The cusum statistic and log of the sr statistic for

Info icon This preview shows pages 43–46. Sign up to view the full content.

The CUSUM statistic and log of the SR statistic for the Gaussian model (pre- change mean µ = 0, post-change mean θ = 1, standard deviation σ = 1, changepoint ν = 200). Fig. 2.4. ADD versus changepoint ν for CUSUM and SR with various initialization strategies. 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
Image of page 43

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

44 A. G. Tartakovsky numerically integral equations for operating characteristics (Moustakides et al. , 2011). The SRP and SR- r procedures outperform both SR and CUSUM with respect to SADD . However, this comes with the additional effort of finding an appropriate initialization – not a trivial task. Besides, the difference is not dramatic. For this reason, in this chapter, we will focus on the “pure” CUSUM and SR procedures with zero initialization as well as on their semiparametric and nonparametric modifications more suitable for intrusion detection applications. Note also that the SR procedure outper- forms all its counterparts when the change occurs at a distant time horizon. The following theorem, whose proof may be found in Tartakovsky et al. (2014, Ch. 9), establishes asymptotic properties of the CUSUM and SR procedures for the low FAR (large γ , i.e., as h, A → ∞ ). We need additional notation: λ n = n i =1 Z i , τ a = min { n : λ n a } , κ = lim a →∞ E 0 ( λ τ a a ) , ζ = lim a →∞ E 0 e ( λ τa a ) , Q CS ( y ) = P 0 min n 0 λ n y , Q SR ( y ) = P 0 log 1 + n =1 e λ n y , Q CS st ( y ) = lim n →∞ P ( W n y ) , Q SR st ( y ) = lim n →∞ P ( R n y ) , C CS 0 = 0 −∞ y d Q CS ( y ) , C SR 0 = 0 y d Q SR ( y ) , C CS = 0 z −∞ y d Q CS ( y )d Q CS st ( z ) , C SR = 0 z 0 y d Q SR ( y )d Q SR st ( z ) . Define also the Kullback–Leibler (KL) information number I = E 0 Z 1 = log g ( x ) f ( x ) g ( x ) d µ ( x ) . Theorem 2.1. Consider the i.i.d. case. Assume that E 0 | Z 1 | 2 < and that Z 1 is P 0 -nonarithmetic. (i) If h = log( 2 γ ) , then ARL2FA ( T CS ) γ as γ → ∞ and SADD ( T CS ) = 1 I (log γ + C CS ) + o (1) , STADD ( T CS ) = 1 I (log γ + C CS ) + o (1) , where C CS = κ log(1 /Iζ 2 ) C CS 0 and C CS = κ log(1 /Iζ 2 ) C CS . 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
Image of page 44
Rapid Detection of Attacks by Quickest Changepoint Detection Methods 45 (ii) If A = log( ζγ ) , then ARL2FA ( T SR ) γ as γ → ∞ and SADD ( T SR ) = 1 I (log γ + C SR ) + o (1) , STADD ( T SR ) = 1 I (log γ + C SR ) + o (1) , where C SR = κ log(1 ) C SR 0 and C CS = κ log(1 ) C SR .
Image of page 45

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

Image of page 46
This is the end of the preview. Sign up to access the rest of the document.
  • Spring '12
  • Kushal Kanwar
  • Graph Theory, Statistical hypothesis testing, Imperial College Press, applicable copyright law

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern