Multivariate charts for multivariate poisson-distributed data

There has been much research involving simultaneous monitoring of several correlated quality characteristics that rely on the assumptions of multivariate normality and independence. In real world applications, these assumptions are not always met, particularly when small counts are of interest. In general, the use of normal approximation to the Poisson distribution seems to be justified when the Poisson means are large enough. A new two-sided Multivariate Poisson Exponentially Weighted Moving Average (MPEWMA) control chart is proposed, and the control limits are directly derived from the multivariate Poisson distribution. The MPEWMA and the conventional Multivariate Exponentially Weighted Moving Average (MEWMA) charts are evaluated by using the multivariate Poisson framework. The MPEWMA chart outperforms the MEWMA with the normal-theory limits in terms of the in-control average run lengths. An extension study of the two-sided MPEWMA to a one-sided version is performed; this is useful for detecting an increase in the count means. The results of comparison with the one-sided MEWMA chart are quite similar to the two-sided case. The implementation of the MPEWMA scheme for multiple count data is illustrated, with step by step guidelines and several examples. In addition, the method is compared to other model-based control charts that are used to monitor the residual values such as the regression adjustment. The MPEWMA scheme shows better performance on detecting the mean shift in count data when positive correlation exists among all variables.