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
• ne.
• Wrong Isolation: Identifying the wrong correlated pair when there is
• ne.

Problem formulation

Problem Setup

– p information sources: {Xi(t) : t ∈ N}, i = 1 . . . p.

• For a fixed source i ∈ {1, . . . , p}, Xi(t)

iid

∼ N(0, 1), t ∈ N.

• The set of all (unordered) pairs: E := {(i, j) : 1 ≤ i < j ≤ p}
• At each time t ∈ N, Corr(Xk(t), Xl(t)) = ρe, where e ∈ E such that

e = (k, l).

– Given a user-specified value ρ∗ ∈ (0, 1), we perform multiple testing

• for each e ∈ E, H0 : ρe = 0 vs. H1 : |ρe| = ρ∗,
• when at most one of the p

2

• nulls should be rejected.

Problem Setup

– Ft = σ(X(1), . . . , X(t)), where X(t) = (X1(t), X2(t), . . . , Xp(t)). – A sequential test (τ, d) consists of:

• an {Ft}-stopping time, τ, at which we stop sampling,
• and an {Fτ}-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

• P0 : prob. measure when all sources are independent.
• Pe+ (resp. Pe−): 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:

P0(d = ∅) ≤ α,

• Missed detection: for all e ∈ E,

Pe+(d = ∅), Pe+(d = ∅) ≤ β,

• Wrong Isolation: for all e ∈ E,

Pe+(d = ∅, d = {e}), Pe−(d = ∅, d = {e}) ≤ γ.

– Problem: Find (τ, d) ∈ ∆(α, β, γ) that minimizes E[τ] under P0 and Pe+, Pe− 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 Λe+(n) := dPe+ dP0 (F(n)), Λe−(n) := dPe− dP0 (F(n)). – Mixture likelihood ratio statistic for the two sided testing problem: Λe(n) := Λe+(n) + Λe−(n) 2 . – At time n, the ordered mixture likelihood ratio statistics are: Λ(1)(n) ≥ . . . Λ(K)(n), and Λik(n) ≡ Λ(k)(n), k = 1 . . . K := p 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{τ1, τ2}, with

• τ1 := inf{n ≥ 1 : Λ(1)(n) ≤ 1/A},
• τ2 := inf{n ≥ 1 : Λ(1)(n) ≥ B, Λ(1)(n)/Λ(2)(n) ≥ C}.

– d∗ :=

• ∅

if τ1 < τ2, i1(τ∗) if τ2 < τ1.

Illustration

Σ =

• 1

0.8 0.8 1 1

• .

Σ =

• 1

1 1

• .

5 10 15 20 25 30 −15 −10 −5 5 10 15 sample size log(statistic) log(C) −log(A) log(B) stop sampling

(1,2) (2,3) (3,1)

5 10 15 20 25 30 −15 −10 −5 5 10 15 sample size log(statistic) −log(A) log(B) stop sampling

(1,2) (2,3) (3,1)

Error Control

Recall, K = p

2

• .

Theorem For any A, B, C > 1, we have P0(d∗ = ∅) ≤ K/B, Pe+(d∗ = ∅) = Pe−(d∗ = ∅) ≤ 1/A, Pe+(d∗ = ∅, d∗ = {e}) = Pe−(d∗ = ∅, d∗ = {e}) ≤ (K − 1)/C. In particular, (τ∗, d∗) ∈ ∆(α, β, γ) when A = 1 β , B = K α and C = K − 1 γ . (1)

Asymptotic Upper Bound

– For each e ∈ E, the KL information numbers D0 := E0[− log Λe+(1)] = E0[− log Λe−(1)], D1 := Ee+[log Λe+(1)] = Ee−[log Λe−(1)]. – Let x ∧ y := min{x, y}, x ∨ y := max{x, y}. Lemma Let e ∈ E. As A, B, C → ∞ we have E0[τ∗] ≤ log A D0 (1 + o(1)), Ee−[τ∗], Ee+[τ∗] ≤ log B D1

• log C

D0 + D1

• (1 + o(1)).

Asymptotic Optimality

Universal Lower Bound

• Let

h(x, y) := x log

• x

1 − y

• + (1 − x) log

1 − x y

• ,

x, y ∈ (0, 1). Lemma If α, β, γ ∈ (0, 1) such that α + β < 1 and β + 2γ < 1, e ∈ E, and (τ, d) ∈ ∆(α, β, γ), then E0[τ] ≥ h(α, β) D0 , Ee+[τ], Ee−[τ] ≥ h(β, α) D1 h(β + γ, γ) ∨ h(γ, β + γ) D0 + D1 .

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 E0[τ∗] ∼ inf

(τ,d)∈∆(α,β,γ) E0[τ] ∼ | log β|

D0 , Ee+[τ∗] ∼ inf

(τ,d)∈∆(α,β,γ) Ee+[τ] ∼ | log α|

D1 | log γ| D0 + D1 , Ee−[τ∗] ∼ inf

(τ,d)∈∆(α,β,γ) Ee−[τ] ∼ | log α|

D1 | log γ| D0 + D1 .

Simulation Study

An Alternate Rule

– An alternate rule (τint, dint) is a modification of the intersection rule proposed in [De and Baron, 2012], where

• τint := inf{n ≥ 1 : 0 ≤ p(n) ≤ 1 and Λe(n) /

∈ (1/A, B) for all e ∈ E},

• dint :=
• ∅

if p(τint) = 0, i1(τint)

• therwise.

,

• p(n) = |{e ∈ E : Λe(n) > 1}|.

– (τint, dint) ∈ ∆(α, β, γ) when the thresholds are A = 1 β and B = max K α , K − 1 γ

• .

Illustration

Σ =

• 1

0.8 0.8 1 1

• .

Σ =

• 1

1 1

• .

5 10 15 20 25 30 −15 −10 −5 5 10 15 sample size log(statistic) log(C) −log(A) log(B) proposed rule stops intersection rule (modified) stops

(1,2) (2,3) (3,1)

5 10 15 20 25 30 −15 −10 −5 5 10 15 sample size log(statistic) −log(A) log(B) proposed rule stops intersection rule (modified) stops

(1,2) (2,3) (3,1)

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).

20 40 60 80 100 True value of correlation in the correlated pair Expected Sample Size −0.7 0.0 0.7 Intersection Rule Proposed Rule

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

• f 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.

References ii

Kowalski, C. J. (1971). The oc and asn functions of some sprt’s for the correlation coefficient. Technometrics, 13(4):833–841. Pradhan, M. and Sathe, Y. S. (1975). An unbiased estimator and a sequential test for the correlation coefficient. Journal of the American Statistical Association, 70(349):160–161. Song, Y. and Fellouris, G. (2017). Asymptotically optimal, sequential, multiple testing procedures with prior information on the number of signals.

• Electron. J. Statist., 11(1):338–363.

References iii

Wald, A. (1945). Sequential tests of statistical hypotheses. The Annals of Mathematical Statistics, 16(2):117–186. Wolde-Tsadik, G. (1976). A generalization of an sprt for the correlation coefficient. Journal of the American Statistical Association, 71(355):709–710. Woodroofe, M. (1979). Repeated likelihood ratio tests. Biometrika, 66(3):453–463.

Thank you!