Sequential Detection Overview & Open Problems

1. Sequential DetectionOverview & Open Problems George V. Moustakides, University of Patras, GREECE

2. Outline GV MOUSTAKIDES: Sequential Detection, Overview & Open Problems, ISR, Dec. 2011 • Sequential hypothesis testing • SPRT • Application from databases • Open problems • Sequential detection of changes • The Shiryaev test • The Shiryaev-Roberts test • The CUSUM test • Open problems • Decentralized detection (sensor networks)

3. Sequential Hypothesis testing Conventional binary hypothesis testing: Fixed sample size observation vector X=[x1,...,xK]X satisfies the following two hypotheses: Given the data vector X, decide between the two hypotheses. Decision rule: GV MOUSTAKIDES: Sequential Detection, Overview & Open Problems, ISR, Dec. 2011

4. Bayesian formulation Neyman-Pearson formulation Likelihood ratio test: For i.i.d.: GV MOUSTAKIDES: Sequential Detection, Overview & Open Problems, ISR, Dec. 2011

5. Sequential binary hypothesis testing Observations x1,...,xt,... become available sequentially Time Observations Decision Decide Reliably Decide as soon as possible! GV MOUSTAKIDES: Sequential Detection, Overview & Open Problems, ISR, Dec. 2011

6. We apply a two-rule procedure 1st rule: at each time instant t, evaluates whether the observed data can lead to a reliable decision Time Observations Can x1 lead to a reliable decision? Can x1,x2 lead to a reliable decision? No No Can x1,x2,..., xT lead to a reliable decision? Yes STOP getting more data. T is a stopping rule Random! 2nd rule: Familiar decision rule GV MOUSTAKIDES: Sequential Detection, Overview & Open Problems, ISR, Dec. 2011

7. Why Sequential ? In average, we need significantly less samples to reach a decision than the fixed sample size test, for the same level of confidence (same error probabilities) For the Gaussian case it is 4 - 5 times less samples. GV MOUSTAKIDES: Sequential Detection, Overview & Open Problems, ISR, Dec. 2011

8. SPRT (Wald 1945) Here there are two thresholds A < 0 < B Stopping rule: Decision rule: GV MOUSTAKIDES: Sequential Detection, Overview & Open Problems, ISR, Dec. 2011

9. H1 B Infinite Horizon ut T 0 t A H0 GV MOUSTAKIDES: Sequential Detection, Overview & Open Problems, ISR, Dec. 2011

10. Amazing optimality property!!! SPRT solves both problemssimultaneouslyA,B need to be selected to satisfy the two error probability constraints with equality • I.i.d. observations (1948, Wald-Wolfowitz) • Brownian motion (1967, Shiryaev) • Homogeneous Poisson (2000, Peskir-Shiryaev) Open Problems: Dependency, Multiple Hypotheses GV MOUSTAKIDES: Sequential Detection, Overview & Open Problems, ISR, Dec. 2011

11. H1 B Finite Horizon ut T 0 K t A H0 GV MOUSTAKIDES: Sequential Detection, Overview & Open Problems, ISR, Dec. 2011

12. ... ... ... ... ... Record linkage Data Base (records) attribute 2 attribute 1 attribute 3 attribute K ... ... We assume known probabilities for xi =0 or 1 under match (Hypothesis H1) or nonmatch (Hypothesis H0) for each attribute. x1 x2 x3 xK 0 0 1 1 ... GV MOUSTAKIDES: Sequential Detection, Overview & Open Problems, ISR, Dec. 2011

13. Data from: Statistical Research Division of the US Bureau of the Census GV MOUSTAKIDES: Sequential Detection, Overview & Open Problems, ISR, Dec. 2011

14. Total number of record comparisons: 3703 GV MOUSTAKIDES: Sequential Detection, Overview & Open Problems, ISR, Dec. 2011

15. Total number of record comparisons: 3703 GV MOUSTAKIDES: Sequential Detection, Overview & Open Problems, ISR, Dec. 2011

16. In collaboration with V. Verykios GV MOUSTAKIDES: Sequential Detection, Overview & Open Problems, ISR, Dec. 2011

17. Sequential change detection Change in statistics xt ¿ t Detect occurrence as soon as possible GV MOUSTAKIDES: Sequential Detection, Overview & Open Problems, ISR, Dec. 2011

