32 the scan statistic in this section we describe the

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3.2. The Scan Statistic In this section, we describe the methodology behind scanning for local anomalies in a graph over time. Windowing in this space is then discussed, followed by the definition of the scan statistic. 3.2.1. Windows in the cross product space We are interested in examining sets of windows in the T ime × Graph prod- uct space. We define these sets of windows as follows. We have a graph G = ( V, E ) with node set V and edge set E . For each edge e E , at discrete time points t ∈ { 1 , . . . , T } , we have a data process X e ( t ). We denote the set of time windows on edges e over discretized time intervals ( s, s + 1 , . . . , k ) as Ω = { [ e, ( s, s + 1 , . . . , k )] : e E, 0 s < k T } . The set of all subsets of windows, Γ = {{ w 1 , w 2 , . . . } : w j } , is usually very large, and we are normally interested in only a subset, Γ s Γ, that contains locality constraints in time and in graph space. We therefore restrict our attention to sets of windows γ Γ s . For convenience, we denote X ( γ ) as the data in the window given by γ . Next, we assume that for any time point t and edge e , we can describe X e ( t ) with a stochastic process (specific examples are given in Section 3.4) with parameter function given by θ e ( t ). We denote the values of the parameter functions evaluated in the corresponding set of windows γ by θ ( γ ). Finally, we denote the likelihood of the stochastic process on γ as L ( θ ( γ ) | X ( γ )). At this point, it is worth returning to our discussion in Section 3.1.3 of the 3-path used to detect traversal. In this example, X e ( t ) are the directed time series of counts of connections between the pair of hosts that define each edge e . Then, Ω is the set of all (edge, time interval) pairs. We would like to combine edges to form shapes, so we take all subsets of Ω and call that Γ. For this example, we now restrict our set of shapes to sets consisting of three (edge, time interval) pairs such that the edges form a directed 3-path, and the time interval is selected to be the same on each edge. In the simulations and real network example, the time intervals are 30 minutes long, and overlap by ten minutes with the next time window, and are identical on each edge in the shape. These are then the windows γ that are used in the 3-path scan shape. 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 79 3.2.2. A scan statistic for windows in the T ime × Graph space We would like to examine whether the data in a window γ was likely to have been produced by some known function of the parameters θ 0 ( γ ), versus alternatives indicating that the parameters have changed. That is, given that we observe X ( γ ) = x ( γ ), we would like to test whether H 0 : θ ( γ ) = θ 0 ( γ
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