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Generating Code-128 in Visual Studio .NET Mathematical and statistical models
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1 D C Montgomery, and G C Runger, Applied Statistics and Probability for Engineers New York: John Wiley & Sons, Ltd 1999 2 G E P Box, and G M Jenkins, Time Series Analysis: Forecasting and Control, 2nd edn San Francisco, CA: Holden-Day, 1976 3 R A Yaffee, Introduction to Time Series Analysis and Forecasting San Diego, CA: Academic Press, 2000 4 Statistica, wwwstatsoftcom
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Cuscore-based attack norm separation models
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The attack norm separation methodology aims at separating the effects of the attack data and the normal use data in their mixture to enhance the performance of cyber attack detection Attack norm separation can be considered as a signal-noise separation problem if the normal use data is considered as noise and the attack data is considered as the signal to detect Many signal processing techniques exist to perform noise cancellation and signal detection This chapter focuses on a speci c technique, called the cumulative score (cuscore) [1, 2], which is used to carry out the attack norm separation methodology The attack and normal use data models described in 16 are employed in the cuscore-based attack norm separation models In Section 171, the cuscore chart is introduced Section 172 describes the application of the cuscore-based attack norm separation models, or simply called the cuscore models, to cyber attack detection Section 173 shows the detection performance of the cuscore models in comparison with that of the ANN technique for signature recognition in 13 and the EWMA control chart technique for anomaly detection in 14
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171 THE CUSCORE
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In [1, 2], the following statistical model is considered: t = (yt , xt , ), (171)
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where yt and xt are the observation values of the two random variables at time t, is an unknown parameter capturing the relationship of y with x, t is the residual obtained by subtracting yt from yt , and yt is the predicted value of yt based on xt and When = 0 which is the true value of the unknown parameter, the resulting t0 s are a white noise sequence, that is, a sequence of independently identically normally distributed random variables with the mean of zero and the variance of 2 Thus, the random variables, 1 , 2 , , n , in the white noise sequence have
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Secure Computer and Network Systems: Modeling, Analysis and Design Nong Ye C 2008 John Wiley & Sons, Ltd
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Cuscore-based attack norm
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a joint multivariate normal distribution with the joint probability density function as follows [3]: p( 1 , 2 , , n | = 0 ) =
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t0 1 1 2 2 e t=1 , n/2 (2 ) n 2
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(172)
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where t0 denotes t at time t when = 0 The natural log likelihood, l( 1 , 2 , , n | = 0 ), is: l( 1 , 2 , , n | = 0 ) = 1 (2 )n/2 1 2 2
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n 2 t0 = t=1
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1 1 n/2 2 2 (2 )
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2 t0 (173)
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Note that n depends on Hence, l( 1 , 2 , , n ) is a function of When = 0 , l( 1 , 2 , , n ) should gain the maximum likelihood value, or l( 1, 2,, n,) = 0, where 1 l( 1 , 2 , , n | = 0 ) = (2 )n/2 1 2 2
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n 2 t0 = t=1
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1 1 n/2 2 (2 )
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t0
t0
(174)
The cumulative score (cuscore) is de ned as follows [1, 2]:
Q0 =
t0 dt0 ,
(175)
where dt0 = t0 (176)
If = 0 , the cuscore should remain zero or randomly uctuate around zero due to the modeling error Hence, the cuscore can be used to detect when changes from 0 by monitoring when the cuscore departs from zero not randomly but in a consistent manner