Monitoring Processes That Wander Using Integrated Moving Average Models

TLDR

Traditional control charts assume random samples, yet process data are often temporally, spatially, or nested correlated, making the random‑sample assumption inappropriate. Monitoring such correlated data can be achieved by charting forecast errors from a time‑series model—e.g., an integrated moving average—and by evaluating four schemes (CUSUM, EWMA, Shewhart individuals, likelihood ratio) to design charts that respond to level shifts. After a level shift, the mean of forecast errors initially equals the shift but then decays geometrically to zero, and CUSUM charts can be tuned to perform at least as well as, and often better than, the other schemes, while Shewhart individuals perform much worse, graphical aids are provided for CUSUM design.

Abstract

Often the least appropriate assumption in traditional control-charting technology is that process data constitute a random sample. In reality most process data are correlated—either temporally, spatially, or due to nested sources of variation. One approach to monitoring temporally correlated data uses a control chart on the forecast errors from a time series model of the process with, possibly, a transfer-function term to model compensatory adjustments. If the time series term is an integrated moving average, then a sudden level shift in the process results in a patterned shift in the mean of forecast errors. Initially the mean shifts by the same amount as the process level, but then it decays geometrically back to 0 corresponding to the ability of the forecast to “recover” from the upset. We study four monitoring schemes—umulative sums (CUSUM's), exponentially weighted moving averages, Shewhart individuals charts, and a likelihood ratio scheme. Comparisons of signaling probabilities and average run lengths show that CUSUM's can be designed to perform at least as well as, and often better than, any of the other schemes. Shewhart individuals charts often perform much worse than the others. Graphical aids are provided for designing CUSUM's in this context.