Using the glr statistic in cusum leads to the

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Using the GLR statistic in CUSUM leads to the generalized CUSUM pro- cedure T GCS ( A ) = min n 1 : max 0 k n sup θ Θ Λ k n ( θ ) A , (2.10) while using the weighted LR statistic in SR leads to the weighted (mixture) SR procedure T WSR ( A ) = min n 1 : n 1 k =0 Θ Λ k n ( θ )d π ( θ ) A . (2.11) Selecting A = γ guarantees that both procedures belong to the class C γ , i.e., the ARL2FA for both procedures is lower-bounded by γ for every γ > 1. With this threshold, both detection procedures are uniformly first-order asymptotically minimax under quite general conditions: as γ → ∞ for all θ Θ inf T C γ SADD θ ( T ) SADD θ ( T GCS ) SADD θ ( T WSR ) log γ I θ , where I θ = lim n →∞ 1 n E 0 [log Λ 0 n ] (cf. Tartakovsky et al. , 2014). In particular, consider the i.i.d. case and a multivariate exponential family of distributions with density (with respect to some non-degenerate 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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Rapid Detection of Attacks by Quickest Changepoint Detection Methods 47 σ -finite measure µ (d x )) f θ ( x ) = exp θ x ψ ( θ ) , θ Θ : ψ ( θ ) = log Θ e θ x µ (d x ) < , assuming that both X = ( X 1 , . . . , X ) R and θ = ( θ 1 , . . . , θ ) Θ R are -dimensional vectors. The pre-change parameter is θ = 0 and the post- change parameter is θ Θ ε = Θ B ε , where B ε is the ball of radius ε in R . It can be shown (Tartakovsky et al. , 2014) that ARL2FA ( T GCS ) h / 2 e h /K ε ( 1 + o (1) ) as h → ∞ with K ε = (2 π ) / 2 Θ ε ζ θ det[ 2 ψ ( θ )] /I θ d θ . Therefore, setting h = log[ K ε γ (log γ ) / 2 ] yields ARL2FA ( T GCS ) γ (1 + o (1)) and, as γ → ∞ , SADD θ ( T GCS ) = 1 I θ log γ + 2 log log γ + C GCS ( θ ) + o (1) , where C GCS ( θ ) = 2 [1 + log(2 π )] + log Θ ε ζ t det[ 2 ψ ( t )] /I t d t µ θ + κ θ (cf. Tartakovsky et al. , 2014). Here I θ = θ ψ ( θ ) ψ ( θ ) is the KL number, is the gradient, 2 is the Hessian, µ θ = E θ [min n 0 λ n ( θ )], λ n ( θ ) = θ n k =1 X k ψ ( θ ) n and κ θ = lim a →∞ E θ [ λ τ a ( θ ) a ] is the limiting average overshoot in the one-sided test τ a = min { n : λ n ( θ ) a } . A similar approximation holds for the weighted SR procedure with an arbi- trary continuous weight (of course with a different constant C WSR ( θ )). The second term 2 log log γ that goes to infinity when γ becomes large is the price one pays for prior uncertainty. While these procedures have a nice uniform asymptotic optimality property, it is difficult to implement them online, since even in the i.i.d.
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  • Spring '12
  • Kushal Kanwar
  • Graph Theory, Statistical hypothesis testing, Imperial College Press, applicable copyright law

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