for the charts design parameters along with an illustrative example. The in- and out-of-control performances of the chart are studied through extensive simulations on the basis of the average run-length (ARL), the standard deviation of run-length (SDRL), the median run-length (MDRL), and some percentiles of run-length. Further, a comparison with a number of existing control charts, including the parametric CUSUM chart and a recent nonparametric CUSUM chart based on the Wilcoxon rank-sum statistic, called the rank-sum CUSUM chart, is made. It is seen that the exceedance CUSUM chart performs well in many cases and thus can be a useful alternative chart in practice. A summary and some concluding remarks are given.

Design and implementation of CUSUM exceedance control charts for unknown location

Nonparametric control charts provide a robust alternative in practice when the form of the underlying distribution is unknown. Nonparametric CUSUM (NPCUSUM) charts blend the advantages of a CUSUM with that of a nonparametric chart in detecting small to moderate shifts. In this paper, we examine efficient design and implementation of Phase II NPCUSUM charts based on exceedance (EX) statistics, called the NPCUSUM-EX chart. We investigate the choice of the order statistic from the reference (Phase I) sample that defines the exceedance statistic. We see that choices other than the median, such as the 75th percentile, can yield improved performance of the chart in certain situations. Furthermore, observing certain shortcomings of the average run-length, we use the median run-length as the performance metric. The NPCUSUM-EX chart is compared with the NPCUSUM-Rank chart based on the popular Wilcoxon rank-sum statistic. We also study the choice of the reference value, k, of the CUSUM charts. An illustration with real data is provided.

A nonparametric exponentially weighted moving average signed-rank chart for monitoring location

Nonparametric control charts can provide a robust alternative in practice to the data analyst when there is a lack of knowledge about the underlying distribution. A nonparametric exponentially weighted moving average (NPEWMA) control chart combines the advantages of a nonparametric control chart with the better shift detection properties of a traditional EWMA chart. A NPEWMA chart for the median of a symmetric continuous distribution was introduced by Amin and Searcy (1991) using the Wilcoxon signed-rank statistic (see Gibbons and Chakraborti, 2003). This is called the nonparametric exponentially weighted moving average Signed-Rank (NPEWMA-SR) chart. However, important questions remained unanswered regarding the practical implementation as well as the performance of this chart. In this paper we address these issues with a more in-depth study of the two-sided NPEWMA-SR chart. A Markov chain approach is used to compute the run-length distribution and the associated performance characteristics. Detailed guidelines and recommendations for selecting the charts design parameters for practical implementation are provided along with illustrative examples. An extensive simulation study is done on the performance of the chart including a detailed comparison with a number of existing control charts, including the traditional EWMA chart for subgroup averages and some nonparametric charts i.e. runs-rules enhanced Shewhart-type SR charts and the NPEWMA chart based on signs. Results show that the NPEWMA-SR chart performs just as well as and in some cases better than the competitors. A summary and some concluding remarks are given.

A Nonparametric EWMA Sign Chart for Location Based on Individual Measurements

ABSTRACT Nonparametric control charts are useful when there is limited or complete lack of knowledge about the form of the underlying distribution. Though traditional statistical process control (SPC) applications of control charts involve subgrouped data, recent advances have led to more and more instances where individual measurements (data) are collected over time. A two-sided nonparametric exponentially weighted moving average (EWMA) control chart for i.i.d. individual data is proposed based on the sign (SN) statistic. A Markov chain approach is used to determine the run-length distribution of the chart and some associated performance characteristics. An important advantage of the nonparametric EWMA-SN chart is its inherent in-control robustness. In fact, the in-control run-length distribution and hence all of its associated characteristics (e.g., false alarm rate, average, standard deviation, median, etc.) of the chart remain the same for all unknown continuous distributions. In order to aid practical implementation, tables are provided for the charts design parameters. An extensive simulation study shows that on the basis of minimal required assumptions, robustness of the in-control run-length distribution and out-of-control performance, the proposed nonparametric EWMA-SN chart can be a strong contender in many applications where traditional parametric charts are currently used.

A Phase I nonparametric Shewhart-type control chart based on the median

A nonparametric Shewhart-type control chart is proposed for monitoring the location of a continuous variable in a Phase I process control setting. The chart is based on the pooled median of the available Phase I samples and the charting statistics are the counts (number of observations) in each sample that are less than the pooled median. An exact expression for the false alarm probability (FAP) is given in terms of the multivariate hypergeometric distribution and this is used to provide tables for the control limits for a specified nominal FAP value (of 0.01, 0.05 and 0.10, respectively) and for some values of the sample size (n) and the number of Phase I samples (m). Some approximations are discussed in terms of the univariate hypergeometric and the normal distributions. A simulation study shows that the proposed chart performs as well as, and in some cases better than, an existing Shewhart-type chart based on the normal distribution. Numerical examples are given to demonstrate the implementation of the new chart.

On the performance of Shewhart-type synthetic and runs-rules charts combined with an X chart

Part of this work was supported by the SARCHI Chair at the University of Pretoria. Sandile Shongwe’s research was supported by National Research Foundation (NRF) and Department of Science and Technology’s Innovation Doctoral Scholarship (SFH14081591713) and Marien Graham’s research was supported in part by the NRF’s (Thuthuka program: TTK14061168807; grant number: 94102).

A modified side-sensitive synthetic chart to monitor the process mean

Abstract In this paper, we propose a modified side-sensitive (MSS) synthetic chart which signals only if all the consecutive plotting statistics that lead to an out-of-control event fall on one side of the centre line, unlike the non-side-sensitive, standard and revised side-sensitive synthetic charts that also signal even when some of the plotting statistics fall on opposite sides of the centre line. Moreover, we use the Markov chain imbedding technique to study and compare the zero-state and steady-state average run-length (ARL), extra quadratic loss, average ratio of the ARLs and performance comparison index of the proposed MSS chart with other Shewhart-type synthetic and runs-rules charts. The synthetic chart with this MSS feature has a better overall zero-state and steady-state performance than the existing synthetic charts and hence makes it a strong contender in many applications where existing synthetic charts are currently used.

Design and implementation issues for a class of distribution-free Phase II EWMA exceedance control charts

Distribution-free (nonparametric) control charts can play an essential role in process monitoring when there is dearth of information about the underlying distribution. In this paper, we study various aspects related to an efficient design and execution of a class of nonparametric Phase II exponentially weighted moving average (denoted by NPEWMA) charts based on exceedance statistics. The choice of the Phase I (reference) sample order statistic used in the design of the control chart is investigated. We use the exact time-varying control limits and the median run-length as the metric in an in-depth performance study. Based on the performance of the chart, we outline implementation strategies and make recommendations for selecting this order statistic from a practical point of view and provide illustrations with a data-set. We conclude with a summary and some remarks.