Time-varying ARMA models for residual control charts

Control charts play a key role in process surveillance. In early schemes, the plotted quantities (often batch summary statistics) were assumed independent. With the advent of high frequency monitoring, this assumption became untenable and model-based approaches appeared. Here, a model is fitted to the original process to take into account the dependence structure and the process is controlled by monitoring its residuals. These residuals are typically assumed to be (Gaussian) white noise and so standard charting procedures can be applied.

However, it is apparent that in many cases, the residuals are not uncorrelated. Typically, this is because the time series models employed are insufficiently flexible to cover the full structure of the data. Often this might be because it exhibits only local stationarity. In practice, many processes can function satisfactorily in the presence of slowly-varying means or variances. Therefore there is considerable interest in establishing control procedures which can accommodate this level of manageable change and identify only more worrying disturbances. We propose here the use of charts based on the residuals of time-varying ARMA processes (specifically those with a time-varying AR component, but static MA part) for this purpose. Parameter estimation for these models is addressed by recasting them in the state space framework and applying the EM algorithm. We demonstrate that this provides a suitably flexible and computationally feasible model for practical application.