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Statistical Process Monitoring with Integrated Moving Average Noise 263

This paper has addresscd statistical process monitoring for the case when the stochastic component of process data is well modeled as an integrated moving average (IMA) process. First, two examples were used to make the case that a sensible approach to monitoring will depend upon details particular to each application. Sometimes, monitoring autocorrelated process data can be handled adequately with a standard control chart that has wider-than-usual action limits. In other cases a morę refined approach is needed. An idea supported by many authors is to apply standard monitoring techniques to forecast errors from time series models. We have studied this approach for the important class of IMA models.

In Section 2 we sought to understand the effect of a sudden change in the process level on IMA forecast errors. It was shown that a level shift leaves forecast errors independent with the same variance but causes their mean to shift—initially by the same amount as the change in level, but thereafter, the mean decays geometrically back to 0. The decay ratę is \-X where X can rangę from 0 representing an iid process to 1 representing random walk. This result shows that, unless the process is iid, evidence of a level shift is transitory. If the shift is not detected early, it will be missed. The closer an IMA is to a random walk, the less that evidence of a shift can build up.

ARL performance was given for four classes of monitoring schemes for detecting level shifts in IMAs. We showed how to design CUSUM and EWMA charts that have given "in control" ARLs and Iow ARLs for shifts of various sizes. We compared these schemes to a traditional Shewhart individuals chart and a scheme based on likelihood ratio statistics. Two broad conclusions came from the study: (1) unless X is smali only large level shifts can be detected within a reasonable number of periods; and (2) of the 4 classes of schemes studied, CUSUM charts have the best ARL properties. For any given X, a CUSUM chart can be designed to perform at least as well as (and sometimes better than) any of the other schemes.

Finally, in Section 4 we studied the nuli distribution of the likelihood ratio statistic used to test whether a level shift has occurred in a given set of data. These hypothesis tests are appropriate for ‘bff-line” data analysis as opposed to the "on-line" monitoring schemes studied in the rest of the paper. We developed a set of recursive equations that can be used to compute the exact nuli distribution of the likelihood ratio statistic. The recursions are computationally intensive, however, and an excellent approximation is available to calculate upper taił areas. The approximation is based on an extension of Bonferroni's ineąuality and is easily calculated using only univariate and bivariate normal probabilities.