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For the text editing data model of this variable, the following differencing transformations are rst performed on the time series data of the text editing in the training data set before tting an ARMA model: yt = xt xt 32 z t = yt yt 30 f t = z t z t 10 The ARMA model tted to the transformed text editing data, ft , is the following: f t = 06867 f t 1 + et 05773et 1 (1615) (1612) (1613) (1614)
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For the web browsing data model of this variable, the following differencing transformations are rst performed on the time series data of the web browsing in the training data set before tting an ARMA model: yt = xt xt 32 z t = yt yt 30 The ARMA model tted to the transformed web browsing data, zt , is the following: z t = 04788z t 1 + et + 09919 1et 1 (1618) (1616) (1617)
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165 THE WAVELET-BASED MATHEMATICAL MODEL FOR THE WAVELET FEATURE
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As illustrated in 11 through the example of the Haar wavelet, the following function, f (x), which is de ned by Formula 114 in 11 and is repeated below as Formula 1619, is used to represent a data sample of ai for all is: f (x) =
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Formulas 1111 and 1112, which are repeated below as Formulas 1620 and 1621, can be used to transform the scaling functions in Formula 1619 into the wavelet functions at various frequencies and a series of time locations along with the wavelet coef cient 1 2k 1 x i + 2k 1 x i 2 1 2k 1 x i 2k 1 x i 2k x 1 i = 2 2k x i = (1620) (1621)
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Hence, f (x) can be de ned using those wavelet functions and corresponding wavelet coef cients Formulas 119 and 1110, which are repeated here as Formulas 1622 and 1623,
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The wavelet-based mathematical model for the wavelet feature
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are used to reconstruct the data sample using the wavelet coef cients from the wavelet transform 2k 1 x i = 2k x i + 2k x i 1 2k 1 x i = 2k x i 2k x i 1 (1622) (1623)
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Different scaling functions and wavelet functions are used for the different wavelet transforms along with different data reconstruction methods (see 11) For the data variable involved in a given wavelet change attack characteristic, the attack data model and two normal use data models for the two normal use activities, respectively, are developed in the following steps 1 For the attack data sample of the data variable in the training data set: (a) Select a wavelet transform from the Paul, DoG, Haar, Daubechies and Morlet wavelet transforms, and apply the wavelet transform to the data sample (b) Initialize the target set of the wavelet coef cients to empty, and the original set of the wavelet coef cients to include all the resulting wavelet coef cients from the wavelet transform (c) Take out the wavelet coef cient with the largest absolute value from the original set of the wavelet coef cients, and add this wavelet coef cient to the target set of the wavelet coef cients (d) Reconstruct the data sample using only the wavelet coef cients in the target set of the wavelet coef cients (e) Compute the Mean Squared Error (MSE) of the reconstructed data sample from the original data sample as follows: MSE = 1 n 1
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where xi represents the original data sample, and x i represents the reconstructed data sample (f) Plot this pair of the MSE value and the number of the wavelet coef cients in the target set as a data point in the MSE chart (see examples in Figure 163) (g) Repeat Steps 1(c) 1(f) until the curved line connecting the data points in the MSE chart approximately levels off (h) Select the number of wavelet coef cients and the corresponding target set of wavelet coef cients at the elbow point of the curved line in the MSE chart when the leveling-off occurs, because this target set of wavelet coef cients gives the best- t data model to the original data sample using the smallest number of the largest (in absolute value) wavelet coef cients (i) Repeat Steps 1(a) 1(h) until the best- t data models are selected for all the ve wavelet transforms
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