On the Phase I analysis for monitoring time-dependent count processes

In designing a control chart for online process monitoring, a Phase I analysis is often conducted as a first step to estimate the unknown process parameters. It is based on historical data, where parameter estimation, chart design, and data filtering are iterated until a stable and reliable chart design is obtained. Researchers sometimes neglect the effects of Phase I analysis by assuming that process parameters are known but directly evaluate the performance of a control chart in Phase II (online process monitoring). In this research, the Phase I analysis of time-dependent count data stemming from Poisson INAR(1) and binomial AR(1) processes is considered. Due to data filtering, parameter estimation in Phase I analysis is challenging, especially when the data are autocorrelated. In this regard, solutions on how to modify the method of moments, least squares, and maximum likelihood estimation are presented. Performance of these estimators in Phase I as well as performance of designed control charts in Phase II are evaluated, and recommendations are provided. A real-world example on IP counts is used to illustrate the Phase I and Phase II control chart implementations.