Daubechies wavelet

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Morlet wavelet

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Figure 112 The Paul wavelet, the Derivative of Gaussian wavelet, the Daubechies wavelet, and the Morlet wavelet

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Discovering the wavelet change characteristics for attacks

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112 DISCOVERING THE WAVELET CHANGE CHARACTERISTICS FOR ATTACKS

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For the collected data of the Windows performance objects described in 7, the same data screening procedure as described in Section 82 is performed to eliminate the data variables which have the observations of the same value under all three conditions: the inactive, attack and norm conditions Each of the remaining data variables is analyzed to extract the wavelet feature and discover the wavelet change characteristics of attack and normal use data For the data sample of a given data variable under each condition (inactive, attack and norm) of the collected data, the wavelet transform is performed using each of the ve wavelet forms The statistical toolbox of MATLAB Version 650180913a (R13) is used to perform the wavelet transforms and obtain the wavelet coef cients For the wavelet transform using the Haar and Daubechies wavelets, the k value of 8 is applied to a data sample of 256 data observations Three frequency bands are de ned with the low frequency band containing the three lowest frequencies, the high frequency band containing the three highest frequencies, and the medium frequency band containing the remaining two frequencies [6] For the Paul, DoG and Morelet wavelet transforms applied to each data variable, there are 29 frequencies for 256 data observations These frequencies are considered to fall into three frequency bands: the low frequency band containing the eight lowest frequencies, the high frequency band containing the twelve highest frequencies, and the medium frequency band containing the remaining nine frequencies [6] For each wavelet transform of each variable under each condition (inactive, attack and norm), the Signal Strength (SS) at each frequency band is computed using the wavelet coef cients at that frequency as follows: SS = 1 n

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(1113)

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where w j is a wavelet coef cient and n is the total number of wavelet coef cients at that frequency band The analysis of variance (ANOVA) is then carried out in the following steps: 1 For the 10-minute data under the inactive condition and the attack data for the entire attack period from Run 1 of the data collection, perform an ANOVA with two independent variables of frequency band and condition and the dependent variable of the signal strength The condition has two levels: inactive and attack The frequency band has three levels: low, medium, and high The SS value for a frequency in a given frequency band under a given condition is a data observation for that combination of the frequency and the condition For example, there are eight data observations of SS from a Haar transform of a data variable under a given condition The ANOVA test along with the Tukey test, which is carried out using the statistical toolbox of MATLAB, reveals whether or not there is a signi cant difference or change of SS from the inactive condition to the attack condition at each frequency band If there is a signi cant change of the signal strength at a given frequency band, this change of the wavelet signal strength is considered a wavelet change attack characteristic For example, if the signal strength from the Haar transform of a given data variable at the low frequency has a signi cant increase from the inactive condition to

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Wavelet change characteristics

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the attack condition, the data variable is marked with an attack characteristic denoted by WHL+ which stands for Wavelet, Haar transform, Low frequency, and increase (+) The ve wavelet transforms are denoted by:

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r P for the Paul transform r D for the DoG transform r H for the Haar transform r Da for the Daubechies transform r M for the Morlet transform

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The three frequency bands are denoted by:

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