## Cusum

Another detection algorithm taken from statistical quality control is CUSUM (Page, 1954), which stands for cumulative sum. The CUSUM algorithm is able to detect small shifts from the mean more quickly than a control chart. As its name suggests, CUSUM maintains a cumulative sum of deviations from a reference value r, which is usually the mean of the process. Let Xi be the ith observation and let \$ be the ith cumulative sum. The calculation of the cumulative sum is as follows:

If the observations Xi are close to the mean, then the cumulative sums \$ will be around zero. However, once a shift from the mean occurs, the \$ values will either increase or decrease quickly. Since we are usually only concerned with an increase in counts in biosurveillance, we can rewrite the formula for \$i to keep the cumulative sums positive. The max term in Eq. 5 ensures that the cumulative sum never goes below 0.

In the equations above, we have used the mean as the reference value r. Typically, the reference value consists of the mean plus or minus a slack value or allowance variable called K. We can rewrite Eq. 5 as:

In Eq. 6, any value within K units of r is effectively ignored. The value of K is typically set to be the midpoint between the in control process mean and the out-of-control process mean h table 14.1 Experiments with Univariate Methods expressed in terms of the standard deviation o of the in-control process: