A Bayesian Approach for Interpreting Mean Shifts in Multivariate Quality Control

Abstract

Multivariate quality characteristics are often monitored using a single statistic or a few statistics. However, it is difficult to determine the causes of an out-of-control signal based on a few summary statistics. Therefore, if a control chart for the mean detects a change in the mean, the quality engineer needs to determine which means shifted and the directions of the shifts to facilitate identification of root causes. We propose a Bayesian approach that gives a direct answer to this question. For each mean, an indicator variable that indicates whether the mean shifted upward, shifted downward, or remained unchanged is introduced. Prior distributions for the means and indicators capture prior knowledge about mean shifts and allow for asymmetry in upward and downward shifts. The mode of the posterior distribution of the vector of indicators or the mode of the marginal posterior distribution of each indicator gives the most likely scenario for each mean. Evaluation of the posterior probabilities of all possible values of the indicators is avoided by employing Gibbs sampling. This renders the computational cost more affordable for high-dimensional problems. This article has supplementary materials online.

KEY WORDS:

• Fault isolation
• Gibbs sampling
• Hierarchical Bayes
• Interpretation of out-of-control signal
• Multivariate statistical process control
• Variable selection

ACKNOWLEDGMENTS

Tan's research was supported by NSF grants DMS0705261 and DMS1007574, and Shi's research was partially supported by NSF grant CMMI-0927574. We thank the editor and two anonymous referees for many valuable comments that helped improve the article significantly.