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Quickest detection and the problem of two-sided alternatives

Quickest detection and the problem of two-sided alternatives. Olympia Hadjiliadis. Outline. The change detection problem Overview of existing results Lorden’s criterion & the CUSUM stopping time The Brownian motion model with multiple alternatives A modified Lorden criterion

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Quickest detection and the problem of two-sided alternatives

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  1. Quickest detection and the problem of two-sided alternatives Olympia Hadjiliadis

  2. Outline • The change detection problem • Overview of existing results • Lorden’s criterion & the CUSUM stopping time • The Brownian motion model with multiple alternatives • A modified Lorden criterion • Optimality of the CUSUM rule for 1-sided alternatives • Two sided 2-CUSUM test & the HMR • Further optimality issues for 2-sided alternatives & Optimality of Equalizer Rules • Discussion and Open Problems

  3. We are observing sequentially a process with the following characteristics: The change detection problem • Change time : deterministic (but unknown) or random • In control distribution is known • Out of control can be known or unknown Detect the change “as soon as possible” Applications include: systems monitoring, quality control, financial decision making, remote sensing (radar, sonar, seismology), occurrence of industrial accidents, epidemiology etc

  4. The observation process »t becomes available sequentially;This can be expressed through the filtration: For detecting the change we are interested in sequential schemes Any sequential detection scheme can be represented by a stopping time T (the time we stop & declare the change) The stopping time T is adapted to In other words, at every instant t, we perform a test (whether to stop or continue sampling) using only available information up to time t

  5. : the probability measure induced by , when the change takes place at time : the corresponding expectation : all data under nominal (in-control) regime : all data under alternative (out-of-control) regime Optimality Criteria • They are basically a trade-off of two parts: • The first is the detection delay (out-of-control A.R.L.) • The second is the frequency of false alarms (in-control A.R.L.) Overview of Existing Results Possible approaches are Bayesian & Min-Max

  6. is random and exponentially distributed The Shiryaev test consists of computing the statistics and stop when • is optimum (Shiryaev 1978): • In discrete time: when are i.i.d. before and after the change • In continuous time: when is a Brownian Motion with constant drift before and after the change Bayesian Approach (Shiryaev):

  7. is deterministic and unknown subject to Optimality results exist only for discrete time when are i.i.d before and after the change. Specifically if we define the statistics where , the common pdf of the data before and after the change then (Yakir 1997) the stopping time is optimum Min-Max Approach (Shiryayev-Roberts-Pollak)

  8. subject to Remark: In seeking solutions need only Lorden’s Criterion An alternative min-max approach: (Lorden 1971) And solve the min-max problem The test closely related to Lorden’s criterion and being the most popular one used in practice is the Cumulative Sum (CUSUM) test

  9. The CUSUM statistic process • Optimality results • Discrete time: when are i.i.d. before and after the change (Moustakides 1986, Ritov 1990) • Continuous time: when is a Brownian Motion with constant drift before and after the change (Shiryayev 1996, Beibel 1996) Define the CUSUM statistics as follows: ; The CUSUM stopping time (Page 1954): where the threshold º is selected so that

  10. ut mt yt TC

  11. WHY THE CUSUM? X1,X2,…,Xt i.i.d. -Composite hypothesis H0 : no change vs H1 : change at time 1 or H2 : change at time 2 or ... Ht : change at time t testing

  12. The Brownian motion model with one alternative We observe sequentially the process with the following dynamics: where the drift ¹ is known.

  13. Lorden’s criterion & the CUSUM

  14. We observe sequentially the process with the following dynamics: where the are known Remark:We are not interested in identifying which of the occurs, just in detecting the change time . Due to the symmetry of the BM, it suffices to consider the following cases: Case 1: Case 2 (Symmetric): Case 3 (Asymmetric): The Brownian motion model with two/multiple alternatives

  15. We want to solve the min-max problem with the following optimality criterion: subject to A modified Lorden Criterion One possible way to extend Lorden’s criterion, to include multiple changes, is in the following way: Remark:We are interested solely in the detection problem. No special care is taken for estimating the type of change (i.e. identifying i).

  16. Key point in the proof is the following inequality: Optimality Issues Case 1: We can show that the 1-sided CUSUM test, run for is optimal (in a sense detecting is the worst case scenario.

