A Multivariate CUSUM Procedure and a Most Active Bivariate Chart

Abstract

We propose a new multivariate cumulative sum (CUSUM) scheme whose components are the difference between the Page [10] univariate CUSUM statistic for detecting an increase and that for detecting a decrease. Our procedure also applies when monitoring principal components. We then suggest a new quality chart procedure where, at each new observation, the results for the two most active components are graphed. These two components are selected to have the largest value of the appropriate bivariate CUSUM statistic. This novel graphical approach, intended to augment the information in the chart based on all of the variables, is illustrated with an application to data from an automobile assembly process. We compare our new scheme with the classic T² charts and a number of other existing multivariate CUSUM schemes, including the scheme proposed in Crosier[2]. In our simulations, among existing procedures, Crosier[2] has the best overall average run length (ARL) performance. Our new multivariate scheme has ARL performance that is usually comparable with that of Crosier’s scheme.

Key Words:

• CUSUM statistic
• graphical procedure
• most active bivariate plot process capability plots
• multivariate quality monitoring

Additional information

Notes on contributors

Ruojia Li

Richard A. Johnson is a Professor of Statistics at the University of Wisconsin. His research and consulting interests include reliability and life length analysis, applied multivariate analysis, and applications to engineering. He is a Fellow of the American Statistical Association, a Fellow of the Institute of Mathematical Statistics, and an elected member of the International Statistical Institute. He has been editor of Statistics and Probability Letters, since in began 25 years ago, and co-author of six books and several book chapters, and over one-hundred papers in the statistical and engineering literature.

Richard A. Johnson

Ruojia Li is a research scientist at Eli Lilly & Company. She recently completed her Ph. D. degree in Statistics from the Department of Statistics at the University of Wisconsin. This research was conducted while she was a student, under the supervision of Professor Johnson. Her research interests include multivariate quality monitoring schemes and statistical applications in phamecutical research.