Robust design of residual-based EWMA control charts

Residual-based control charts for autocorrelated processes are known to be sensitive to autoregressive moving average (ARMA) modeling errors and often suffer from inflated false alarm rates. We propose a design method that mitigates this problem by modifying the control limits based on the level of model uncertainty. Compared to existing robust statistical process control (SPC) methods, this approach achieves a more suitable balance between false alarms and control chart power. Modeling errors exist in practice because a model is estimated from collected data Residuals generated from an estimated model with parameter modeling errors are autocorrelated. When an exponentially weighted moving average (EWMA) filter is applied to the autocorrelated residuals, the actual EWMA variance will be different from the ideal EWMA variance that assumes no modeling errors. Our design approach quantifies the differences between the actual and ideal variances and modifies the control limits accordingly. The actual variance of the EWMA statistic is represented using a second-order Taylor approximation in this proposed method. After taking the expectation of the second-order approximation, with respect to the parameter uncertainty, the result is an expression for the expected EWMA variance as a function of (among other things) the parameter estimation error covariances. When designing the control limits, we use the square root of the expected EWMA variance instead of the normal EWMA standard deviation. To calculate the expected EWMA variance, the only information that is needed is the parameter estimates and their error covariance matrix. Most commercial software packages for time series mo deling will produce this information automatically. Moreover, for low order ARMA processes that have simple, closed-form expressions for the parameter covariance, closed-form expressions for the expected EWMA variance can also be determined. To evaluate the proposed method, it is compared to two existing robust SPC methods. We compare the following aspects: control limits width for the same sample size, sensitivity of in-control ARLs, performance of out-of-control ARLs. A Shewhart individual chart, which is a special case of an EWMA chart, is included in the comparison of out-of-control performances. The proposed method was more robust in terms of the sensitivity analysis of the incontrol ARLs for the same levels of parameter uncertainty. The interpretation is that the proposed method is more likely to provide the desired in-control ARL and reduce the risk of excessive false alarms. The comparisons of out-of-control ARL performance indicate that the proposed method generally performs better because it provides in-control ARLs that are closer to the desired values, in addition to possessing a less severe loss of power in detecting mean shifts.