[PPT] - Unstructured Sequential Change Detection in Sensor Networks Grigory PowerPoint Presentation

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Unstructured Sequential Change Detection in Sensor Networks

Grigory Sokolov Department of Mathematics University of Southern California Los Angeles, California United States of America gsokolov@usc.edu Joint work with Georgios Fellouris and Alexander Tartakovsky Fourth International Workshop in Sequential Methodologies University of Georgia, Athens, Georgia, USA July 20, 2013

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Multi-sensor Change-point Detection

Given: K sensors , observing series {Xk

t }t≥1, k = 1, 2, · · · , K of independent

data, collected sequentially, one at a time at each sensor.

Assumption: the change can occur in an unknown subset of sensors N:

Xk

t i.i.d.

∼ fk(x), t ≤ ν, Xk

t i.i.d.

∼

gk(x) ≡ fk(x),

k ∈ N, fk(x), k / ∈ N, t > ν,

X



∼ f

...

Sensor 1:

X



∼ f

X

ν ∼ f

X

ν +1 ∼ g

X

ν +2 ∼ g

...

X



∼ f

...

Sensor 2:

X



∼ f

X

ν ∼ f

X

ν +1 ∼ f

X

ν +2 ∼ f

...

X K



∼ fK

...

Sensor K:

X K



∼ fK

X K

ν ∼ fK

X K

ν +1 ∼ gK X K ν +2 ∼ gK

...

Change occurs time

and the change-point, 0 ≤ ν ≤ ∞, is unknown (deterministic).

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Multi-sensor Change-point Detection (Cont’d)

Let Fk

t = σ(Xk 1, . . . , Xk t ) for t 0, k = 1, · · · , K with Fk 0 = {∅, Ω}.

Notation: for t 0 let Pt(·) = P( · |ν = t) and Et[ · ] = E[ · |ν = t]; in particular,

P∞(·) = P( · |ν = ∞) and E∞[ · ] = E[ · |ν = ∞].

Let T be a stopping time, adapted to {Fk

n}n1. Define ARL(T) = E∞[T].

Use Lorden’s detection measure (Lorden 1971)

J (T) = sup

ν≥0

ess sup Eν[(T − ν)+|Fν].

Goal: To find stopping time T, such that J (T) is minimized within class

{T : ARL(T) γ} for every γ 1.

Say that a stopping rule is first-order asymptotically optimal if

J (T) infT : ARL(T)γ J (T) → 1, γ → ∞.

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Multi-sensor Change-point Detection (Cont’d)

If the set of affected sensors, N, is known, the CUSUM stopping rule

TCUSUM = inf {t ≥ 1 : Wt ≥ a} , where Wt = ut − min

s≤t us,

ut =

t

i=1
k∈N

log gk f k(Xi), is optimal with respect to Lorden’s detection measure, when a is chosen so that ARL(TCUSUM) = γ.

When N is unknown, Mei (2010’11) proposed the following procedure:

TMei = inf

t ≥ 1 :

K

k=1

W k

t 1

l{W k

t ≥bk} ≥ a

,

where uk

t = t

i=1

log gk f k(Xi), W k

t = uk t − min s≤t uk s.

This procedure is first-order asymptotically optimal.

A different approach was suggested by Xie and Siegmund (2013).

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Decentralized Change-point Detection

Sensor 

Fusion Center . . . . .. ... ... . ....

Sensor  Sensor K

. . . . .. ... ... . ....

Sensor K–1

In a decentralized setting two types of constraints are usually considered:

a) Sensors communicate with the fusion center at a given rate (e.g. Banerjee and Veeravalli 2012). b) Only a certain number of bits is permissible per transmission (e.g. Mei 2005, Fellouris and Moustakides 2013).

We will address both types of communication constraints.

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Proposed Communication Scheme

Transmit information to the fusion center at times τ k
n. At any moment t, let

τ k(t) be the last transmission time prior to t.

Random sampling (Fellouris and Moustakides 2013):

τ k

n = inf

t > τ k

n−1 : uk t − uk τk

n−1 /

∈ (−∆k, ∆

k)

,

where ∆k, ∆

k are design parameters.

∆

k

∆k τ k



τ k



∆

k

∆k ∆

k

∆k

Communication times

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Proposed Communication Scheme (Cont’d)

Due to communication constraints, use uk

τk(t) instead of uk t .

