Copyright 2014 imperial college press all rights

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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 59 0 0.5 1 1.5 2 2.5 0 50 100 150 200 250 300 number of packets per sample period attack starts sample number x 10 5 Fig. 2.9. Time series of the total number of UDP packets in a sample period 0.015 msec. Observe that the attack is not visible to the naked eye. ADD (sec) -4 -3 -2 -1 0 1 2 3 4 5 6 7 0 5 10 15 20 25 - log(FAR) c = 0 c = 2 c = 6 c = 8 c = 4 (a) Operating characteristic of score CUSUM. ADD (sec) -1 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 - log(FAR) (b) Operating characteristic of binary CUSUM. Fig. 2.10. SADD (sec) versus - log FAR for the score-based and binary CUSUM algo- rithms. Operating characteristics ( SADD versus log FAR = log ARL2FA ) of the score-based and binary CUSUM algorithms are shown in Figure 2.10. Specifically, Figure 2.10(a) illustrates the operating characteristic of the linear score-based CUSUM procedure (i.e., C 1 = 1 and C 2 = 0 in (2.15)) for different values of C 3 = c , the tuning parameter. Note that the range of log FAR = log ARL2FA from 4 to 7 is equivalent to the frequency of false alarms from every 0 . 018 sec to every 1096 sec ( 18 min). Note that in the left-most region of Figure 2.10(a) the ADD is very small but this results in a very large FAR. On the other hand, the right-most region in Figure 2.10(a) has the lowest FAR and, hence, a bigger ADD. For example, the ADD is 0 . 015 sec for ARL2FA = 0 . 018 sec (the left-most region) and 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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60 A. G. Tartakovsky 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 1.5 2 2.5 3 3.5 4 4.5 5 ADD( ) ADD( b ) - log(FAR) Fig. 2.11. Relative efficiency versus - log FAR. the ADD is 26 sec for ARL2FA = 945 sec (the right-most region). Varying the value of c allows us to optimize the algorithm. The optimal value of c is c opt = 6. For this c , we get the best performance in the sense that, for the same FAR, we obtain the smallest ADD as compared to other values of c . Figure 2.10(b) shows the operating characteristic of the binary CUSUM algorithm defined in (2.21). The optimal quantization threshold is t opt = 105 and the corresponding maximum KL number I b ( t opt ) = 0 . 163. Figure 2.11 illustrates the efficiency of the binary CUSUM detection procedure with respect to the optimized score-based CUSUM procedure (with c opt = 6) in terms of the relative efficiency, which is defined as SADD ( T sc CS ) / SADD ( T b CS ). In this case, the binary CUSUM algorithm per-
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