Transcript Multi-dimensional quickest detection

Multi-dimensional
quickest
detection
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 Si is sequentially observing continuous observations
S3
S2
S1
SN
T2
T3
T1
TN
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
Multiple sources set-up
subject to
d
Define
The optimal stopping rule satisfies
Proof: Let N=2 and consider S be s.t.
Let U stop as S, when observing {ξt1} instead of {ξt2} & 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→∞
where

If

If

If
Asymptotic optimality (equal strengths)
N=2, µ=1
Asymptotic optimality
(unequal strengths)
N=2, µ1=1, µ2=1.2µ1
Asymptotic optimality
(unequal strengths)
N=2, µ1=1, µ2=1.5µ1
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
subject to
d
Define
The optimal stopping rule satisfies
ASSUMPTION
are the same in law across all i
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
Suppose
through independent sources i=1,2
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
Suppose
through independent sources i=1,2
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.
Thanks to all
collaborators

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)

THE END