Sequential Detection and Isolation of a Correlated Pair Anamitra - PowerPoint PPT Presentation

Sequential Detection and Isolation of a Correlated Pair Anamitra Chaudhuri Department of Statistics University of Illinois, Urbana-Champaign Joint work with Georgios Fellouris 2020 IEEE International Symposium on Information Theory Los Angeles, California 21-26 June, 2020

Introduction

Motivation – Quickest inference about the underlying dependence structure. – Environmental monitoring, sensor networks, fault detection in power grid, neural coding etc. – In this context, – data are observed sequentially and the sample size is not fixed in advance, – there are multiple hypotheses regarding the dependence structure. – Goal: stop sampling as quickly as possible and identify the true hypothesis while controlling the probability of errors .

Related works – Detection and isolation of the correlation structure in a p − variate Gaussian random vector. – p = 2: Sequential hypothesis testing for the correlation coefficient ρ in bivariate Gaussian - Binary hypothesis testing [Choi, 1971, Kowalski, 1971, Pradhan and Sathe, 1975, Wolde-Tsadik, 1976, Wald, 1945, . . . ] - Two sided version [Woodroofe, 1979] – p > 2: Sequential multiple testing and design - Observation from only one component is taken at each time, temporal dependence [Heydari and Tajer, 2017] – Sequentially observed data from independent streams, simultaneous testing of multiple binary hypotheses. [Song and Fellouris, 2017]

Goal In this work, – data from all sources are observed sequentially, – the observations are independent over time , – at most one pair of its components is correlated. Goal: – stop sampling as quickly as possible, – identify the correlated pair, if there is any, – control three kinds of errors: - False Alarm: Detecting a correlated pair when there is none. - Missed Detection: Failing to detect a correlated pair when there is one. - Wrong Isolation: Identifying the wrong correlated pair when there is one.

Problem formulation

Problem Setup – p information sources: { X i ( t ) : t ∈ N } , i = 1 . . . p . iid - For a fixed source i ∈ { 1 , . . . , p } , X i ( t ) ∼ N (0 , 1) , t ∈ N . - The set of all (unordered) pairs: E := { ( i , j ) : 1 ≤ i < j ≤ p } - At each time t ∈ N , Corr ( X k ( t ) , X l ( t )) = ρ e , where e ∈ E such that e = ( k , l ). – Given a user-specified value ρ ∗ ∈ (0 , 1), we perform multiple testing - for each e ∈ E , H 0 : ρ e = 0 vs. H 1 : | ρ e | = ρ ∗ , - when at most one of the � p � nulls should be rejected. 2

Problem Setup – F t = σ ( X (1) , . . . , X ( t )), where X ( t ) = ( X 1 ( t ) , X 2 ( t ) , . . . , X p ( t )). – A sequential test ( τ, d ) consists of: - an {F t } -stopping time, τ , at which we stop sampling , - and an {F τ } -measurable decision rule d , which denotes the subset of pairs declared to be correlated upon stopping. – Since there is at most one correlated pair, let - P 0 : prob. measure when all sources are independent. - P e + (resp. P e − ): when the pair e has correlation ρ ∗ (resp. − ρ ∗ ) and all other sources are independent.

Problem Setup – ∆( α, β, γ ): the class of sequential tests ( τ, d ) for which - False alarm: P 0 ( d � = ∅ ) ≤ α, - Missed detection: for all e ∈ E , P e + ( d = ∅ ) , P e + ( d = ∅ ) ≤ β, - Wrong Isolation: for all e ∈ E , P e + ( d � = ∅ , d � = { e } ) , P e − ( d � = ∅ , d � = { e } ) ≤ γ. – Problem: Find ( τ, d ) ∈ ∆( α, β, γ ) that minimizes E [ τ ] under P 0 and P e + , P e − for every e ∈ E to a first order asymptotic approximation as α, β, γ → 0.

Notations and Statistics – For each e ∈ E , the likelihood ratios Λ e + ( n ) := dP e + Λ e − ( n ) := dP e − ( F ( n )) , ( F ( n )) . dP 0 dP 0 – Mixture likelihood ratio statistic for the two sided testing problem: Λ e ( n ) := Λ e + ( n ) + Λ e − ( n ) . 2 – At time n , the ordered mixture likelihood ratio statistics are: � p � Λ (1) ( n ) ≥ . . . Λ ( K ) ( n ) , and Λ i k ( n ) ≡ Λ ( k ) ( n ) , k = 1 . . . K := . 2

Proposed Procedure

Proposed Rule Inspired by the gap-intersection rule proposed in [Song and Fellouris, 2017], our proposed procedure is ( τ ∗ , d ∗ ), where – τ ∗ := min { τ 1 , τ 2 } , with - τ 1 := inf { n ≥ 1 : Λ (1) ( n ) ≤ 1 / A } , - τ 2 := inf { n ≥ 1 : Λ (1) ( n ) ≥ B , Λ (1) ( n ) / Λ (2) ( n ) ≥ C } . � ∅ if τ 1 < τ 2 , – d ∗ := i 1 ( τ ∗ ) if τ 2 < τ 1 .

