Distribution-free multivariate process control based on log-linear modeling

Abstract

This paper considers Statistical Process Control (SPC) when the process measurement is multivariate. In the literature, most existing multivariate SPC procedures assume that the in-control distribution of the multivariate process measurement is known and it is a Gaussian distribution. In applications, however, the measurement distribution is usually unknown and it needs to be estimated from data. Furthermore, multivariate measurements often do not follow a Gaussian distribution (e.g., cases when some measurement components are discrete). We demonstrate that results from conventional multivariate SPC procedures are usually unreliable when the data are non-Gaussian. Existing statistical tools for describing multivariate non-Gaussian data, or transforming the multivariate non-Gaussian data to multivariate Gaussian data, are limited, making appropriate multivariate SPC difficult in such cases. In this paper, we suggest a methodology for estimating the in-control multivariate measurement distribution when a set of in-control data is available, which is based on log-linear modeling and which takes into account the association structure among the measurement components. Based on this estimated in-control distribution, a multivariate CUSUM procedure for detecting shifts in the location parameter vector of the measurement distribution is also suggested for Phase II SPC. This procedure does not depend on the Gaussian distribution assumption; thus, it is appropriate to use for most multivariate SPC problems.

Keywords:

• Discrete measurements
• log-linear modeling
• multivariate distribution
• non-Gaussian data
• nonparametric procedures
• statistical process control

Acknowledgement

We thank two referees for some constructive comments and suggestions, which greatly improved the quality of the paper. This research was supported by an NSF grant.