Multi-target Detection
in Sensor Networks
Xiaoling Wang
ECE691, Fall 2003
Target Detection in Sensor Networks

Single target detection



Eobs
Energy decay model:
Constant false-alarm rate (CFAR)
Esource
 
d
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
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:

m  arg max P( H m | X)
m

Available source number estimation
algorithms



Sample-based approach: RJ-MCMC (reversiblejump Markov Chain Monte Carlo) method
Variational learning
Bayesian source number estimation
Bayesian Source Number Estimation
(BSNE) Algorithm
X : sensor observation matrix
S : source matrix
A : mixing matrix, X  AS
P( H m | X) 
p ( X | H m ) P( H m )
allH p(X | H ) P( H )
W : unmixing matrix, S  WX and W  ( AT A) 1 AT
H m: hypothesis of the number of sources
a : latent variable, a  WX and S   (a)
 : non-linear transformation function
Rn : noise, with variance 1 / 
 () : marginal distribution of a
L(m)  log p(x(t ) | H m )
t
p(x(t ) | A,  , Rn )   p(x(t ) | A,  , Rn , a) (a)da


 
1

1

 log (a(t ))  (n  m) log( )  log A A  (x(t )  A a(t ))2
2
 2
2
 2
2
m

mn

n
[
log( )  ( log a j )  m nlog ]
2
2
2 j 1

Detailed derivation
p( X | A,  , Rn )   p(x(t ) | A,  , Rn )
T 
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



Structure of the distributed hierarchy
M-ary hypothesis testing
Fusion of detection
probabilities
Distributed structure
Posterior Probability Fusion
Based on Bayes Theorem
Since X  X1  X2  XL ,
P( Η m | X) 
p( X | H m ) P( H m ) p( X1  X2    X L | H m ) P( H m )

p( X)
p( X1  X2    X L )
Since X1 , X2 ,, X L are independent, p(Xi  X j )  0, for i  j
Therefore,
L
L
p( X1  X2    X L | H m )   p( Xi | H m )   p( Xi  X j | H m )
i 1
i , j 1
i j
L
  p(X L | H m )
i 1

| X) 
L
 p( H m
i 1
p ( Xi | H m ) P ( H m )

L
i 1
p( Xi )
p ( Xi )
1

where
L
 j 1 p(X j ) Ki
L

i 1
Ki

1
E


k
Ki
k 1
p ( Xi )
L
j 1
Ki
p( X j )
1

2
k 1 d k
p ( H m | Xi )
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
P( H m

| X) 
1 
allH f  H g  H m ,i  j
P( H f | X i )  P ( H g | X j )
allH f  H g  ,i  j
P ( H f | Xi )  P( H g | X j )
Performance Evaluation of Multiple
Target Detection
Target types
Sensor laydown
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
p ( X | H m ) P( H m )
P( H m | X) 
allH p(X | H ) P( H )
p( X | A,  , Rn )   p(x(t ) | A,  , Rn )
t
p(x(t ) | A,  , Rn )   p(x(t ) | A,  , Rn , a) (a)da (1)
Choose  (a)   tanh(a), then since  (a)  d log (a) / da
1
(cosh(a))1/ 
Z ( )
where log(Z ( ))  a log(c /   1)  b, a, b, c are constants.
 (a) 
Suppose noise on each component has same variance,1 /  , then
p (x(t ) | A,  ,  , a) 
1

exp{  (x(t )  Aa) 2 }
H
2
where
1
 n/2
(
)
H
2

Assume the integral in (1) is dominated by a sharp peak at a  a, then by using
Laplace approximation of the marginal integral,
1
H



1
4 m / 2  2 (x(t )  Aa) 2 1/ 2

|
exp{ (x(t )  A a) 2 }
2
 exp{ 2 (x(t )  Aa) } (a)da  H  (a)(  ) |
a
2
2
Therefore,

log p (x(t ) | A,  ,  )  log  (a(t )) 


1

1

(n  m) log(
)  log | AT A |  (x(t )  A a(t )) 2
2
2
2
2
where a(t )  Wx(t )
Then p (x(t ) |  ,  )   P ( A) p (x(t ) | A,  ,  )dA

 12 ( n m )

  (a(t ))( )

(
A
)
exp{

(
x
(
t
)

A
a
(t ))2 }dA

2
2

where  ( A) 
(2)
P( A)
| AT A |1/ 2

Assume the density function of A is sharply peaked at A and use Laplace approximation,

p(x(t ) |  ,  )   (a(t ))( )
2

1
( nm)
2

(
)
4
 12 ( n m )   mn
  (a(t ))( )
(
) 2
2
4


 ( A)
mn

2
exp{

| (x(t )  A a(t ))2 |1/ 2

 ( A)
2
( j 1 (a j ) n )1/ 2
m
exp{

2

2
 
(x(t )  A a(t ))2 }
 
(x(t )  A a(t ))2 }

Using the maximum-likelihood estimation,
p (x(t ) |  , A,  )
 0 gives


 
1

(x(t )  A a(t ))2
nm

Assume P( A)   mn ,   2 || A || , and  (a)   tanh(a),
Then
L(m)  log p(x(t ) | H m )


 
1

1

 log (a(t ))  (n  m) log( )  log A A  (x(t )  A a(t ))2
2
2
2
2


T 
2
mn

n m
[
log( )  ( log a j )  m nlog ]
2
2
2 j 1