Els described in heard et al 2010 and elsewhere with

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els described in Heard et al. (2010) and elsewhere, with one important distinction: we allow the high state to emit zeros. We believe that this is important in modeling our data. Again, referring to Figure 3.1, we see that zero counts are interspersed with the nonzero data, but are still clearly a part of the “active” state. Intuitively, we think of the active state as “the user is present at the machine,” and therefore likely to make communica- tions, not as “the user is making a communication on this edge.” Next, the estimation of the HMM is discussed in this setting. Copyright © 2014. Imperial College Press. All rights reserved. May not be reproduced in any form without permission from the publisher, except fair uses permitted under U.S. or applicable copyright law. EBSCO Publishing : eBook Collection (EBSCOhost) - printed on 2/16/2016 3:37 AM via CGC-GROUP OF COLLEGES (GHARUAN) AN: 779681 ; Heard, Nicholas, Adams, Niall M..; Data Analysis for Network Cyber-security Account: ns224671
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Statistical Detection of Intruders Within Computer Networks 83 Notation and Likelihood. At a set of T discrete time points we observe counts x = [ x 1 , . . . , x T ] , with x t ∈ { 0 , 1 , . . . } for t = 1 , . . . , T . In this model, the counts are viewed as coming from one of two distributions, as governed by Z = [ Z 1 , . . . , Z T ] , a latent two-state Markov process. Letting p 01 = Pr( Z n = 1 | Z n 1 = 0) and p 10 = Pr( Z n = 0 | Z n 1 = 1), we denote the latent transition matrix as A = 1 p 01 p 01 p 10 1 p 10 . The initial state distribution is denoted π = Pr( Z 1 = 1). The marginal distribution of the count at time t , given that Z t = 0 is degenerate at 0, i.e. Pr( X t = x t | Z t = 0) = I ( X t = 0) where I ( · ) is the indicator function. When Z t = 1, we assume that the counts are distributed according to a negative binomial distribution with mean and size parameters given by φ = [ µ, s ] , i.e. Pr( X t = x t | Z t = 1 , φ ) = Γ( s + x t ) Γ( s )Γ( x t + 1) s µ + s s µ µ + s x t . A useful fact is that the joint probability distribution over both latent and observed variables can be factored in a way that is useful for compu- tation, since it separates the different parameter types: Pr( X = x , Z = z | θ ) = Pr( Z 1 = z 1 | π ) T t =2 Pr( Z t = z t | Z t 1 = z t 1 , A ) × T t =1 Pr( X t = x t | Z t = z t , φ ) where θ = ( π, A , φ ) . Finally, the likelihood is Pr( X = x | θ ) = 1 z 1 =0 · · · 1 z t =0 Pr( X = X , Z = z | θ ) . (3.4) Maximum Likelihood Estimates. Equation (3.4) involves 2 T terms, making it computationally infeasible to work with directly, for even mod- erately large T . Hence, we look to expectation maximization (EM) as Copyright © 2014. Imperial College Press. All rights reserved. May not be reproduced in any form without permission from the publisher, except fair uses permitted under U.S. or applicable copyright law. EBSCO Publishing : eBook Collection (EBSCOhost) - printed on 2/16/2016 3:37 AM via CGC-GROUP OF COLLEGES (GHARUAN) AN: 779681 ; Heard, Nicholas, Adams, Niall M..; Data Analysis for Network Cyber-security Account: ns224671
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84 J. Neil, C. Storlie, C. Hash and A. Brugh an iterative approach for calculating the maximum likelihood estimates.
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