Quickest Detection With Post-Change Distribution Uncertainty

Transcription

1 Quickest Detection With Post-Change Distribution Uncertainty Heng Yang City University of New York, Graduate Center Olympia Hadjiliadis City University of New York, Brooklyn College and Graduate Center Michael Ludkovski University of California Santa Barbara June 24th, 2015 IWSM (Columbia University) Detection With Post-Change Uncertainty June 24, / 33

2 Content of Topics 1 Introduction 2 Notations And Detection Problem Notations Detection Delay Criterion And False Alarm the Λ-CUSUM stopping time 3 A Composite Stopping Time Third-Order Asymptotic Optimality An Identification Function Numerical Illustration IWSM (Columbia University) Detection With Post-Change Uncertainty June 24, / 33

3 Introduction Introduction This quickest detection problem is concerned with detecting an abrupt change in the distribution of a stochastic process. In this work, we assume that there is uncertainty about the post-change distribution. This problem is different from the estimation of the change point when given the information that the change has already happened in a time window. IWSM (Columbia University) Detection With Post-Change Uncertainty June 24, / 33

4 Contribution Introduction We develop a novel family of composite stopping times that combines multiple CUSUM stages along with the CUSUM reaction period (CRP). We design such composite stopping times to establish third-order optimality for the problem of detecting a Wiener disorder with uncertain post-change drift. We also use such composite stopping times to analyze the question of distinguishing the different values of post-change drift and show an asymptotic identification result. IWSM (Columbia University) Detection With Post-Change Uncertainty June 24, / 33

5 Mathematical Setup Notations And Detection Problem Notations On a probability space (Ω, F,P), we observe the sequential process {Z t } t 0 with Z 0 = 0. Suppose that this process changes from a standard Brownian motion to a Brownian motion with drift m at a fixed but unknown change time τ 0, that is: { dw t t < τ dz t := mdt + dw t t τ where W is a standard Brownian motion. IWSM (Columbia University) Detection With Post-Change Uncertainty June 24, / 33

6 Post-Change Drift Notations And Detection Problem Notations We have no probability structure on the change time τ. Suppose that the post-change drift m comes from a Bernoulli distribution, which is independent of the pre-change Brownian motion process {W s } s<τ : { m 1 with likelihood p m = with likelihood 1 p m 2 where 0 < m 1 < m 2 and 0 < p < 1 are known constants. We can generalize to the discrete range {m 1,m 2,...,m N }. IWSM (Columbia University) Detection With Post-Change Uncertainty June 24, / 33

7 Filtration Notations And Detection Problem Notations To facilitate our analysis, we introduce the measures and filtration. Assume that the probability space supports a Bernoulli random variable U that is independent of {Z t } t=0. Let G t correspond to the σ algebra generated by both {Z s } s t and U. Note that G 0 = σ(u). This enlargement of the natural filtration of Z is to enable randomization. IWSM (Columbia University) Detection With Post-Change Uncertainty June 24, / 33

8 Measures Notations And Detection Problem Notations P m i τ is the measure defined on this filtration, under which we have dz t = dw t for t < τ, and dz t = m i dt + dw t for t τ, where τ [0, ) and i = 1,2. P represents the measure under which there is no change, and so dz t = dw t for all t. The uncertainty regarding m is modeled by a probability measure P m τ = pp m 1 τ + (1 p)p m 2 τ. IWSM (Columbia University) Detection With Post-Change Uncertainty June 24, / 33

9 Notations And Detection Problem Detection Delay Criterion And False Alarm Detection Delay Criterion And False Alarm Our objective is to find an stopping time to detect the change point that balances the trade off between a small detection delay and the penalty on the frequency of false alarms. IWSM (Columbia University) Detection With Post-Change Uncertainty June 24, / 33

10 Notations And Detection Problem Detection Delay Criterion And False Alarm Detection Delay Criterion And False Alarm Our objective is to find an stopping time to detect the change point that balances the trade off between a small detection delay and the penalty on the frequency of false alarms. For any G-stopping time R, the false alarm can be represent by E [R]. IWSM (Columbia University) Detection With Post-Change Uncertainty June 24, / 33

