figure 14.5 Synthetic data that account for seasonality,noise from a finite sample, day-of-week effects, and the effects of holidays.

cumulative distribution function for a standard normal-that is, a normal distribution with mean 0 and variance 1.The max term in Eq. 4 ensures that only counts greater than the mean will have an alarm value greater than 0.5.

Alarm Level = O

The constant k determines how sensitive the control chart will be. With a lower value of k, the control chart becomes more sensitive as it will trigger an alarm with less of a deviation from the mean of the process. Two typical values used in industrial applications of k are 2 or 3. Setting the constant k to be 2 corresponds to a 5% chance that an in control process is mistakenly said to be out of control while setting k to be 3 corresponds to a 1% chance. These percentages are obtained from the assumption that the observations X1, X2, ..., XN follow a normal distribution.

Figure 14.6 shows the performance of a control chart on our example from Figure 14.1. Figure 14.7 shows the same data (with the same injected ramp) on a large scale. This figure shows many interesting phenomena. The height of the dark gray "predict'' line at time t shows the historical average of the count based on all data previous to time t. The light gray line shows the upper limit of the control chart.The good news is that the counts do indeed exceed the upper limit during the synthetic outbreak. But the bad news is that there are many other occasions where this also happens, especially during the peaks in fall 2003 and winter 2004. The major problem with the control chart in this form is that it does not anticipate the seasonal effects, and when it is in the middle of a winter peak, it issues an alarm frequently. The next section alleviates this problem.

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