A Bayesian Approach to Change Point Estimation in Multivariate SPC

Abstract

A Bayesian procedure is developed to estimate the time of a change in the process mean vector for a multivariate process, given that an out-of-control signal was raised on a multivariate control chart. In addition, we can infer simultaneously which variable(s) had a change in mean value, when the change occurred, and the value of the changed mean. All three problems (inferring change point time, variables that shifted, and new values for the shifted variables) are addressed in a single statistical model. Markov chain Monte Carlo (MCMC) methods, through the software WinBUGS, are used to estimate parameters of the change point models. To identify the mean shift in a process with more than two variables, we propose a branch-and-bound search algorithm so that MCMC can be carried out with a predictable computing time in each search step. A simulation study shows that the Bayesian approach has similar performance compared to the maximum likelihood estimation (MLE) in terms of identifying the true change point location when a noninformative prior is assumed; however, it can perform better when proper prior knowledge is incorporated into the estimation procedure. The Bayesian approach provides full posterior distributions for the model and change point, which can contain information that is not available in a likelihood analysis.