11 Notations And Detection Problem Detection Delay Criterion And False Alarm Detection Delay Criterion And False Alarm Our objective is to find an stopping time to detect the change point that balances the trade off between a small detection delay and the penalty on the frequency of false alarms. For any G-stopping time R, the false alarm can be represent by E [R]. The worst detection delay of R given the post-change drift m i is of Lorden s form: J i (R) := supesssup ω E m i τ [(R τ) + G τ ]. τ 0 Since m is unknown, we average over J i s according to the distribution of m to define the detection delay criterion: J(R) := pj 1 (R) + (1 p)j 2 (R). IWSM (Columbia University) Detection With Post-Change Uncertainty June 24, / 33

12 Notations And Detection Problem The Detection Problem Detection Delay Criterion And False Alarm We would like to find a stopping time T S to detect τ, where for a constant γ > 1. S = {G stopping time T: E [T] γ} Given γ,m 1,m 2,p, our quickest detection problem is inf J(T) T S IWSM (Columbia University) Detection With Post-Change Uncertainty June 24, / 33

13 Notations And Detection Problem The Λ-CUSUM Stopping Time the Λ-CUSUM stopping time A Λ-CUSUM stopping time (Cumulative Sum) is a stopping time with the parameters Λ and K > 0 defined as { } T(ξ,Λ,K) := inf t > 0 : V t inf V s K 0 s t where V t = Λξ t 1 2 Λ2 t and ξ := {ξ t } t 0 is a stochastic process. IWSM (Columbia University) Detection With Post-Change Uncertainty June 24, / 33

14 Notations And Detection Problem The Λ-CUSUM Stopping Time the Λ-CUSUM stopping time When m 1 = m 2 = Λ is a constant, the log-form of maximum likelihood ratio becomes dp Λ τ log sup 0 τ t dp = V t inf V τ Gt 0 τ t where V t = ΛZ t 1 2 Λ2 t. The CUSUM-form stopping time provides an optimal solution in the detection problem in this case. IWSM (Columbia University) Detection With Post-Change Uncertainty June 24, / 33

15 Notations And Detection Problem the Λ-CUSUM stopping time Detection Problem Without Post-change Uncertainty Theorem (Beibel 96; Shiryaev 96; Moustakides 04) When the post-change drift is a constant m, for any stopping time T S, we have J(T) J(T m K ), where T m K is the CUSUM stopping time satisfying E [T m K ] = γ. In our work, we suppose that the post-change drift has uncertainty, that is 0 < m 1 < m 2. We can show that the Λ-CUSUM stopping time is no longer optimal stopping time or even asymptotic optimal stopping time. IWSM (Columbia University) Detection With Post-Change Uncertainty June 24, / 33

16 A Composite Stopping Time Construction Of A Composite Stopping Time We consider a composite stopping time, namely T com, that is constructed in two stages. In the first stage, we apply a CUSUM stopping time, where 0 < λ < 2m 1 and ν > 0. T λ ν := T(Z,λ,ν) We define the CUSUM reaction period (CRP) with respect to the first stage. IWSM (Columbia University) Detection With Post-Change Uncertainty June 24, / 33

17 CUSUM Reaction Period (CRP) A Composite Stopping Time Up to the first stage, there is the last reset time (last running minimum position): { } ρ(z,λ,ν) := sup t [0,Tν λ ) : V t = inf V s. s t The time length between the last reset and the end of the first stage is S(Z,λ,ν) := T(Z,λ,ν) ρ(z,λ,ν). Denote S λ ν = S(Z,λ,ν) and we call it as the CUSUM reaction period (CRP) with respect to T λ ν. IWSM (Columbia University) Detection With Post-Change Uncertainty June 24, / 33

18 Motivation of CRP A Composite Stopping Time Conditional on the case that the change has happened before the beginning of CRP Sν λ, CRP has the following pattern. Density of S λ ν S λ ν Figure: The parameters are λ = 1 and ν = 5. The solid curve (left) gives the density function under the case {m = 5, τ < ρ λ ν }; the dashed curve (right) gives the density under the case {m = 2, τ < ρ λ ν}. IWSM (Columbia University) Detection With Post-Change Uncertainty June 24, / 33

