Differential Smoothing in the Bivariate Exponentially Weighted Moving Average Chart

The multivariate exponentially weighted moving average (MEWMA) control chart proposed by Lowry et al. (1992) has become one of the most widely used charts to monitor multivariate processes. Its simplicity, combined with its high sensitivity to small and moderate process mean jumps, is at the core of its appeal. Lowry et al. (1992) advocated equal smoothing of each quality variable unless there is an a priori reason to weight quality characteristics differently. However, one may have situations where differential smoothing may be justified. For instance: (a) departures in process mean may be different across quality variables, (b) some variables may evolve over time at a much different pace than other variables, and (c) the level of correlation between variables could vary substantially. Here, we assess the performance of the differentially smoothed MEWMA chart. The case of two quality variables (BEWMA) is discussed in detail. A bivariate Markov-chain method that uses conditional distributions is developed for average run-length (ARL) calculations. The proposed chart is shown to perform at least as well as Lowry et al. (1992)'s chart and noticeably better in many mean-jump directions. Comparisons with the recently introduced double-smoothed BEWMA chart and the use of univariate charts for the independent case show that the proposed differentially smoothed BEWMA chart has superior performance.