Against alternatives that can be formed by

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) against alternatives that can be formed by restricting the overall parameter space, Θ , to a subset Θ A Θ . The generalized likelihood ratio test statistic (GLRT) is a natural statistic to use. Let λ γ = 2 log L ( θ 0 ( γ ) | x ( γ )) sup θ Θ A L ( θ ( γ ) | x ( γ )) . (3.1) The size of λ γ depends on the number of parameters being tested in the window, making it difficult to use directly. To address this issue, we nor- malize λ γ by converting it into a p -value, p γ . These p -values are discussed in detail in Section 3.4.5. To scan for anomalies in the T ime × Graph product space, we must slide over all windows γ , keeping track of the scan statistic Ψ = min γ p γ . In practice, thresholding of the set of p -values is performed so more than just the minimum p -value can be considered. For online monitoring, we set a threshold on the p -values to control the false discovery rate (Benjamini and Hochberg, 1995). This threshold setting is described in more detail in Section 3.4.6, but we emphasize that generally, when a detection occurs, a set of windows (not just one) are detected, and so the union of these windows is the detected anomaly produced by the system. 3.3. Independence Among Edges in a Path In order to scan for anomalous shapes, it is necessary to have models that describe the behavior of the data in the window γ under normal conditions. The number of enumerated subgraphs tends to scale exponentially with the number of nodes, however, and an assumption of independence among the edges in the shape facilitates scaling the computations required to process large graphs at line speeds, under reasonable memory requirements. While the general approach discussed in Section 3.2.1 does not require indepen- dence among the edges in the window γ , independence will be assumed among the stars and paths discussed in Section 3.1.3 and used in the simulation and real-data sections. Note that for those shapes, no edge is repeated. 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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80 J. Neil, C. Storlie, C. Hash and A. Brugh Edge independence ensures the ability to scale, since models (and the storage of their parameters) for each edge are sufficient to construct models for subgraphs of edges under this assumption, whereas non-independence might require models for each shape , of which there may be many hundreds of millions, if not billions. Therefore, an examination of the assumption of independence among edges connected in a path is conducted.
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  • Spring '12
  • Kushal Kanwar
  • Graph Theory, Statistical hypothesis testing, Imperial College Press, applicable copyright law

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