Multivariate Statistical Process Control Schemes for Controlling a Mean

Abstract

The quality of products produced and services provided can only be improved by examining the process to identify causes of variation. Modern production processes can involve tens to hundreds of variables, and multivariate procedures play an essential role when evaluating their stability and the amount of variation produced by common causes. Our treatment emphasizes the detection of a change in level of a multivariate process.

After a brief introduction, in Sect. 18.1 we review several of the important univariate procedures for detecting a change in level among a sequence of independent random variables. These include Shewhartʼs X −bar chart, Pageʼs cumulative sum, Crosierʼs cumulative sum, and exponentially weighted moving-average schemes.

Multivariate schemes are examined in Sect. 18.2. In particular, we consider the multivariate T ² chart and the related bivariate ellipse format chart, the cumulative sum of T chart, Crosierʼs multivariate scheme, and multivariate exponentially weighted moving-average schemes.

An application to a sheet metal assembly process is discussed in Sect. 18.3 and the various multivariate procedures are illustrated.

Comparisons are made between the various multivariate quality monitoring schemes in Sect. 18.4. A small simulation study compares average run lengths of the different procedures under some selected persistent shifts.

When the number of variables is large, it is often useful to base the monitoring procedures on principal components. Section 18.5 discussesthis approach. An example is also given using the sheet metal assembly data.

Finally, in Sect. 18.6, we warn against using the standard monitoring procedures without first checking for independence among the observations. Some calculations, involving first-order autoregressive dependence, demonstrate that dependence causes a substantial deviation from the nominal average run length.