Olympia Hadjiliadis

Outline: Part I- One Shot Schemes
The multi-source quickest detection problem
The decentralized example &a synchronous communication through
CUSUM one shot schemes (& its optimality)
Multiple sources set-up of trade-off asap vs fa
Asymptotically there is no loss of information
In the decentralized/centralized set-up
(more mathematically….)
Asymptotic optimality of the N-CUSUM rule
Summary

The multi-source quickest detection
We sequentially observe through independent sources i=1,…,N
Information (if all at one location) with special attn to
ASSUMPTION
&
1) The onset of signal can take place at distinct times
2) :unknown constants
3)
OBJECTIVE : Detect the minimum of as soon as possible but controlling false alarms

Optimality of the CUSUM
Min-Max approach
Lorden’s criterion (1971) subject to
The Cumulative Sum ( CUSUM ) is optimal
Shiryaev(1996), Beibel(1996)

The decentralized system
Each sensor S i is sequentially observing continuous observations
S
1
T
1
S
2
T
2
S
3
T
3
T
N
S
N
Fusion center

CUSUM & its optimality
The CUSUM statistic process is
The CUSUM stopping rule is
… and is optimal in subject to
Asap Mean time to false alarm

d
Multiple sources set-up subject to
Define
The optimal stopping rule satisfies
Proof: Let N=2 and consider S be s.t.
Let U stop as S, when observing { ξ t
1 } instead of { ξ t
2 } & vice versa
Then and
Consider T stops as S when Heads and as U when Tails while

Multiple sources set-up
Natural candidate: N-CUSUM
Hence the best N-CUSUM
Which translates to (as γ→ ∞) satisfies

Asymptotic optimality as γ→∞
 If where
 If
 If

Asymptotic optimality (equal strengths)
N=2, µ=1

Asymptotic optimality
(unequal strengths)
N=2, µ
1
=1, µ
2
=1.2
µ
1

Asymptotic optimality
(unequal strengths)
N=2, µ
1
=1, µ
2
=1.5
µ
1

Outline: Part II-Coupled systems
The multi-source quickest detection problem
Models of general dependencies
Objective: Detect the first instance of a signal;
Meaning:
Detect the min of N change points in Ito processes
Set-up the problem as a stochastic optimization w.r.t. a Kullback Leibler divergence
Asymptotic optimality of the N-CUSUM rule
Summary

The multi-source quickest detection
We sequentially observe through independent sources i=1,…,N
Information (if all at one location) with special attn to &
ASSUMPTION
1) The onset of signal can take place at distinct times
2) : unknown constants
OBJECTIVE : Detect the minimum of as soon as possible but controlling false alarms

Examples
A system with AR behavior in each component and additive feedback from other sources
Such a system with signals of different strengths in each sensor

CUSUM & its optimality N=1
The CUSUM statistic process is
The CUSUM stopping rule is is optimal in subject to
Asap
Mean time to false alarm

Multiple sources set-up d subject to
Define
The optimal stopping rule satisfies

Multiple sources set-up
Natural candidate: N-CUSUM
Since are the same across i…
ALL ABOVE ARE EQUAL
Therefore…

To solve this problem we need…
Take N=2. It is possible to show that satisfy

To solve this…
 G is the probability that a particle placed at (x,y) will leave D after t.

Asymptotic optimality as γ→∞ where
NOTE
:
 If

Non-symmetric signals
We sequentially observe through independent sources i=1,2
Suppose where subject to

Multiple sources set-up
Natural candidate: 2-CUSUM
In order to have an equalizer rule, or equivalently we need
 If as

Non-symmetric signals
We sequentially observe through independent sources i=1,2
Suppose where subject to

Summary


Asymptotic optimality of the N-CUSUM rule in the case are the same across I
In the case of Brownian motions with const drift
MESSAGE:
If you want to detect the first instance of onset of a signal, let the sensors do the work!
(Lose almost nothing in efficiency)
 Extensions to the case different in law across i
 What if the noises across sources are correlated.

 H. Vincent Poor
 Tobias Schaefer
 Hongzhong Zhang


“One-shot schemes for decentralized quickest change detection”, O. Hadjiliadis, H.
Zhang and H. V. Poor,
IEEE Transactions on Information Theory 55(7)
2009.
 “Quickest Detection in coupled systems”, O.
Hadjiliadis, T. Schaefer and H. V. Poor ,
Proceedings the 48th IEEE Conference on
Decisions and Control, (2009)
Submitted to the SIAM Journal on control and optimization (2010)