Mathematical and statistical models in Visual Studio .NET

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Mathematical and statistical models
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DoG wavelet transform:
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1400 1200 Mean square error 1000 800 Elbow point at 32 coefficients 600 400 200 0 0 50 100 150 200 Number of wavelet coefficients 250 300
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Paul wavelet transform:
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3000 2500 Mean square error 2000 1500 1000 500
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100 150 200 Number of wavelet coefficients
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Haar wavelet transform:
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1235 1230 Mean square error 1225 1220 1215 1210 1205
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150 200 100 Number of wavelet coefficients
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Figure 163 The MSE charts for developing a wavelet-based attack data model of Process( Total)\Page Faults/sec under the ARP Poison attack
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The wavelet-based mathematical model for the wavelet feature
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Daubechies wavelet transform:
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4500 4000 Mean square error 3500 3000 2500 2000 1500 1000 0 50 100 150 200 Number of wavelet coefficients 250 300
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Morlet wavelet transform:
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5500 5000 Mean square error 4500 4000 3500 3000 2500 2000 1500 0 50 100 150 200 Number of wavelet coefficients 250 300
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Figure 163 (Continued)
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(j) Select the best- t data model from the data models produced in Step 1(i) that gives the smallest MSE, and this data model is used as the attack data model 2 Repeat Step 1 but replace the attack data sample with the normal use data sample for the text editing in the training data set to develop the text editing data model 3 Repeat Step 1 but replace the attack data sample with the normal use data sample for the web browsing in the training data set to develop the web browsing data model Figures 163, 164 and 165 show the MSE charts produced to develop the attack data model, two normal use data models for the text editing and the web browsing, respectively, for the wavelet change attack characteristic of Process( Total)\Page Faults/sec under the ARP Poison attack, WDL- This attack characteristic shown in Table 111 and Table 131 indicates the signal strength decrease of the DoG wavelet at the low frequency band As indicated in Figure 163,
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Mathematical and statistical models
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DoG wavelet transform:
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110 100 90 Mean square error 80 70 60 50 40 30 20 10 0 100 200 300 400 Number of wavelet coefficients 500 600 Elbow point at 32 coefficients
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Paul wavelet transform:
1200 1000 Mean square error 800 600 400 200 0 150 300 350 200 250 Number of wavelet coefficients 400
Haar wavelet transform:
1850 1800 1750 Mean square error 1700 1650 1600 1550 1500 1450 1400 1350 0 100 200 300 400 Number of wavelet coefficients 500 600
Figure 164 The MSE charts for developing a wavelet-based normal use data model of Process( Total)\ Page Faults/sec under the text editing norm
Summary
Daubechies wavelet transform:
800 700 Mean square error 600 500 400 300 200
100 150 200 Number of wavelet coefficients
Morlet wavelet transform:
1600 1400 Mean square error 1200 1000 800 600 400 200 0 100 300 400 200 Number of wavelet coefficients 500 600
Figure 164 (Continued)
the best- t attack data model is the DoG wavelet-based model with 32 wavelet coef cients and the MSE value around 200 As indicated in Figure 164, the best- t text editing data model is the DoG wavelet-based model with 65 wavelet coef cients and the MSE value around 20 As indicated in Figure 165, the best- t web browsing data model is the DoG wavelet-based model with 64 wavelet coef cients and the MSE value around 40
166 SUMMARY
This chapter describes the statistical and mathematical models that are used to develop the attack data model, the text editing data model, and the web browsing data model for the data variable involved in a given attack characteristic Speci cally, various data features,
Mathematical and statistical models
DoG wavelet transform:
120 110 100 90 Mean square error 80 70 60 50 40 30 20 0 100 200 300 400 Number of wavelet coefficients 500 600 Elbow point at 64 coefficients
Paul wavelet transform:
1000 900 800 Mean square error 700 600 500 400 300 200 100 0 0 100 200 300 400 Number of wavelet coefficients 500 600
Haar wavelet transform:
1100 1000 900 Mean square error 800 700 600 500 400 300 200 100 0 100 200 300 400 Number of wavelet coefficients 500 600
Figure 165 The MSE charts for developing a wavelet-based normal use data model of Process( Total)\ Page Faults/sec under the web browsing norm
Summary
Daubechies wavelet transform:
800 700 Mean square error 600 500 400 300 200 100 0 50 100 150 200 250 300 350 Number of wavelet coefficients 400 450 500
Morlet wavelet transform:
450 400 Mean square error 350 300 250 200 150 100 50 0 0 100 200 300 400 Number of wavelet coefficients 500 600
Figure 165 (Continued)
including the mean, autocorrelation, probability distribution, and wavelet in the time-frequency domain which are described in Part III, require different kinds of data models to capture the data features and represent the data characteristics in those data features The sample average is used to represent a data characteristic in the mean feature The empirical cumulative density function is used to represent a data characteristic in the distribution feature The Box-Jenkins time series model is used to represent a data characteristic in the autocorrelation feature The wavelet-based mathematical model is used to represent a data characteristic in the wavelet feature The attack and normal use data models are required in the cuscorebased detection models for attack-norm separation in 17 to help the cuscore-based detection models to achieve the better detection performance than the ANN technique for signature recognition in 13 and the EWMA control charts for anomaly detection in 14