Identifying the time of polynomial drift in the mean of autocorrelated processes

Control charts are used to detect changes in a process. Once a change is detected, knowledge of the change point would simplify the search for and identification of the special cause. Consequently, having an estimate of the process change point following a control chart signal would be useful to process engineers. This paper addresses change point estimation for covariancestationary autocorrelated processes where the mean drifts deterministically with time. For example, the mean of a chemical process might drift linearly over time as a result of a constant pressure leak. The goal of this paper is to derive and evaluate a maximum likelihood estimator (MLE) for the time of polynomial drift in the mean of autocorrelated processes. It is assumed that the behavior in the process mean over time is adequately modeled by the kth-order polynomial trend model. Further, it is assumed that the autocorrelation structure is adequately modeled by the general (stationary and invertible) mixed autoregressive-moving-average model. The estimator is intended to be applied to data obtained following a genuine control chart signal in efforts to help pinpoint the root cause of process change. Application of the estimator is demonstrated using a simulated data set. The performance of the estimator is evaluated through Monte Carlo simulation studies for the k = 1 case and across several processes yielding various levels of positive autocorrelation. Results suggest that the proposed estimator provides process engineers with an accurate and useful estimate for the last sample obtained from the unchanged process.