Charles W. Champ

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 Hotellings 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 Hotellings 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.

Effects of parameter estimation on control chart properties : A literature review

Control charts are powerful tools used to monitor the quality of processes. In practice, control chart limits are often calculated using parameter estimates from an in-control Phase I reference sample. In Phase II of the monitoring scheme, statistics based on new samples are compared with the estimated control limits to monitor for departures from the in-control state. Many studies that evaluate control chart performance in Phase II rely on the assumption that the in-control parameters are known. Although the additional variability introduced into the monitoring scheme through parameter estimation is known to affect the chart performance, many studies do not consider the effect of estimation on the performance of the chart. This paper contains a review of the literature that explicitly considers the effect of parameter estimation on control chart properties. Some recommendations are made and future research ideas in this area are provided.

Exact results for shewhart control charts with supplementary runs rules

This article gives a simple and efficient method, using Markov chains, to obtain the exact run-length properties of Shewhart control charts with supplementary runs rules. Average run-length comparisons are made among the Shewhart chart with supplementary runs rules, the basic Shewhart chart, and the cumulative sum (CUSUM) chart.

The Performance of Exponentially Weighted Moving Average Charts With Estimated Parameters

The exponentially weighted moving average (EWMA) control chart is typically designed assuming that standards are given for the process parameters. In practice, the parameters are rarely known, and control charts are constructed using estimates in place of the parameters. This practice can affect the control charts run-length performance in both in- and out-of-control situations. Specifically, estimation can lead to substantially more frequent false alarms and yet reduce the sensitivity of the chart to detecting process changes. In this article, the run-length distribution of the EWMA chart with estimated parameters is derived. The effect of estimation on the performance of the chart is discussed in a variety of practical scenarios.

Poisson EWMA Control Charts

An exponentially weighted moving average control chart for monitoring Poisson data is introduced. The charting procedure is evaluated using a Markov chain approximation, and its average run length is compared to other procedures for Poisson data. Figure..

The Run Length Distribution of the CUSUM with Estimated Parameters

The CUSUM control chart is a popular method used to monitor the performance of production processes. The performance of the CUSUM is generally evaluated with the assumption that the process parameters are known. In practice, the parameters are rarely known and are frequently replaced with estimates from an in-control reference sample. We discuss the run length distribution of the CUSUM with estimated parameters and provide a method for approximating this distribution and moments. We evaluate the performance of the CUSUM with estimated parameters in a variety of practical situations.

An Overview of Phase I Analysis for Process Improvement and Monitoring

We provide an overview and perspective on the Phase I collection and analysis of data for use in process improvement and control charting. In Phase I, the focus is on understanding the process variability, assessing the stability of the process, investigating process-improvement ideas, selecting an appropriate in-control model, and providing estimates of the in-control model parameters. In our article, we review and synthesize many of the important developments that pertain to the analysis of process data in Phase I. We give our view of the major issues and developments in Phase I analysis. We identify the current best practices and some opportunities for future research in this area.

Properties of the T2 Control Chart When Parameters Are Estimated

Moments of the run length distribution are often used to design and study the performance of quality control charts. In this article the run length distribution of the T2 chart for monitoring a multivariate process mean is analyzed. It is assumed that the in-control process observations are iid random samples from a multivariate normal distribution with unknown mean vector and covariance matrix. It is shown that the in-control run length distribution of the chart does not depend on the unknown process parameters. Furthermore, it is shown that the out-of-control run length distribution of the chart depends only on the statistical distance between the in-control and out-of-control mean vectors. It follows that a performance analysis can be given without knowledge of the in-control values of the parameters or their estimates. The performance of charts constructed using traditional F-distribution–based control limits is studied. Recommendations are given for sample size requirements necessary to achieve desired performance. Corrected control limits are given for designing charts with estimated parameters when large sample sizes are not available.

Distribution-free Phase I Control Charts for Subgroup Location

Much of the work in statistical quality control is dependent on the proper completion of a Phase I study. Many Phase I control charts are based on an implicit assumption of normally distributed process observations. In the beginning stages of process control, little information is available about the process and the normality assumption may not be reasonable. Existing robust and distribution-free control charts are concerned with the establishment of Phase II control limits that are robust to nonnormality or outliers from the Phase I sample. Our literature review revealed no purely distribution-free Phase I control-chart methods. We propose a distribution-free method for defining the in-control state of a process and identifying an in-control reference sample. The resultant reference sample can be used to estimate the process parameters for the Phase II procedure of choice. The proposed rank-based method is compared with the traditional X chart using Monte Carlo simulation. The rank-based method compares favorably to the X chart when the process is normally distributed and performs better than the X chart in many situations when the process distribution is skewed or heavy tailed.

Design Strategies for Individuals and Moving Range Control Charts

Individuals and moving range charts are often used when production volume is too low to justify rational subgrouping. This article considers several aspects of the design strategy for these charts. We have found that the X chart alone is nearly as effic..