Observe that uτ(t) can be written as

uτ(t) = (uτ1 − uτ0) + (uτ2 − uτ1) + · · · + (uτmt − uτmt−1) = ℓ1 + ℓ2 + · · · + ℓmt, where ℓn = uτn −uτn−1 is the accumulated log-likelihood ratio in the time-interval [τn, τn−1], and mt is the number of the last transmission.

It suffices to transmit ℓn to the fusion center.
Statistic is updated at communication times, and the stopping rule is

T = inf

t ≥ 1 :

K

k=1

V k

t ≥ a

,

V k

t = uk τk(t) − min m≤mk

t

uk

τk

m.

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Proposed 1-bit Procedure

Due to quantization constraints, approximate uk

t with ˜

uk

τk(t) from the last trans-

mission time: uk

τ(t) = ℓk 1 + ℓk 2 + · · · + ℓk mt,

≈ ˜ ℓk

1 + ˜

ℓk

2 + · · · + ˜

ℓk

mt = ˜

uk

τ(t),

where ˜ ℓk are approximations to ℓk.

Let each sensor transmit one-bit messages zn, the n-th message of whether the

threshold was crossed up on down: zk

n =

1,

if ℓk

n ≥ ∆ k

−1, if ℓk

n− ≤ ∆k

˜

ℓ is then defined following partial likelihood approach: ˜ ℓk

n = log P0(zk n = 1)

P∞(zk

n = 1) 1

l{zk

n=1} + log P0(zk

n = −1)

P∞(zk

n = −1) 1

l{zk

n=−1} .

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A Case Study: Gaussian Scenario

Let Xk

t ’s be standard Gaussian N (0, 1) before the change, and N (0.5, 1) for

k ∈ N after the change. Put K = 5, |N| = 2.

Goal: Examine how the performance is affected by the choice of δ = E∞(τ k

1 ),

the expected time to transmission.

We will examine several cases:

a) δ ≈ 1.5 (frequent transmissions), b) δ ≈ 4.0 (moderate transmission rate), and c) δ ≈ 10.0 (infrequent transmis- sions).

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Gaussian Scenario: 1-bit

15 25 35 45 55 65 100 1000 10000 100000 E

0(T)

ARL δ ≈ 1.5 δ ≈ 10.0 Figure 1: Dependency on average time between communications, δ.

One-bit procedure has two main disadvantages:

a) its performance drops for frequent transmissions, and b) few levels of ARL are attainable for low transmis- sion rates.

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Gaussian Scenario: 1-bit

15 25 35 45 55 65 100 1000 10000 100000 E

0(T)

ARL δ ≈ 1.5 δ ≈ 10.0 Full Figure 2: Dependency on average time between communications, δ.

One-bit procedure has two main disadvantages:

a) its performance drops for frequent transmissions, and b) few levels of ARL are attainable for low transmis- sion rates.

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Proposed Multi-bit Procedure

Remedy: let each sensor transmit with alphabet of the form {−d, · · · , −1, 1, · · · , d},

as opposed to 1-bit messages.

∆

k

∆k τ k



τ k



∆

k

∆k ∆

k

∆k

Overshoot Overshoot

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Proposed Multi-bit Procedure (Cont’d)

∆

k

∆k τ k



τ k



∆

k

∆k ∆

k

∆k

Overshoot Overshoot percentiles



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Proposed Multi-bit Procedure (Cont’d)

Transmit messages zk

n:

zk

n =

j,

if ǫk

j−1 ≤ ℓk n − ∆ k < ǫk j

−j, if − ǫk

j−1 < ℓk n + ∆k ≤ −ǫk j

, 1 ≤ j ≤ d, where the thresholds ǫ are the percentiles of ℓn − ∆.

As before, we approximate uk with ˜

uk

τk(t) = ˜

ℓk

1 + ˜

ℓk

2 + · · · + ˜

ℓk

mk

t , where

˜ ℓk

n = d

j=1
log P0(zk

n = j)

P∞(zk

n = j) 1

l{zk

n=j} + log P0(zk

n = −j)

P∞(zk

n = −j) 1

l{zk

n=−j}

.

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Gaussian Scenario, d = 2

15 25 35 45 55 65 100 1000 10000 100000 E

0(T)

ARL δ ≈ 1.5 δ ≈ 10.0 Figure 3: Dependency on average time between communications, δ.

