Generalization of Shewhart X~-chart

In univariate quality control, the X~-Charts have been useful in determining whether the process mean is in control or not. It would be very useful to have a similar chart applied to the multivariate case. The existing methods do not provide all the information that a quality control practitioner would like to possess such as the indication of which variables are causing the process to be out of control. In this paper we propose a method which allows us to simultaneously control the overall process quality characteristics and to identify the responsible variables leading to an out of control condition. This method is based on the adequate selection of the symmetric square root of the correlation matrix. The associated critical region is also discussed. The process considered is assumed to be multivariate normal with parameters known from historical data or estimated from a large sample. We call this method, "The Generalized Shewhart Chart (GS Chart)", because it reduces to the Shewhart Chart when the process involves only one variable. The procedure has been illustrated with the help of two examples. The effectiveness of this method is compared with existing methods in this paper.