Identifying the time of step change in the mean of autocorrelated processes

Abstract

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 analysts. Change-point methods for the uncorrelated process have been studied extensively in the literature; however, less attention has been given to change-point methods for autocorrelated processes. Autocorrelation is common in practice and is often modeled via the class of autoregressive moving average (ARMA) models. In this article, a maximum likelihood estimator for the time of step change in the mean of covariance-stationary processes that fall within the general ARMA framework is developed. The estimator is intended to be used as an “add-on” following a signal from a phase II control chart. Considering first-order pure and mixed ARMA processes, Monte Carlo simulation is used to evaluate the performance of the proposed change-point estimator across a range of step change magnitudes following a genuine signal from a control chart. Results indicate that the estimator provides process analysts with an accurate and useful estimate of the last sample obtained from the unchanged process. Additionally, results indicate that if a change-point estimator designed for the uncorrelated process is applied to an autocorrelated process, the performance of the estimator can suffer dramatically.

Keywords:

• ARMA(p, q) models
• autocorrelated processes
• change-point estimation
• stationary processes
• statistical process control (SPC)

Notes

As mentioned previously, although the proposed change-point estimator can be applied following signals from any control chart, we restrict its application in this article to the EWMA control chart. An important future study might involve assessing whether the type of control chart used significantly affects the performance of the proposed change-point estimator.

In this article, the steady-state control limits for the EWMAST control chart are employed.

These methods might include the Box–Jenkins method or an information-based methodology such as the AIC.

An important future study might be to assess the effects of phase I estimation error on the performance of τˆ.

for AR(1), for MA(1), and for ARMA(1,1).

for AR(1), for MA(1), and for ARMA(1,1).

φ∈(0.75, 1) for AR(1), for MA(1), and for ARMA(1,1).