Let λ v have the distribution of the ? v we model λ

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Let Λ v have the distribution of the λ v . We model Λ v as Λ v = B v X v where B v Bernoulli( p v ) and X v Gamma( τ v , η v ). Since all λ e in the sum could be zero, Λ v must have a point mass at zero. This is captured by B v . To model the positive part of the distribution for Λ v , the Gamma distribution is attractive since it is equal to a χ 2 distribution with degrees of freedom ν when τ v = ν 2 and η v = 2. The asymptotic distribution of λ v is then the sum of independent zero-inflated χ 2 distributed random variables. Thus, we expect the zero-inflated Gamma to be able to model the distribution of λ v fairly well. The log-likelihood of N i.i.d. samples is given by l ( p, τ, η ) = N i =1 I ( λ i = 0) log(1 p ) + I ( λ i > 0)[( τ 1) log λ i λ i log Γ( τ ) τ log η ] . (3.13) To estimate τ v and η v , we use direct numerical optimization of (3.13) over 10 days of non-overlapping 30-minute windows, for each star centered at node v . We denote the MLEs as (ˆ p v , ˆ τ v , ˆ η v ). Then for an observed λ v , the upper p -value is calculated by P v > λ v ) = ˆ p v (1 F Γ ( λ v | ˆ τ v , ˆ η v ) where F Γ is the Gamma CDF. Path p -values. Unlike stars, the large number of paths makes modeling λ γ for each path prohibitively expensive, both in computation time and memory requirements. Instead, we build a model for each individual edge, and then combine them during the path likelihood calculation. For each edge e , let Λ e have the null distribution of e ’s GLRT scores, λ e . Again, we use a zero-inflated Gamma distribution to model this. Now, however, it will be on a per-edge basis. Once again, this model is motivated by the fact that 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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90 J. Neil, C. Storlie, C. Hash and A. Brugh asymptotically, the null distribution of λ e is a zero-inflated χ 2 (with 50% mass at zero if testing one parameter). Let Λ e = B e X e where B e Bernoulli( p e ), and X e Gamma( τ e , η ), with edge-specific shape τ e and shared scale η . That is, we have two free parameters for each edge, p e and τ e , and a common scale parameter for all edges, η . The importance of the common scale parameter will become clear shortly. We estimate MLEs ˆ p e , ˆ τ e , and ˆ η using λ e s from non-overlapping 30-minute windows. The likelihood is similar to (3.13), but since each edge has its own τ e , and a shared η , we have developed an iterative scheme that alternates between estimating η for all edges, and then, for that fixed η , esti- mating individual τ e . Since each step of the iteration increases likelihood, the overall procedure increases likelihood.
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