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176 174 172 170 168
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450 400 350 300 No of obs 250 200 150 100 50 0
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1 58 115 172 229 286 343 400 457 514 571 628 685 742 799
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b The random fluctuation pattern of Processor(_Total)\Interrupts/sec under the Hardware Keylogger attack
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Figure 91 The data plots and histograms of the variables with ve data patterns
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Probability distribution change characteristics
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Histogram (PureAttack 340v*30c) 30 28 26 24 22 20 No of obs 18 16 14 12 10 8 6 4 2 0 7E7 8E7 9E7 1E8 11E8 12E8 13E8 14E8 15E8 16E8 17E8
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16E8 \\ALPHA02-VICTIM\Process(_Total)\Private Bytes 15E8 14E8 13E8 12E8 11E8 1E8 9E7 8E7 7E7
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c The step change pattern of Memory\Write Copies/sec under the Fork Bomb attack
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Line Plot (PureTSOnly_1_17 1106v*221c) 50 \\ALPHA02-VICTIM\TCP\Connections Passive 45 40 35 30
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25 20 15 10 5 0
106 121 136 151 166 181 196 211
5 01 15 20 25 30 35 40 45 \\ALPHA02-VICTIM\TCP\Connections Passive
d The steady change (increase) pattern of TCP\Connections Passive under the Remote Dictionary attack
Line Plot (PureTSOnly_1_17 1106v*221c) \\ALPHA02-VICTIM\IP\Datagrams Received Delivered/sec 40 35 30 25 20 15 10 5 0
0 No of obs 60 50 40 30 20 10 5 0 5 10 15 20 25 30 35 40 \\ALPHA02-VICTIM\IP\Datagrams Received Delivered/sec 45 100 90 80 70 Histogram (PureTSOnly_1_17 1106v*221c)
106 121 136 151 166 181 196 211
e The sine-cosine wave with noise pattern of IP\Datagrams Received Delivered/sec under the Remote Dictionary attack
Figure 91 (Continued )
Observation of data patterns
a The data plot and historgram of a spike pattern
b The data plot and historgram of a random fluctuation pattern
c The data plot and historgram of a step change pattern
d The data plot and historgram of a steady change pattern
Figure 92 An illustration of data patterns in association with probability distributions
Probability distribution change characteristics
92 SKEWNESS AND MODE TESTS TO IDENTIFY FIVE TYPES OF PROBABILITY DISTRIBUTIONS
For the ve types of probability distributions described in Section 91, skewness and mode tests can be used in combination to uniquely identify each of the ve distributions Both positive and negative deviations from the mean contribute to the variance in the same way since the variance squares both positive and negative deviations The skewness cubes the deviations from the mean to measure if the deviations are largely symmetric, from the right side of the mean, or from the left side of the mean, as follows: skewness = E(x )3 , 3 (91)
where and are the population mean and variance, respectively The skewness of a normal distribution or any symmetric distribution is expected to be zero A left skewed distribution with a long tail to the left of the mean has a negative skewness value A right skewed distribution with a long tail to the right of the mean has a positive skewness value Given a data sample, x 1 , x2 , , xn , the sample skewness is computed as follows in [1, 2]: skewness =
n n i=1 (xi x )3 (n 1)(n 2)s 3
(92)
where x and s are the sample average and standard deviation, and n is the sample size The skewness value is computed using Statistica [2] Statistica computes the standard error of the skewness value If a skewness value is greater than three times of the standard error of the skewness value, the data variable is considered to be right skewed If a skewness value is smaller than minus three times of the standard error of the skewness value, the data variable is considered to be left skewed The mode in the probability density indicates clustering in the data [3] A probability distribution can have one mode or multiple modes For examples, a normal distribution has one mode, a skewed distribution has one mode, and a bimodal distribution has two modes Both the DIP test [3 6] and the mode test [7] are used together to determine the modality of data because through testing the data on only one of the tests is not adequate to distinguish the unimodality and multimodality of data The DIP test is a test of the unimodality of data as a whole [3, 4] The DIP test is performed using the diptest package [5] for R statistical software [6] with the signi cance level set to 005 The mode test [7] is a test for signi cance of each individual potential mode rather than a test on the overall unimodality of data The mode test program by Minnotte [7] is used For each mode tested, the program results show its location along with its p-value The signi cant level is set to 005 Based on the test results, the number of signi cant modes can be counted Through the testing on the Windows performance objects data, the results of the skewness test, the DIP test and the mode test, which indicate the ve types of probability distributions, are obtained and summarized in Table 91 For example, a data variable is considered to have a uniform distribution if:
r the DIP test rejects the null hypothesis of unimodality at the signi cance level of 005;