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Optimal One-Shot Schemes for Multi-Source Quickest Detection

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This paper explores the multi-source quickest detection problem utilizing one-shot schemes, focusing on the decentralized model of signal detection. It presents an optimal strategy using the CUSUM method while addressing trade-offs between the speed of detection and false alarm rates. The analysis includes asymptotic optimality of the N-CUSUM rule across independent sources. An extension to coupled systems is also discussed, addressing various signal strengths and dependencies, with the objective of efficiently detecting minimal change points while controlling for false alarms.

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Optimal One-Shot Schemes for Multi-Source Quickest Detection

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  1. Multi-dimensionalquickest detection Olympia Hadjiliadis

  2. Outline: Part I- One Shot Schemes • The multi-source quickest detection problem • The decentralized example &a synchronous communication through • CUSUM one shot schemes (& its optimality) • Multiple sources set-up of trade-off asap vs fa • Asymptotically there is no loss of information In the decentralized/centralized set-up (more mathematically….) • Asymptotic optimality of the N-CUSUM rule • Summary

  3. We sequentially observe through independent sources i=1,…,N Information (if all at one location) with special attn to & The multi-source quickest detection OBJECTIVE: Detect the minimum of as soon as possible but controlling false alarms ASSUMPTION 1) The onset of signal can take place at distinct times :unknown constants

  4. Lorden’s criterion (1971) subject to Optimality of the CUSUM Min-Max approach The Cumulative Sum (CUSUM) is optimal Shiryaev(1996), Beibel(1996)

  5. T3 T2 T1 TN The decentralized system Each sensor Si is sequentially observing continuous observations S3 S2 … S1 SN Fusion center

  6. subject to The CUSUM stopping rule is … and is optimal in CUSUM & its optimality The CUSUM statistic process is Asap Mean time to false alarm

  7. Then and Multiple sources set-up subject to d Define The optimal stopping rule satisfies Proof: Let N=2 and consider S be s.t. Let U stop as S, when observing {ξt1} instead of {ξt2} & vice versa Consider T stops as S when Heads and as U when Tails while

  8. Multiple sources set-up Natural candidate: N-CUSUM Hence the best N-CUSUM satisfies Which translates to (as γ→∞)

  9. Asymptotic optimality as γ→∞ • If • If • If where

  10. Asymptotic optimality (equal strengths) N=2, µ=1

  11. Asymptotic optimality (unequal strengths) N=2, µ1=1, µ2=1.2µ1

  12. Asymptotic optimality(unequal strengths) N=2, µ1=1, µ2=1.5µ1

  13. Outline: Part II-Coupled systems • The multi-source quickest detection problem • Models of general dependencies • Objective: Detect the first instance of a signal; Meaning: • Detect the min of N change points in Ito processes • Set-up the problem as a stochastic optimization w.r.t. a Kullback Leibler divergence • Asymptotic optimality of the N-CUSUM rule • Summary

  14. We sequentially observe through independent sources i=1,…,N Information (if all at one location) with special attn to & The multi-source quickest detection OBJECTIVE: Detect the minimum of as soon as possible but controlling false alarms ASSUMPTION 1) The onset of signal can take place at distinct times 2) : unknown constants

  15. Examples A system with AR behavior in each component and additive feedback from other sources Such a system with signals of different strengths in each sensor

  16. CUSUM & its optimality N=1 The CUSUM statistic process is The CUSUM stopping rule is subject to is optimal in Asap Mean time to false alarm

  17. ASSUMPTION are the same in law across all i Multiple sources set-up subject to d Define The optimal stopping rule satisfies

  18. Since are the same across i… ALL ABOVE ARE EQUAL Multiple sources set-up Natural candidate: N-CUSUM Therefore…

  19. To solve this problem we need… Take N=2. It is possible to show that satisfy

  20. To solve this… • G is the probability that a particle placed at (x,y) will leave D after t.

  21. NOTE: Asymptotic optimality as γ→∞ • If where

  22. We sequentially observe through independent sources i=1,2 subject to Non-symmetric signals Suppose where

  23. Multiple sources set-up • If as Natural candidate: 2-CUSUM In order to have an equalizer rule, or equivalently we need

  24. We sequentially observe through independent sources i=1,2 subject to Non-symmetric signals Suppose where

  25. Summary • Asymptotic optimality of the N-CUSUM rule in the case are the same across I • In the case of Brownian motions with const drift MESSAGE: If you want to detect the first instance of onset of a signal, let the sensors do the work! (Lose almost nothing in efficiency) • Extensions to the case different in law across i • What if the noises across sources are correlated.

  26. Thanks to all collaborators • H. Vincent Poor • Tobias Schaefer • Hongzhong Zhang

  27. “One-shot schemes for decentralized quickest change detection”, O. Hadjiliadis, H. Zhang and H. V. Poor, IEEE Transactions on Information Theory 55(7) 2009. • “Quickest Detection in coupled systems”, O. Hadjiliadis, T. Schaefer and H. V. Poor , Proceedings the 48th IEEE Conference on Decisions and Control, (2009) Submitted to the SIAM Journal on control and optimization (2010)

  28. THE END

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