Control charts for variability monitoring in high-dimensional processes

Monitoring process variability is associated with detecting changes in the covariance matrix of a multivariate normal process. Most monitoring methods estimate the sample covariance matrix and compare it with the in control covariance matrix that is mostly priori known based on the sufficient historical data. However, when the sample size is smaller than the number of variables, the sample covariance matrix is not applicable to estimate the covariance matrix, since the matrix may not be positive semi-definite. In this paper, we propose a new control chart for monitoring changes in the covariance matrix when the sample size is smaller than the number of variables. The proposed chart is based on the ridge penalized likelihood ratio. It detects general changes, without sparsity assumption, in the covariance matrix efficiently when the sample size is small, while other existing penalized likelihood-based methods are expected to detect only sparse changes in the covariance matrix. The superiority of the proposed chart is demonstrated through an average run length performance to variety in shift patterns. The proposed chart also maintains a low computational complexity. These differentiated properties of the proposed chart were proved through numerous simulation studies and in a real example from the semiconductor industry.