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BSS in sensor networks BSS problem involves Source number estimation Source separation Assumptions Linear, instantaneous mixture model Number of sources = number of observations This equality assumption is not the case in sensor networks due to the large amount of sensors deployed
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Multi-target Detection in Sensor Networks
Xiaoling WangECE691, Fall 2003
Target Detection in Sensor Networks Single target detection
Energy decay model: Constant false-alarm rate (CFAR)
Multiple target detection Blind source separation (BSS) problem
Targets are considered as the sources “Blind”: there is no a-priori information on
the number of sources the probabilistic distribution of source
signals the mixing model
Independent component analysis (ICA) is common technique to solve the BSS problem
dEE source
obs
source
BSS in sensor networks BSS problem involves
Source number estimation Source separation
Assumptions Linear, instantaneous mixture model Number of sources = number of observations This equality assumption is not the case in sensor
networks due to the large amount of sensors deployed
Source Number Estimation Source number estimation:
Available source number estimation algorithms Sample-based approach: RJ-MCMC (reversible-
jump Markov Chain Monte Carlo) method Variational learning Bayesian source number estimation
)|(maxarg XmmHPm
Bayesian Source Number Estimation (BSNE) Algorithm
allH
mmm HPHp
HPHpHP)()|(
)()|()|(X
XX
)|)((log)( mHtpmL x2))()((
2log
21)
2log()(
21))((log tAtAAmnt
T
axa
aaaxx
xX
dRAtpRAtp
RAtpRAp
nn
tnn
)(),,,|)((),,|)((
),,|)((),,|(
: sensor observation matrixX
: hypothesis of the number of sourcesmH
: source matrixS: mixing matrix,A SX A: unmixing matrix,W and TT AAAW 1)( XS W
: latent variable,a Xa W and )(aS : non-linear transformation function
: noise, with variancenR: marginal distribution of)( a
/1
m
jj mnanmn
1
2
]log)log(2
)2
log(2
[
Detailed derivation
Centralized vs. Distributed Schemes Centralized scheme: long
observed sequences from all the sensors are available for source number estimation
Centralized processing is not realistic in sensor networks due to: Large amount of sensor
nodes deployed Limited power supply on
the battery-powered sensor node
Distributed scheme: Data is processed locally Only the local decisions are
transferred between sensor clusters
Advantages of distributed target detection framework: Dramatically reduce the
long-distance network traffic
Therefore conserve the energy consumed on data transmissions.
Distributed Source Number Estimation Scheme
Sensor nodes clustering
The distributed scheme includes two levels of processing: An estimation of source number is obtained from each
cluster using the Bayesian method The local decisions from each cluster are fused using the
Bayesian fusion method and the Dempster’s rule of combination.
Distributed Hierarchy
Unique features of the developed distributed hierarchy M-ary hypothesis testing Fusion of detection
probabilities Distributed structure
Structure of the distributed hierarchy
Posterior Probability Fusion Based on Bayes Theorem
)()()|(
)()()|()|(
21
21
L
mmLmmm p
HPHpp
HPHpΗPXXX
XXXX
XX
ii K
k ki
K
kk
iL
j j
i
dKE
Kpp
12
11
111)(
)(X
X
)|()(
)()(
)()|()|(
111
1im
L
iL
j j
iL
i i
L
i mmim Hp
pp
p
HPHpHp X
XX
X
XX
Since ,21 LXXXX
where
LXXX ,,, 21 Since are independent, ,0)( jip XX for ji Therefore,
L
jiji
mji
L
imimL HpHpHp
1,121 )|()|()|( XXXXXX
L
imL Hp
1
)|(X
Dempster’s Rule of Combination
Utilize probability intervals and uncertainty intervals to determine the likelihood of hypotheses based on multiple evidence
Can assign measures of belief to combinations of hypotheses
jiHallH jgif
jiHHallH jgif
m
gf
mgf
HPHP
HPHPHP
,
,
)|()|(1
)|()|()|(
XX
XXX
Performance Evaluation of Multiple Target Detection
Sensor laydown
Target types
Results Comparison: Log-likelihood and Histogram
Results Comparison: Kurtosis, Detection Probability, and Computation Time
Discussion
The distributed hierarchy with the Bayesian posterior probability fusion method has the best performance, because: Source number estimation is only performed within each
cluster, therefore, the effect of signal variations are limited locally and might contribute less in the fusion process
The hypotheses of different source numbers are independent, exclusive, and exhaustive set which is in accordance with the condition of the Bayesian fusion method.
The physical characteristics of sensor networks are considered, such as the signal energy captured by each sensor node versus its geographical position
Derivation of the BSNE Algorithm
aaaxx
xX
dRAtpRAtp
RAtpRAp
nn
tnn
)(),,,|)((),,|)((
),,|)((),,|(
allH
mmm HPHp
HPHpHP)()|(
)()|()|(X
XX
Choose ),tanh()( aa then since aaa dd /)(log)(
/1))(cosh(
)(1)( aaZ
where ,)1/log())(log( bcaZ cba ,, are constants.
Suppose noise on each component has same variance, ,/1 then
}))((2
exp{1),,,|)(( 2axax AtH
Atp where 2/)
2(1 n
H
(1)
Assume the integral in (1) is dominated by a sharp peak at ,
aa
Laplace approximation of the marginal integral,
}))((2
exp{|))((|)4)((1)(}))((2
exp{1 22/12
222/
2
ax
aaxaaaax AtAt
HdAt
Hm
then by using
Therefore,
2))()((2
||log21)
2log()(
21))((log),,|)((log tAtAAmntAtp T
axax
where )()( tWt xa
Then dAAtpAPtp ),,|)(()(),|)(( xx
dAtAtAtmn
}))()((2
exp{)()2
))((( 2)(21
axa
where 2/1||)()(
AAAPA T
(2)
Assume the density function of A is sharply peaked at
A and use Laplace approximation,
}))()((2
exp{|))()((|
)()4
()2
))(((),|)(( 2
2/12
2)(
21
tAttAt
Attpmnmn
axax
ax
}))()((2
exp{))((
)()4
()2
))((( 2
2/11
22
)(21
tAta
Atm
jn
j
mnmn
axa
Assume ,)( mnAP ,||||2
A
Then )|)((log)( mHtpmL x
2))()((2
log21)
2log()(
21))((log tAtAAmnt
T
axa
m
jj mnanmn
1
2
]log)log(2
)2
log(2
[
and ),tanh()( aa
Using the maximum-likelihood estimation, 0),,|)((
Atp x gives
2))()((1 tAtmn
ax
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