For d = 2 procedure both effects are mitigated:

a) it performs well for frequent transmissions, and b) for low transmission rates more levels of ARL are attainable.

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Gaussian Scenario, d = 2

15 25 35 45 55 65 100 1000 10000 100000 E

0(T)

ARL δ ≈ 1.5 δ ≈ 10.0 Full Figure 4: Dependency on average time between communications, δ.

For d = 2 procedure both effects are mitigated:

a) it performs well for frequent transmissions, and b) for low transmission rates more levels of ARL are attainable.

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Proposed procedures offer discrete set of run lengths

15 25 35 45 55 65 100 1000 10000 100000 E

0(T)

ARL d = 1 d = 2 d = 3 Full Figure 5: Proposed procedures, δ = 10.0.

For practical purposes d = 3 procedure

a) performs well for frequent trans- missions, and b) offers reasonably dense set of ARL levels even at very low transmission rates.

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Proposed procedures offer discrete set of run lengths

15 25 35 45 55 65 100 1000 10000 100000 E

0(T)

ARL d = 1 Full Figure 6: Proposed procedures, δ = 10.0.

For practical purposes d = 3 procedure

a) performs well for frequent trans- missions, and b) offers reasonably dense set of ARL levels even at very low transmission rates.

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Proposed procedures offer discrete set of run lengths

15 25 35 45 55 65 100 1000 10000 100000 E

0(T)

ARL d = 2 Full Figure 7: Proposed procedures, δ = 10.0.

For practical purposes d = 3 procedure

a) performs well for frequent trans- missions, and b) offers reasonably dense set of ARL levels even at very low transmission rates.

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Proposed procedures offer discrete set of run lengths

15 25 35 45 55 65 100 1000 10000 100000 E

0(T)

ARL d = 3 Full Figure 8: Proposed procedures, δ = 10.0.

For practical purposes d = 3 procedure

a) performs well for frequent trans- missions, and b) offers reasonably dense set of ARL levels even at very low transmission rates.

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Performance Comparison

Let Cδ denote the class of procedures where the probability that a sensor com-

municates at any given time is 1/δ.

Recall that for Mei’s procedure k-th sensor communicates with the fusion center
nly if its CUSUM statistic exceeds a given threshold bk:

TMei = inf

t ≥ 1 :

K

k=1

W k

t 1

l{W k

t ≥bk} ≥ a

.
Note: Transmission rate is controlled via bk, whereas performance is controlled

via detection threshold a, and Cδ it is equivalent to E∞{# of transmitting sensors} = K/δ at any given time t.

For proposed procedure it is equivalent to E∞(τ k

1 ) = δ, i.e.

E∞{time between transmissions} = δ for any given sensor k.

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Gaussian Scenario: Mei vs. proposed, δ = 1.5

10 15 20 25 30 35 40 45 50 55 60 100 1000 10000 100000 ARL Mei E

0(T)

Proposed

γ = 500 γ = 5000 a ARL ADD0 a ARL ADD0 Mei 9.65 520.5 25.9 13.1 4992 39.4 d = 3 9.5 514.1 26.3 13.0 5327 39.8

Table 1: Performance for δ = 1.5.

For frequent transmissions the loss in performance is negligible for d = 3 procedure

as compared to Mei’s.

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Gaussian Scenario: Mei vs. proposed, δ = 10.0

10 15 20 25 30 35 40 45 50 55 60 100 1000 10000 100000 ARL Mei E

0(T)

Proposed

γ = 500 γ = 5000 a ARL ADD0 a ARL ADD0 Mei 8.02 503.9 25.7 11.9 4882 39.0 d = 3 5.8 495.2 27.2 9.6 4948 39.4

Table 2: Performance for δ = 10.0.

For infrequent transmissions the loss in performance is negligible for d = 3 pro-

cedure as compared to Mei’s.

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Conclusion

Proposed procedures address the two aspects of decentralized sequential change

detection problem, when the change happens in an unknown subset of sensors.

Even under low transmission rates when the proposed procedures only offer a

discrete set of run lengths, for as few as d = 3 this set is dense enough for practical purposes.

When no restrictions on the number of bits is imposed, proposed procedure enjoys

first-order asymptotic optimality for any choice of communication thresholds ∆: J (T) infT : ARL(T)γ J (T) → 1, γ → ∞.

If d is fixed and ∆ → ∞, or d → ∞ and ∆ is fixed, the first-order asymptotic
ptimality is retained.

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