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against the sample number i. Now, Si is called the cumulative sum up to and including the ith sample. Because they combine information from several samples, cumulative sum charts are more effective than Shewhart charts for detecting small process shifts. Furthermore, they are particularly effective with samples of n 1. This makes the cumulative sum control chart a good candidate for use in the chemical and process industries where rational subgroups are frequently of size 1, as well as in discrete parts manufacturing with automatic measurement of each part and online control using a microcomputer directly at the work center. If the process remains in control at the target value 0, the cumulative sum de ned in equation 16-29 should uctuate around zero. However, if the mean shifts upward to some value 1 0, say, an upward or positive drift will develop in the cumulative sum Si. Conversely, if the mean shifts downward to some 1 0, a downward or negative drift in Si will develop. Therefore, if a trend develops in the plotted points either upward or downward, we should consider this as evidence that the process mean has shifted, and a search for the assignable cause should be performed. This theory can easily be demonstrated by applying the CUSUM to the chemical process concentration data in Table 16-3. Since the concentration readings are individual measurements, we would take Xj Xj in computing the CUSUM. Suppose that the target value for the concentration is 0 99. Then the CUSUM is Si a 1Xj 992 992

To get back at the data inside the kernel, use the sample function sampleNearest() as you might use an array lookup in a 2D array. If you decided to store different information in the channels of each pixel, you can get this out, and back in, by swizzling. Remember that you can use multiple image inputs. Simply pass the image you want to use as input to the sample function, and you can mix in data from multiple sources. After the next section, you ll see how to read and execute shaders, in Example 38-3.

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Table 16-7 shows the computation of this CUSUM, where the starting value of the CUSUM, S0, is taken to be zero. Figure 16-19 plots the CUSUM from the last column of Table 16-7. Notice that the CUSUM uctuates around the value of 0. The graph in Fig. 16-19 is not a control chart because it lacks control limits. There are two general approaches to devising control limits for CUSUMS. The older of these two methods is the V-mask procedure. A typical V mask is shown in Fig. 16-20(a). It is a V-shaped notch in a plane that can be placed at different locations on the CUSUM chart. The decision procedure consists of placing the V mask on the cumulative sum control chart with the point O on the last value of si and the line OP parallel to the horizontal axis. If all the previous cumulative sums, s1, s2, . . . , si 1, lie within the two arms of the V mask, the process is in control. However, if any si lies outside the arms of the mask, the process is considered to be out of control. In actual use, the V mask would be applied to each new point on the CUSUM chart as soon as it was plotted. In the example shown in Fig. 16-20(b), an upward shift in the mean is indicated, since at least one of the points that have occurred earlier than sample 22 now lies below the lower arm of the mask, when the V mask is centered on the thirtieth observation. If the point lies above the upper arm, a downward shift in the mean is

Use ShaderJob to control how a kernel is run. With ShaderJob, it s up to you to provide all the proper inputs and an appropriate target for the kernel s output. These may be in any combination of the types discussed in Preparing data for Pixel Bender. To create a ShaderJob, initialize it with a Shader parameter. You can set the target, width, and height in the constructor or afterward using those parameter names. Before running the job,

Table 16-7

you need to specify a valid target for the output, but width and height may be omitted if this is a BitmapData instance.

CUSUM Computations for the Chemical Process Concentration Data in Table 16-3 xi 102.0 94.8 98.3 98.4 102.0 98.5 99.0 97.7 100.0 98.1 101.3 98.7 101.1 98.4 97.0 96.7 100.3 101.4 97.2 101.0 xi 99 3.0 4.2 0.7 0.6 3.0 0.5 0.0 1.3 1.0 0.9 2.3 0.3 2.1 0.6 2.0 2.3 1.3 2.4 1.8 2.0 si (x i 99) 3.0 1.2 1.9 2.5 0.5 0.0 0.0 1.3 0.3 1.2 1.1 0.8 2.9 2.3 0.3 2.0 0.7 1.7 0.1 1.9 si

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