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Statistical Process Monitoring with Integrated Moving Average Noise 2S1 2. CUSUM: A (two sided) CUSUM scheme is based on a high side statistic H, and a Iow side statistic L,:

H, = max{0,a, - k +    }

L_t = max{0,-a_l - k +    }

where H₀ and L₀ are initialized at 0. H, is sensitive to changes causing an increase in the mean of a_t while L_t is sensitive to changes causing a decrease. The scheme signals when max{L_t,H,\ exceeds an action limit h. The

reference level k and the action limit h are design parameters. Typically k is set between 0.25a and 1.5ct. For a given k, h can be selected to produce a desired ARL when no shift is present. The left panel of Figurę 4 shows curves of h versus k for three values of ARL₀: 100, 250, and 500. Given ARL₀ and k, the

appropriate curve can be used to find h. The choice of k gives some flexibility in ARL performance when shifts occur. Setting h = 0 mimics a Shewhart individuals chart with “&-sigma” limits applied to the forecast errors.

Figurę 5 is an aid for choosing a value of k for a particular application. The panels in Figurę 5 show ARLs for various values of X, p (the shift size), and k with h selected (from Figurę 4 ) to provide a given ARL₀. The panels are arranged so that columns represent values of X from 0 to 0.5 and rows represent values of p from 0.5 to 4. Each panel has 3 curves. The upper, middle and lower curves correspond to charts designed to have ARL₀ values of

100, 250 and 500 respectively. Each curve shows ARLs for steps of size p for various values of k.

To choose a value for k im particular application, look at the column of plots corresponding to X nearest the estimated value. Now focus on the ARL curve in each panel of that column that represents a value of ARL₀ closest to

the one desired. Finally, visually choose a value of k that gives Iow ARLs for the sizes of shifts that are most important. This step will possibly involve trading off performance for shifts of one size for better detection of shifts of another.

As an example, suppose X is estimated to be near 0.1. From the curves in the second column of plots, we see that the middle panel is most interesting. Shifts of 0.5 standard deviations are nearly impossible to detect no matter what value of k is used. Their ARL curves drop only slightly below ARL₀. On the other hand shifts of 4 standard deviations are detected ąuickly as long as k is at