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Table 11-1 Oxygen and Hydrocarbon Levels Observation Number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Hydrocarbon Level x (%) 0.99 1.02 1.15 1.29 1.46 1.36 0.87 1.23 1.55 1.40 1.19 1.15 0.98 1.01 1.11 1.20 1.26 1.32 1.43 0.95 Purity y (%) 90.01 89.05 91.43 93.74 96.73 94.45 87.59 91.77 99.42 93.65 93.54 92.52 90.56 89.54 89.85 90.39 93.25 93.41 94.98 87.33

n.- 1

100 98 96 Purity ( y) 94 92 90 88 86 0.85

1.15 1.25 1.35 Hydrocarbon level ( x)

Figure 11-1 Scatter diagram of oxygen purity versus hydrocarbon level from Table 11-1.

(3.3)

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of the data in Table 11-1. This is just a graph on which each (xi, yi) pair is represented as a point plotted in a two-dimensional coordinate system. This scatter diagram was produced by Minitab, and we selected an option that shows dot diagrams of the x and y variables along the top and right margins of the graph, respectively, making it easy to see the distributions of the individual variables (box plots or histograms could also be selected). Inspection of this scatter diagram indicates that, although no simple curve will pass exactly through all the points, there is a strong indication that the points lie scattered randomly around a straight line. Therefore, it is probably reasonable to assume that the mean of the random variable Y is related to x by the following straight-line relationship: E1Y 0 x2

where the slope and intercept of the line are called regression coef cients. While the mean of Y is a linear function of x, the actual observed value y does not fall exactly on a straight line. The appropriate way to generalize this to a probabilistic linear model is to assume that the expected value of Y is a linear function of x, but that for a xed value of x the actual value of Y is determined by the mean value function (the linear model) plus a random error term, say, Y

One problem with the sequence in (3.3) is that the linear complexity is, in a sense, concentrated in just a single bit. That is, the linear complexity is zero, until the last bit is processed, then the complexity jumps from the minimum to the maximum possible value. Recognizing this, Rueppel [125] proposes the linear complexity profile as a practical measure of the quality of a keystream. This profile is simply the graph of the linear complexity L of S O , s1,. . , s k for each k = 0 , 1 , 2 , .. .. The required L values are obtained . when the linear complexity of s is computed using the Berlekamp-Massey Algorithm, so it is efficient to determine such a profile. Rueppel has shown that most sequences have a linear complexity profile that follows the n/2 line closely but irregularly, and he proposes this as a criteria for judging the quality of a keystream. Figure 3.4 illustrates a linear complexity profile that satisfies Rueppel s criteria and would therefore be considered cryptographically strong by his definition.

0 1x

(11-1)

SET Select Time Zone 11.25.2009 10:00

where is the random error term. We will call this model the simple linear regression model, because it has only one independent variable or regressor. Sometimes a model like this will arise from a theoretical relationship. At other times, we will have no theoretical knowledge of the relationship between x and y, and the choice of the model is based on inspection of a scatter diagram, such as we did with the oxygen purity data. We then think of the regression model as an empirical model. To gain more insight into this model, suppose that we can x the value of x and observe the value of the random variable Y. Now if x is xed, the random component on the righthand side of the model in Equation 11-1 determines the properties of Y. Suppose that the mean and variance of are 0 and 2, respectively. Then E1Y 0 x2 E1

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