In multivariate statistical process control, it is recommendable to run two individual charts: one for the process mean vector and another one for the covariance matrix. The resulting joint scheme provides a way to satisfy “Shewhart's dictum” that proper process control implies monitoring both process location and spread.
The multivariate quality characteristic is deemed to be out-of-control whenever a signal is triggered by either individual chart of the joint scheme. Consequently, a shift in the mean vector can be misinterpreted as a shift in the covariance matrix and vice versa. Compelling results are provided to give the quality control practitioner an idea of how joint schemes for the mean vector and covariance matrix are prone to trigger misleading signals that will likely lead to a incorrect diagnostic of which parameter has changed.

CEMAT - Center for Computational and Stochastic Mathematics