Process monitoring for covariance matrices with latent structures

The monitoring of process variability is generally achieved by prompt detection of changes in the covariance matrix of multiple correlated quality characteristics. In the literature, there have been many approaches successfully developed for this purpose. Notably, in many chemical processes or some with a special or symmetric structure, involved high-dimensional data are strongly linearly correlated, so that they can be regarded to be determined by a number of low-dimensional latent variables, and hence, have a small intrinsic dimension. In this sense, the observed high-dimensional data have actually a latent structure, formed by latent variables and sensor errors. If such a latent structure can be fully exploited, more efficient monitoring of covariance matrices can be achieved. To this end, this article refines the structure of covariance matrices and develops a series of charting statistics, which are able to efficiently detect shifts in the covariance matrices of latent variables and of sensor errors. Based on the sensitivity analysis of the proposed charting statistics, we provide their choices, as well as a diagnostic scheme to determine the source of shifts. Monte Carlo simulations have demonstrated their superiority over existing alternatives in detecting covariance matrix shifts in latent variables or in sensor errors.