Illustration � � � � 1 0 . 8 0 1 0 0 Σ = . Σ = . 0 . 8 1 0 0 1 0 0 0 1 0 0 1 15 15 (1,2) 10 10 log(B) log(B) 5 5 log(C) log(statistic) log(statistic) 0 0 (2,3) (2,3) −5 −5 −log(A) −log(A) (1,2) −10 −10 (3,1) (3,1) stop sampling stop sampling −15 −15 0 5 10 15 20 25 30 0 5 10 15 20 25 30 sample size sample size

Error Control � p � Recall, K = . 2 Theorem For any A , B , C > 1 , we have P 0 ( d ∗ � = ∅ ) ≤ K / B , P e + ( d ∗ = ∅ ) = P e − ( d ∗ = ∅ ) ≤ 1 / A , P e + ( d ∗ � = ∅ , d ∗ � = { e } ) = P e − ( d ∗ � = ∅ , d ∗ � = { e } ) ≤ ( K − 1) / C . In particular, ( τ ∗ , d ∗ ) ∈ ∆( α, β, γ ) when A = 1 α and C = K − 1 B = K β , . (1) γ

Asymptotic Upper Bound – For each e ∈ E , the KL information numbers D 0 := E 0 [ − log Λ e + (1)] = E 0 [ − log Λ e − (1)] , D 1 := E e + [log Λ e + (1)] = E e − [log Λ e − (1)] . – Let x ∧ y := min { x , y } , x ∨ y := max { x , y } . Lemma Let e ∈ E . As A , B , C → ∞ we have E 0 [ τ ∗ ] ≤ log A (1 + o (1)) , D 0 � log B log C � � E e − [ τ ∗ ] , E e + [ τ ∗ ] ≤ (1 + o (1)) . D 0 + D 1 D 1

Asymptotic Optimality

Universal Lower Bound - Let � x � � 1 − x � + (1 − x ) log x , y ∈ (0 , 1) . h ( x , y ) := x log , 1 − y y Lemma If α, β, γ ∈ (0 , 1) such that α + β < 1 and β + 2 γ < 1 , e ∈ E , and ( τ, d ) ∈ ∆( α, β, γ ) , then E 0 [ τ ] ≥ h ( α, β ) , D 0 � h ( β + γ, γ ) ∨ h ( γ, β + γ ) E e + [ τ ] , E e − [ τ ] ≥ h ( β, α ) . D 1 D 0 + D 1

Main Result: Asymptotic Optimality The definition of the function h allows us to have, when x , y → 0, - h ( x , y ) ∼ | log y | , - h ( x , y ) ∨ h ( y , x ) ∼ | log( x ∧ y ) | . Theorem Suppose the thresholds in ( τ ∗ , d ∗ ) are selected according to (1) . Then, for every e ∈ E , as α, β, γ → 0 we have ( τ, d ) ∈ ∆( α,β,γ ) E 0 [ τ ] ∼ | log β | E 0 [ τ ∗ ] ∼ inf , D 0 � | log γ | ( τ, d ) ∈ ∆( α,β,γ ) E e + [ τ ] ∼ | log α | E e + [ τ ∗ ] ∼ inf , D 0 + D 1 D 1 � | log γ | ( τ, d ) ∈ ∆( α,β,γ ) E e − [ τ ] ∼ | log α | E e − [ τ ∗ ] ∼ inf . D 0 + D 1 D 1

Simulation Study

An Alternate Rule – An alternate rule ( τ int , d int ) is a modification of the intersection rule proposed in [De and Baron, 2012], where - τ int := inf { n ≥ 1 : 0 ≤ p ( n ) ≤ 1 and Λ e ( n ) / ∈ (1 / A , B ) for all e ∈ E} , � ∅ if p ( τ int ) = 0 , - d int := , i 1 ( τ int ) otherwise . - p ( n ) = |{ e ∈ E : Λ e ( n ) > 1 }| . – ( τ int , d int ) ∈ ∆( α, β, γ ) when the thresholds are A = 1 � K α , K − 1 � and B = max . β γ

Illustration � � � � 1 0 . 8 0 1 0 0 Σ = . Σ = . 0 . 8 1 0 0 1 0 0 0 1 0 0 1 15 15 (1,2) 10 10 log(B) log(B) 5 5 log(C) log(statistic) log(statistic) 0 0 (1,2) (2,3) −5 −5 (2,3) −log(A) −log(A) −10 −10 (3,1) (3,1) proposed rule stops proposed rule stops intersection rule (modified) stops intersection rule (modified) stops −15 −15 0 5 10 15 20 25 30 0 5 10 15 20 25 30 sample size sample size

Comparison – p = 10 , ρ ∗ = 0 . 7 , α = β = 10 − 2 , γ = 10 − 3 . – only one pair is correlated with correlation coefficient ρ , all others are uncorrelated. – varied the value of ρ in the interval ( − 0 . 9 , 0 . 9). 100 Intersection Rule Proposed Rule 80 Expected Sample Size 60 40 20 −0.7 0.0 0.7 True value of correlation in the correlated pair

Summary

Summary – Proposed the problem of quick detection and isolation of a correlated pair in a Gaussian random vector. – Sequential multiple testing that controls three kinds of error : false alarm, missed detection and wrong isolation. – Goal: Minimize the average sample size subject to three error constraints. – Proposed a very simple rule based on the mixture likelihood ratios of the pairs and established its asymptotic optimality. – We compared our rule with an alternative one numerically and showed that its performance is significantly better, especially when the true value of the correlation is much higher .

References

References i Choi, S. C. (1971). Sequential test for correlation coefficients. Journal of the American Statistical Association , 66(335):575–576. De, S. K. and Baron, M. (2012). Sequential bonferroni methods for multiple hypothesis testing with strong control of family-wise error rates i and ii. Sequential Analysis , 31(2):238–262. Heydari, J. and Tajer, A. (2017). Quickest search for local structures in random graphs. IEEE Transactions on Signal and Information Processing over Networks , 3(3):526–538.