Statistically having observed the sample x n 1 x 1 x

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Statistically, having observed the sample X n 1 = ( X 1 , . . . , X n ), the prob- lem is to test the hypothesis H k : ν = k 0 that the change has occurred at time k somewhere in the starch of n observations against the alternative H : ν = that there is never a change. The densities of X n 1 given the hypotheses H k and H are p ( X n 1 | H k ) = k i =1 f ( X i | X i 1 1 ) × n i = k +1 g ( X i | X i 1 1 ) , p ( X n 1 | H ) = n i =1 f ( X i | X i 1 1 ) . According to hypothesis testing theory an optimal solution should be based on the likelihood ratio (LR) between these hypotheses given by Λ k n = p ( X n 1 | H k ) p ( X n 1 | H ) = n j = k +1 L j for k < n, L j = g ( X j | X j 1 1 ) f ( X j | X j 1 1 ) , (2.1) where n j = L j = 1 for > n . In the simplest setting, it is assumed that the observations are indepen- dent throughout the entire period of surveillance, so that X 1 , X 2 , . . . , X ν are distributed according to a common pre-change density f , while X ν +1 , X ν +2 , . . . have a common density g f . We refer to this case as the i.i.d. case . Note that due to the i.i.d. assumption, the LR for the n th observation simplifies to L n = g ( X n ) /f ( X n ). The LRs { Λ k n } n 1 are then used by a sequential detection procedure to decide in favor of one of the hypotheses. Given the sequence { X n } n 1 , a sequential detection procedure is defined as a stopping time T adapted to the observations, i.e., the event { T n } is measurable with respect to X 1 , . . . , X n . The stopping time T is nothing but the time instant at which the detection procedure stops and declares that a change is in effect. If the change is declared prematurely, i.e., T ν , then a false alarm is 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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38 A. G. Tartakovsky (a) Process of interest with a change in the mean. (b) Two possible detection scenarios: false alarm (gray) and correct detection (black). Fig. 2.1. Illustration of sequential changepoint detection. raised. Figure 2.1(a) shows a typical example of change in the mean of the process and its detection. The gray trajectory in Figure 2.1(b) illustrates the false alarm situation when the detection statistic exceeds the detection threshold prior to the change occurring, in which case T can be regarded as the (random) run length to the false alarm. The black trajectory in Figure 2.1(b) illustrates true detection when the detection statistic exceeds the detection threshold past the changepoint. Note that the detection delay characterized by the difference T ν is random.
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