### Multivariate SPC

Session Chair: Bill Woodall, Virginia Tech**Multivariate Nonparametric Shewhart-Type Control Charts**

Jeffrey M. Boone and Subhabrata Chakraborti

The University of Alabama

Abstract: In many quality control applications, the distributional assumptions that are necessary to correctly apply the traditional parametric control charts are either not met or there is simply not enough information or evidence to verify the assumption. It is well known that in situations like this the performance of many parametric control charts can be seriously degraded. Thus, control charts that do not require a specific distributional assumption, called nonparametric or distribution-free charts, are desirable. In this paper, two multivariate nonparametric Shewhart-type control charts are proposed. They are relatively simple to use and are based on the multivariate forms of the sign and the Wilcoxon signed-rank tests. We also consider two multiple comparisons-type charts based on the maximum of multiple univariate sign and Wilcoxon signed-rank statistics, respectively. These charts have the added benefit of determining which variable or variables shifted. Performance of the proposed charts is studied in a simulation study. Some observations and recommendations are made.

**Real-Time Multivariate Process and Product Monitoring**

David E. Stevens

Eastman Chemical Company

Modern manufacturing processes are highly automated and have a multitude of process instrument readings and product results available in real-time every few seconds. Many of the critical instrument reading and product results are correlated which makes using the traditional Shewhart Control Chart on the individual readings less effective for monitoring the process. Rather, a real-time multivariate process and product monitoring technique can be implemented using a simple graphical user interface (GUI) that enables the process operator to drill down and determine which readings are causing the process to be "out of statistical control."

This presentation will discuss the Shewhart Control chart pitfalls when monitoring a process and then discuss how principal component analysis (PCA) and partial least squares (PLS) can be used to implement real-time multivariate process and product monitoring. A real-time multivariate process and product monitoring demonstration will also be presented using a software application developed by the author.

**A Multistep, Cluster-Based Multivariate Chart for Retrospective Monitoring of Individuals**

J. Marcus Jobe and Michael Pokojovy

Miami University and Universitaet Konstanz

Abstract: The presence of several outliers in an individuals retrospective multivariate control chart distorts both the sample mean vector and covariance matrix. To overcome the distortion or masking effects on outlier detection, we propose a computer-intensive multistep cluster-based method. Comparisons of our new method with classical and robust estimation procedures are made. Additional comparisons based on real data are given.

**Self-Starting Multivariate Control Charts for Location and Scale**

Edgard M. Maboudou

University of Central Florida

Abstract: Multivariate control charts are very eective when monitoring several correlated characteristics. Proper process control consists of monitoring both mean and variability. In multivariate settings, the multivariate exponentially weighted moving average (MEWMA) is ideal to monitor the mean vector. Similarly, the recently-proposed multivariate exponentially weighted moving covariance matrix (MEWMC) detects changes in the covariance matrix. Using these two charts simultaneously provides a way to satisfy Shewart's dictum of always monitoring both location and variability. However, both charts were established under the assumption that the parameters are known a priori. In practice, the process parameters are commonly unknown so that the assumption of known parameters does not hold. The usual approach to this difficulty is to use estimated values from a Phase I study to calibrate the charts. However, using estimates in place of known parameters adds a random element to the run length distribution, and harms chart performance. This is now known to be true even for large calibration samples. A different approach is to use self-starting methods, which correctly studentize the incoming stream of process readings, and can provide exact control right from startup.