Monitoring multivariate process variability with individual observations via penalised likelihood estimation

Abstract

Excessive variation in a manufacturing process is one of the major causes of a high defect rate and poor product quality. Therefore, quick detection of changes, especially increases in process variability, is essential for quality control. In modern manufacturing environments, most of the quality characteristics that have to be closely monitored are multivariate by the nature of the applications. In these multivariate settings, the monitoring of process variability is considerably more difficult than monitoring a univariate variance, especially if the manufacturing environment only allows for the collection of individual observations. Some recent charts, such as the MaxMEWMV chart, the MEWMS chart and the MEWMC chart, have been proposed to monitor process variability specifically when the subgroup size is equal to 1. However, these methods do not take into account the engineering and operational understanding of how the process works. That is, when the process variability goes out of control, it is often the case that changes only occur in a small number of elements of the covariance matrix or the precision matrix. In this work, we propose a control charting mechanism that enhances the existing methods via penalised likelihood estimation of the precision matrix when only individual observations are available for monitoring the process variability. The average run length of the proposed chart is compared with that of the MaxMEWMV, MEWMS and MEWMC charts. A real example is also presented in which the proposed chart and the existing charts are applied and compared.

Keywords:

• control charts
• quality control
• covariance matrix
• penalty function
• likelihood ratio test
• penalised likelihood function
• phase II monitoring
• sparsity

Acknowledgements

We thank the editor and a referee for their comments and suggestions that improved the presentation of the paper. Dr. Bo Li's work is supported, in part, by the National Natural Science Foundation of China (NSFC) (grants 10801086, 70831003 and 71072012). Dr. Kaibo Wang's work is supported by the NSFC under grant 71072012.