19 A Composite Stopping Time Construction Of A Composite Stopping Time (Cont.) To continue the second stage, we define the shift function θ u of the process for a time u 0 as follows θ u (Z) := {Z t+u Z u } t 0. Note that θ s makes the path re-start at time s from 0. In the second stage, we define G-stopping times T µ 1 h 1 and T µ 2 h 2 : T µ 1 h 1 := T(θ T λ ν (Z),µ 1,h 1 ) and T µ 2 h 2 := T(θ T λ ν (Z),µ 2,h 2 ), where µ i s are constants satisfying 0 < µ i < 2m 1 and h i > 0 are the second-stage thresholds for i = 1,2. IWSM (Columbia University) Detection With Post-Change Uncertainty June 24, / 33

20 A Composite Stopping Time Construction Of A Composite Stopping Time (Cont.) To connect the first and second stages, for a parameter b > 0, when Sν λ b, we continue the second step and stop at T µ 1 h 1 ; when Sν λ < b, we continue the second step and stop at T µ 2 h 2. So the composite stopping time T com is defined as T com = T λ ν + 1 {S λ ν b}t µ 1 h {S λ ν <b}t µ 2 h 2, where 1 {S λ ν b} and 1 {S λ ν <b} are indicator functions. IWSM (Columbia University) Detection With Post-Change Uncertainty June 24, / 33

21 A Composite Stopping Time Idea Of T com The diagram below illustrates the construction of T com. T λ ν S λ ν b S λ ν < b T µ 1 h 1 T µ 2 h 2 IWSM (Columbia University) Detection With Post-Change Uncertainty June 24, / 33

22 Third-Order Asymptotic Optimality A Lower Bound Of Detection Delay Lemma (Lower Bound of Delay) For any G-stopping time T S, we have the lower bound on the detection delay J(T) as where J(T) LB(γ) := 2p ( m1 2 g f 1 ( m2 1 γ ) 2 ) + ( 2(1 p) m2 2 g f 1 ( m2 2 γ ) 2 ) g(x) = e x + x 1 and f (x) = e x x 1, for x > 0. IWSM (Columbia University) Detection With Post-Change Uncertainty June 24, / 33

23 Third-Order Asymptotic Optimality Third-Order Asymptotic Optimality Theorem (Asymptotic Optimality of T com ) The composite G-stopping time the form T com defined above, satisfying the frequency of false alarm constraint E [T com ] = γ, is asymptotically optimal of third order in the sense that lim [J(T com) LB(γ)] = 0 γ when its parameters λ,ν,b,h i,µ i satisfy appropriate conditions. IWSM (Columbia University) Detection With Post-Change Uncertainty June 24, / 33

24 The Behavior of CRP Third-Order Asymptotic Optimality Conditional on the case that the change has happened before the recording of CRP S λ ν, we have the following result. Lemma For any parameter λ (0,2m 1 ) and b,ν > 0 such that b/ν = l is a positive constant that satisfies l > 2 λ(2m 1 λ), there exists a positive constant L = L(m 1,λ,l) such that ( lim ν Pm 1 τ Sν λ b ) τ < ρ λ ν e Lν = 0. IWSM (Columbia University) Detection With Post-Change Uncertainty June 24, / 33

25 The Behavior of CRP (Cont.) Third-Order Asymptotic Optimality When there is no change during the recording of CRP, we have the following result. Lemma For parameters λ,b,ν > 0 such that b/ν is a positive constant, we have lim ν P (S λ ν < b) = 0. Thus, the event {S λ ν < b} can help to distinguish the two cases. IWSM (Columbia University) Detection With Post-Change Uncertainty June 24, / 33

26 An Identification Function Identification Of Post-Change Distribution In our work, we suppose that the post-change drift has uncertainty, that is 0 < m 1 < m 2. We may want not only to detect the change point, but also to identify the post-change distribution. We want to find the additional information about the post-change drift based on the composite stopping time. This requirement is different from the estimation problem solely. IWSM (Columbia University) Detection With Post-Change Uncertainty June 24, / 33

