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Statistical Process Monitoring with Integrated Moving Average Noise 249 where N, is an IMA. If equation (1) (with X = X for simplicity) is applied to Z, the residuals a, will continue to be independent normal variates with

variance a² but their mean will not be zero in periods 150 and following. In fact,

That is, the mean shifts up to p in period 150 and decays exponentially to 0 thereafter. The decay is faster for larger X.

By looking for this pattemed mean in the seąuence of residuals, we can better judge the evidence of a shift in level. This is because we do not have to mentally untangle the effects of autocorrelation from the effects of a possible shift. The residuals are not correlated and a level shift in the IMA creates a simple pattem in the mean of the residuals. The pattem is seen in the third column of plots in Figurę 3 which shows the residuals computed from the IMAs with shifts. The pattemed mean is evident in the residuals and is shown (shifted downward for clarity) in the lower portion of each residual plot.

Monitoring Schemes and ARL Comparisons

This section describes and compares several schemes for using residuals to monitor IMAs for step shifts. For the special cases of iid observations (A. = 0) and a random walk (X = 1) good monitoring schemes seem obvious.

In the iid case the residuals are identical to the process itself. A step in the level of the process is therefore a step in the mean of the residuals. This is the situation traditionally addressed in studies of control chart performance. The literaturę shows that for smali and medium sized shifts (0 to roughly 2.5ct), it is hard to beat the ARL performance of properly designed CUSUM and EWMA charts. For large shifts Shewhart individuals charts perform best. Combining an EWMA or a CUSUM chart with a Shewhart individuals chart results in a control scheme with good ARL performance for both large and smali shifts. A comparable (and even morę traditional) scheme is to supplement a Shewhart individuals (or X) chart with runs rules. ARLs for smali shifts using this scheme are usually between 10% and 50% larger than