A comparison of multivariate control charts for individual observations

For multivariate statistical process control with individual observations the usually recommended procedure in the retrospective phase is the Hotelling's T-2 control chart. All the observations are pooled to estimate the mean vector and covariance matrix. An out-of-control signal results if the T-2 value of any observation exceeds an upper control limit. We show this procedure is not effective in detecting a shift in the mean vector because the covariance matrix is badly estimated. Somewhat surprisingly, the signal probability decreases with increasingly severe shifts in the mean vector. We compare several alternative methods for estimating the covariance matrix and recommend a procedure analogous to the use of moving ranges in the univariate case. This procedure uses the vector differences between successive observations to estimate the in-control covariance matrix of the process. With this more robust estimate, step and ramp shifts in the mean vector are more likely to be detected in the retrospective phase.