27 An Identification Function An Identification Function For the composite rule T com, it is possible to construct an identification function δ Tcom G Tcom taking values in {m 1,m 2 } to serve the purpose of post-change identification of the drift. To this effort, we denote S h2 := S(Z,µ 2,h 2 ) as CRP in the second stage T µ 2 h 2, and denote the CRP S ν = Sν λ first stage. The identification function δ Tcom is defined as follows { m 1 if {S ν b ν } {S h2 b h2,s ν < b ν }; δ Tcom := m 2 if {S h2 < b h2,s ν < b ν }. in the for positive constants b ν and b h2. IWSM (Columbia University) Detection With Post-Change Uncertainty June 24, / 33

28 An Identification Function Asymptotic Identification Performance Proposition If we choose µ 2 = m 1 and a constant l such that 2 m 1 (2m 2 m 1 ) < l < 2 m 2 1 and let b h2 = l h 2. then, as γ, we have lim γ Pm i τ ( δ Tcom m i τ < ρ λ ν ) = 0 for i = 1,2 As γ, the condition is equivalent to {τ < }. IWSM (Columbia University) Detection With Post-Change Uncertainty June 24, / 33

29 The Behavior of CRP An Identification Function The identification performance is based on the CRP in the second stage. Lemma For any parameter µ 2 (0,2m 1 ), when we choose b h2 /h 2 = l to be a positive constant that satisfies l < 2 µ 2 (2m 1 µ 2 ), there exists a positive constant L 1 = L 1 (m 1,µ 2,l) such that ( lim h 2 Pm 1 τ S h2 < b h2 τ < ρ µ 2 h 2 )e L 1h 2 = 0. IWSM (Columbia University) Detection With Post-Change Uncertainty June 24, / 33

30 The Behavior of CRP (Cont.) An Identification Function Lemma For any parameter µ 2 (0,2m 2 ), when we choose b h2 /h 2 = l to be a positive constant that satisfies l > 2 µ 2 (2m 2 µ 2 ), there exists a positive constant L 2 = L 2 (m 2,µ 2,l) such that ( lim h 2 Pm 2 τ S h2 b h2 τ < ρ µ 2 h 2 )e L 2h 2 = 0. IWSM (Columbia University) Detection With Post-Change Uncertainty June 24, / 33

31 Numerical Illustrations J(T com ) LB(γ) Numerical Illustration J(T com )/LB(γ) γ γ 12 h J(T com ) LB(γ) γ γ Figure: Performance of T com in the case m 1 = 2, m 2 = 5, p = 0.4. All graphs are plotted against γ with a logarithmic x-axis. IWSM (Columbia University) Detection With Post-Change Uncertainty June 24, / 33

32 Numerical Illustrations (Cont.) Numerical Illustration Left: P m ( ) ( ) 1 τ δtcom m 1 τ < ρ λ ν. Right: P m 2 τ δtcom m 2 τ < ρ λ ν. All graphs are plotted against γ with a logarithmic x-axis. IWSM (Columbia University) Detection With Post-Change Uncertainty June 24, / 33

33 Future Work Numerical Illustration It is interesting to consider the case where the post-change drift has a more complicated struction. For example, the post-change drift is a piece-wise continuous function µ(t). We may let the post-change drift has a distribution which is unknown but in a family of the given distributions. So we can consider Bayesian estimation with respect to the posterior distribution of the post-change drift. The applications that satisfy such models are interesting. The efficient and robust algorithms will be needed. IWSM (Columbia University) Detection With Post-Change Uncertainty June 24, / 33

34 Reference Numerical Illustration M. Beibel, A note on Ritov s Bayes approach to the minimax property of the CUSUM procedure, Ann. Statist., Vol. 24, No. 4, pp , G. Lorden, Procedures for reacting to a change in distribution, The Annals of Mathematical Statistics, Vol. 42, No. 6, , G. V. Moustakides, Optimality of the CUSUM procedure in continuous time, Annals of Statistics, Vol. 32, No. 1, pp , A.N. Shiryaev, Minimax optimality of the method of cumulative sums (CUSUM) in the case of continuous time, Russian Mathem. Surv. 51, , H. Zhang and O. Hadjiliadis, Drawdowns and the speed of a market crash, Methodol. Comput. Appl. Probab. Vol. 14, no. 3 pp , IWSM (Columbia University) Detection With Post-Change Uncertainty June 24, / 33

35 Numerical Illustration Thank You! IWSM (Columbia University) Detection With Post-Change Uncertainty June 24, / 33