  17. m T2 T1

  18. The process is observed sequentially Change-point detection in B.M. model • is: • and • : are known • deterministic: (Min-Max approach)

  19. subject to Let S be a s.t. & Let U be s.t. declares an alarm at the same time as S but when “-” observations are received Extended Lorden’s criterion The optimal solution has to satisfy… ? Construct a randomized rule V : Flip an unbiased coin. If Heads then follow U. If Tails then follow S.

  20. And the corresponding CUSUM stopping times: Then the 2-sided CUSUM stopping time is: Notice The classical 2-CUSUM rule

  21. The best 2-CUSUM OBJECTIVE: Look in the 2-CUSUM class and find the best rule ( ) in both: • Symmetric case • Non-symmetric case • Harmonic Mean 2-CUSUM rules • Non-harmonic mean 2-CUSUM rules

  22. Harmonic Mean 2-CUSUM rules HMR: Side results ,

  23. The best 2-CUSUM Symmetric case Pick i.e. a Harmonic mean 2-CUSUM rule Hence the best 2-CUSUM rules is a HMR Is it unique? Non-symmetric case ?

  24. Tc1 Tc2 ? Non Harmonic Mean 2-CUSUM rules Case

  25. Therefore, the event is conditioned upon For general ? Non-Harmonic Mean 2-CUSUM (cont.) Suppose that for big.

  26. Andersen (1960) no change change is change is The FIRST MOMENT of a 2-CUSUM rule is Non-Harmonic mean 2-CUSUM (cont’d) The first moment

  27. no change change is change is Taylor (1975) Lehoczky (1977) Non-harmonic mean 2-CUSUM (cont.) Upper and lower bounds < <

  28. ? ? under ; change is under ; no change under ; change is change is no change change is Non-harmonic mean 2-CUSUM

  29. Non-harmonic mean 2-CUSUM Under Under Under The first moment of a general 2-CUSUM ( )

  30. And the corresponding CUSUM stopping times: Define two CUSUM processes for drifts ; EqR.: Case 2 (symmetric) Case 3 (asymmetric) Select a modified 2-CUSUM HMR rule with Modified H.M. Eq. 2-CUSUM rules For small the suggested rule is better than classical 2-CUSUM

  31. Theorem Case 3 (Asymmetric): Define the class of Equalizer Rules:

  32. Proof: LHS of (2) = RHS of (1) Hence: LHS of (1) > RHS of (2)

  33. 2(µ2-µ1) λ1’ λ1 λ1+2µ2 λ2’ λ2 λ1’+2µ2 Equalizer Rules We can rewrite (1) as follows:

  34. EqR.: The best modified H.M. Eq.R 2-CUSUM EqR.: Pick such that RESULT: moderate or big values of Classical better than Modified A comparison -mod. H.M. Eq. 2-CUSUM rules & -classical Eq. 2-CUSUM rules Non-symmetric case (cont.) Mod. H.M. 2-CUSUM OR Classical 2-CUSUM

  35. Symmetric case Classical 2-CUSUM Modified drift λ 2-CUSUM

  36. Non-symmetric case Modified 2-CUSUM HMR λ2=μ2 , λ1=2μ1–μ2 Modified2-CUSUM HMR λ2-λ1=2(μ2–μ1) Classical 2-CUSUM ν1 > ν2

  37. The difference of the two detection delays tends to the constant: . This proves asymptotic optimality.

  38. True drifts: ¹1=1, ¹2= 1.2; Drifts used in 2-CUSUM: ¸ 1=1, ¸ 2= 1.4;

  39. True drifts: ¹1=1, ¹2= 1.5 Drifts used in 2-CUSUM: ¸ 1=1, ¸ 2= 2

  40. Publications • "Optimal and Asymptotically Optimal CUSUM rules for change point detection in the Brownian Motion model with multiple alternatives" O. Hadjiliadis and G. V. Moustakides Theory of Probability and its Applications, 50(1), 2006. • "Optimality of the 2-CUSUM drift equalizer rules for detecting two-sided alternatives in the Brownian motion model" O. Hadjiliadis Journal of Applied Probability, 42(4), 2005. • '‘On the existence and uniqueness of the best 2-CUSUM rules for quickest detection of two-sided alternatives in a Brownian motion model'' O. Hadjiliadis and H. V. Poor [Theory of Probability and its Applications, 53(3), 2008] • '‘A comparison of the best 2-CUSUM rules for quickest detection of two-sided alternatives in a Brownian motion model'' O. Hadjiliadis G. Hernandez-del-Valle, I.Stamos. [Sequential Analysis, 28(1), 2009]

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