Monitoring the covariance matrix via penalized likelihood estimation

In many industrial multivariate quality control applications, based on the engineering and operational understanding of how the process works, when the process variability is out of control it is typically the case that changes only occur in a small number of elements in the covariance matrix. Under such a premise, we propose a new Phase II Shewhart chart for monitoring changes in the covariance matrix of a multivariate normal process. The new control chart is essentially based on calculating the likelihood ratio of testing the hypothesis that the in-control covariance matrix is equal to a known covariance matrix, where the unknown covariance matrix that appears in the likelihood ratio is replaced by an estimate obtained from a penalized likelihood function. The penalized likelihood function is derived by adding an L1 penalty function to the usual likelihood. The performance of the proposed chart is evaluated based on simulations and compared with that of several existing Shewhart charts for monitoring the covariance matrix. The simulation results indicate that the proposed chart outperforms existing charts. A real example from the semiconductor industry is presented and analyzed using the proposed chart and other existing charts. Potential future research directions are also discussed.