Control charts for monitoring a Poisson hidden Markov process

Abstract

Monitoring stochastic processes with control charts is the main field of application in statistical process control. For a Poisson hidden Markov model (HMM) as the underlying process, we investigate a Shewhart individuals chart, an ordinary Cumulative Sum (CUSUM) chart, and two different types of log-likelihood ratio (log-LR) CUSUM charts. We evaluate and compare the charts' performance by their average run length, computed either by utilizing the Markov chain approach or by simulations. Our performance evaluation includes various out-of-control scenarios as well as different levels of dependence within the HMM. It turns out that the ordinary CUSUM chart shows the best overall performance, whereas the other charts' performance strongly depend on the particular out-of-control scenario and autocorrelation level, respectively. For illustration, we apply the HMM and the considered charts to a data set about weekly sales counts.