18. Applications Monitoring of quality of manufacturing process (1930’s) Biomedical Engineering Electronic Communications Econometrics Seismology Speech & Image Processing Vibration monitoring Security monitoring (fraud detection) Spectrum monitoring Scene monitoring Network monitoring and diagnostics (router failures, intruder detection) Databases ..... GV MOUSTAKIDES: Sequential Detection, Overview & Open Problems, ISR, Dec. 2011

19. Mathematical setup We observe sequentially a process {xt} that has the following statistical properties • Changetime ¿ : either random with known prior or deterministic but unknown. • Both pdfs f0, f1 are considered known. Detect occurrence of ¿ as soon as possible GV MOUSTAKIDES: Sequential Detection, Overview & Open Problems, ISR, Dec. 2011

20. We are interested in sequential detection schemes. Whatever sequential scheme one can think of, at every time instant t it will have to make one of the following two decisions: • Either decide that a change didn’t take place before t, therefore it needs to continue taking more data. • Or that a change took place before t and therefore it should stop and issue an alarm. Sequential Detector  Stopping rule GV MOUSTAKIDES: Sequential Detection, Overview & Open Problems, ISR, Dec. 2011

21. Shiryaev test (Bayesian, Shiryaev 1963) Changetime ¿ is random with Geometric prior. If T is a stopping rule then we define the following cost function Optimum T : GV MOUSTAKIDES: Sequential Detection, Overview & Open Problems, ISR, Dec. 2011

22. Define the statistic: There exists such that the following rule is optimum. In discrete time when {xt} are i.i.d. before and after the change. In continuous time when {xt} is a Brownian motion with constant drift before and after the change. In continuous time when {xt} is Poisson with constant rate before and after the change. GV MOUSTAKIDES: Sequential Detection, Overview & Open Problems, ISR, Dec. 2011

23. Shiryaev-Roberts test (Minmax, Pollak 1985) Changetime ¿ is deterministic and unknown. For any stopping rule T define the following criterion: Optimum T : subject to GV MOUSTAKIDES: Sequential Detection, Overview & Open Problems, ISR, Dec. 2011

24. In discrete time, when data are i.i.d. before and after the change with pdfs f0,f1. Compute recursively the following statistic: Pollak (1985) Yakir (AoS 1997) provided a proof of strict optimality. Mei (2006) showed that the proof was problematic. Tartakovsky (2011) found a counterexample. GV MOUSTAKIDES: Sequential Detection, Overview & Open Problems, ISR, Dec. 2011

25. CUSUM test (minmax, Lorden 1971) Changetime ¿ is deterministic and unknown. For any stopping rule T define the following criterion: Optimum T : subject to GV MOUSTAKIDES: Sequential Detection, Overview & Open Problems, ISR, Dec. 2011

26. Compute the Cumulative Sum (CUSUM) statistic yt as follows: Running LLR Running minimum CUSUM statistic CUSUM test For i.i.d. GV MOUSTAKIDES: Sequential Detection, Overview & Open Problems, ISR, Dec. 2011

27. º S º º ut mt GV MOUSTAKIDES: Sequential Detection, Overview & Open Problems, ISR, Dec. 2011

28. Discrete time: i.i.d. before and after the change Lorden (1971) asymptotic optimality. Moustakides (1986) strict optimality. Continuous time Shiryaev (1996), Beibel (1996) strict optimality for BM Moustakides (2004) strict optimality for Ito processes Moustakides (>2012) strict optimality for Poisson processes. Open Problems: Dependent data, Non abrupt changes, Transient changes,… GV MOUSTAKIDES: Sequential Detection, Overview & Open Problems, ISR, Dec. 2011

29. Decentralized detection GV MOUSTAKIDES: Sequential Detection, Overview & Open Problems, ISR, Dec. 2011

30. 1 B1 ut - 1 t1 1 0 t2 t ut ut 1 1 A1 0 If more than 1 bits,quantize overshoot! GV MOUSTAKIDES: Sequential Detection, Overview & Open Problems, ISR, Dec. 2011

31. The Fusion center if at time tn receives information from sensor i it updates an estimate of the global log-likelihood ratio: and performs an SPRT (if hypothesis testing) or a CUSUM (if change detection) using the estimate of the global log-likelihood ratio. GV MOUSTAKIDES: Sequential Detection, Overview & Open Problems, ISR, Dec. 2011

32. GV MOUSTAKIDES: Sequential Detection, Overview & Open Problems, ISR, Dec. 2011

33. Thank you for your attention GV MOUSTAKIDES: Sequential Detection, Overview & Open Problems, ISR, Dec. 2011