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The CRC function in Tahle 3.13 is a cyclic redundancy check, which can be computed as shown in Table 3.14. This is the same CRC calculation used for error detection in t h e ZIP compression process and, undoubtedly, it is reused here for efficiency. A more efficient way t o carry out the CRC calculation in Table 3.14 is discussed in Problem 18, where it is shown that there exists a table, CRCtable[b], where for any byte b, we have

Logistic Regression Linear regression often works very well when the response variable is quantitative. We now consider the situation where the response variable takes on only two possible values, 0 and 1. These could be arbitrary assignments resulting from observing a qualitative response. For

0.4

// update values of X ,Y ,2 update(X, y,2,P ) X = CRC(X,p) Y = (Y (X)24...31) 134,775,813 . Z = CRC(Z,(Y)o 7 ) ... end uDdate

0.6 2

Figure S11-5 Plot of residuals versus tted values yi for the transformed model for the windmill data.

Figure S11-6 Normal probability plot of the residuals for the transformed model for the windmill data.

// //

11-7

The Print menu offers options for controlling which images print when the camera is connected to a printer. The Canon PowerShot G11 supports the DPOF (Digital Print

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example, the response could be the outcome of a functional electrical test on a semiconductor device for which the results are either a success, which means the device works properly, or a failure, which could be due to a short, an open, or some other functional problem. Suppose that the model has the form Yi

b is a byte CRC(X,b) X=X@b for i =0 to 7 i f X is odd then X = ( X >> 1) CE Oxedb88320 else // X is even x=x>>1

(S11-6)

and the response variable Yi takes on the values either 0 or 1. We will assume that the response variable Yi is a Bernoulli random variable with probability distribution as follows:

Problem 19 shows that there is a table, CRCinverse, that is the inverse of CRCtable in the sense that if

0 11

This means that the expected response given by the response function E(Yi) 0 1xi is just the probability that the response variable takes on the value 1. There are some substantive problems with the regression model in Equation S11-6. First, note that if the response is binary, the error terms i can only take on two values, namely,

Consequently, the errors in this model cannot possibly be normal. Second, the error variance is not constant, since

B = (A)0 23 CE CRCtable[(A) ... 2

i 11

since E1Yi 2 0 1xi i. This indicates that the variance of the observations (which is the same as the variance of the errors because i Yi i, and i is a constant) is a function of the mean. Finally, there is a constraint on the response function, because 0 E 1Yi 2

~ 3 CE b], 1

11-8

Order Format) industry standard developed for use among digital camera manufacturers and printers When you use the G11 s DPOF features, you can select the images or range of images to be printed, the number of copies, whether to include the date and lename of each image, and whether to print an index print of the selected images from the memory card Although you may use DPOF with many home photo printers, some retail photo processors support the standard, too DPOF support among photo kiosk NOTE printers is limited Your options on the Print menu include: Print The Print option is dimmed unless the G11 is connected to a printer When the camera is connected to a printer using the supplied USB cable, this option is available and allows you to access the Print menu Select Images & Quantity.

This restriction can cause serious problems with the choice of a linear response function, as we have initially assumed in Equation S11-6. It would be possible to t a model to the data for which the predicted values of the response lie outside the 0, 1 interval. Generally, when the response variable is binary, there is considerable empirical evidence indicating that the shape of the response function should be nonlinear. A monotonically increasing (or decreasing) S-shaped (or reverse S-shaped) function, such as shown in Figure S11-7, is usually employed. This function is called the logit response function, and has the form E1Y 2 or equivalently, E1Y 2 1 exp3 1

(3.14) (3.15)

1 x2 4

(S11-7)

(S11-8)

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