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Strategies for Statistical Monitoring of Integral Control

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where the forecast is declared biased if the S2 value for any time is larger than a critical value K3. In this case, Mastrangelo (1993) recommends a K3 value of 0.5 and that to reduce the number of false alarms, two consecutive points must exceed K3 before declaring that the forecast is biased.

To summarize this SPC control scheme, a predicted error EWMA scheme will be employed on the control actions with an associated "two-points-in-a-row" tracking signal. The predicted error EWMA is morę effective at detecting sustained shifts in the process whereas the tracking signal is morę effective at detecting trends.

In order to evaluate the effectiveness of various control schemes described above, it is necessaiy to develop a performance measure for comparison purposes. In this setting, a weighted performance measure is employed sińce only the number of observations to detect and remove an assignable cause is considered in this study. The weighted performance (PM) is defined as follows

_{pM =} 250 PM1 +PM2+(500-t)PMl

“500    ⁽ *

where PM1 is the performance measure of the first 250 time periods and is defined as PM1 = SSE/250, where SSE is the sum of squared error, PM2 is the performance from time period 251 to t, also measured as average sąuared error, and t is the time period in which the assignable cause is detected and removed. For comparison purposes, we have introduced a change in the mean at observation 251 as this was done in the first study. In Figurę 1, the PM of the EPC scheme alone is 3.23.

Figurę 2 is a plot of the first 252 residuals denoted by from fitting an EWMA with 1 = 0.9 to the control actions. The reason that only 252 residuals are shown instead of 500 is the combined EWMA and tracking signal scheme signaled that the shift in the process was detected and eliminated at this observation. Figurę 3 is a plot of the prediction error EWMA with X, = 0.1 and Figurę 4 is a plot of the smoothed error tracking signal denoted by sts(t) with K3 = 0.5 and a two-in-a-row rule added. This combined scheme quickly detects the assignable cause and the associated performance measure is 2.564 which is an improvement over the EPC scheme. Figures 5 and 6 give the results of the CUSUM scheme using values of K = la and 2a respectively. In these cases