Transcript MILCOM'12 Presentation Template

Distributed Estimation of a Parametric Field
Using Sparse Noisy Data
Presented by Marwan M. Alkhweldi
Co-authors Natalia A. Schmid and Matthew C. Valenti
This work was sponsored by the Office of Naval Research under Award No. N00014-09-1-1189.
November 1, 2012
Outline
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Overview and Motivation
Assumptions
Problem Statement
Proposed Solution
Numerical Results
Summary
November 1, 2012
Overview and Motivation
• WSNs have been used for area monitoring, surveillance, target recognition
and other inference problems since 1980s [1].
• All designs and solutions are application oriented.
• Various constraints were incorporated [2]. Performance of WSNs under the
constraints was analyzed.
• The task of distributed estimators was focused on estimating an unknown
signal in the presence of channel noise [3].
• We consider a more general estimation problem, where an object is
characterized by a physical field, and formulate the problem of distributed
field estimation from noisy measurements in a WSN.
[1] C. Y. Chong, S. P. Kumar, “Sensor Networks: Evolution, Opportunities, and Challenges” Proceeding of the IEEE, vol. 91, no.
8, pp.
1247-1256, 2003.
[2] A. Ribeiro, G. B. Giannakis, “Bandwidth-Constrained Distributed Estimation for Wireless Sensor Networks - Part I:Gaussian
Case,” IEEE Trans. on Signal Processing, vol. 54, no. 3, pp. 1131-1143, 2006.
[3] J. Li, and G. AlRegib, “Distributed Estimation in Energy-Contrained Wireless Sensor Networks,” IEEE Trans. on Signal
Processing, vol. 57, no. 10, pp. 3746-3758, 2009.
November 1, 2012
Assumptions
A
Transmission Channel
Ri
( xc , yc )
Observation Model
Z1
Z2
.
ZK
The object generates
fumes that are modeled as
a Gaussian shaped field.
Fusion Center
* K sensors randomly placed over area A.


* R i  G  xi , yi   Wi , where Wi ~ N 0,  2 .
* Q. is an M Level quantizer.


* Z i  QRi   N i , where N i ~ N 0, 2 .
http://www.classictruckposters.com/wp-content/uploads/2011/03/dream-truck.png
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Problem Statement
Given noisy quantized sensor observations at the Fusion Center,
the goal is to estimate the location of the target and the distribution
of its physical field.
Proposed Solution:
• Signals received at the FC are independent but not i.i.d.
• Since the unknown parameters are deterministic, we take the
maximum likelihood (ML) approach.
• Let l Z : θ  be the log-likelihood function of the observations at
the Fusion Center. Then the ML estimates solve:
θˆ  arg maxl Z : θ .
θΘ
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Proposed Solution
• The log-likelihood function of Z1 , Z 2 ,...,Z K is:
M
 z k v j 2   K
   log 2 2 ,
l z    log  pk v j exp 
2

 2
 j 1
2

k 1



 j 1
 t  Gk 2 
1
dt,
where pk v j   
exp 
2

2

j
2 2


and v1 ,...,vM are reproduction pointsof thequantizerQ(.).

K

• The necessary condition to find the maximum is:
 l Z :   ˆ
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 0.
ML
Iterative Solution
• Incomplete data: Z k
• Complete data: Rk , N k ,
where Rk ~ N Gxk , yk :  , 2 , and Nk ~ N 0, 2 .
• Mapping: Zk  qRk   n.k
where k  1,...,K.
• The complete data log-likelihood:
lcd  
1
2
2




R

G
x
,
y
 terms
i
i
i
2 
K
not function of  .
i 1
A. P. Dempster, N. M. Laird, and D. B. Rubin, “Maximum likelihood from incomplete data via the em algorithm," J. of the Royal Stat. Soc.
Series B, vol. 39, no. 1, pp. 1-38, 1977.
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E- and M- steps
• Expectation Step:
 1
k 1
Q
 E  2
 2
• Maximization Step:
K
 R
k
k 1
 Gk 
2
k  
ˆ
z, .

 1 K
dQ k 1
2 dGi
k  
ˆ
Ri  Gi 
 E 
z, 
 0, t  1,..., L.
2 
d t
d t
 2 i 1
 ˆ  k 1
 k 1
dGi

d t
i 1
K
   G
A Gi


k 
K
i 1
 k 1
i
 
 k 1
dGi
d t
   0,
B Gi
k 
where A Gi( k ) and B Gik are nonlinear in  k .
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t  1,2,..., L.
Experimental Set Up
• Assume the area A is of size 8-by-8;
• K sensors are randomly distributed over A;
• M quantization levels;
• SNR in observation channel is defined as:
2
G
 x, y :  dxdy
SNRO 
A
A
2
.
• SNR in transmission channel is defined as:
2

E
q
 Rx, y dxdy
SNRC 
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A
A
2
.
Performance Measures
Target Localization



* Mean Square ErrorMSE  E[ SE]

* Square ErrorSE  
  ˆ
2
Shape Reconstruction
2
ˆ
* Int egrat edSquare ErrorISE   G ( x, y :  )  G ( x, y :  ) dxdy

A


* Int egrat edMean Square ErrorIMSE  E[ ISE]

* Occurrence of out liersPoutliers ( )  P[ SE   ]
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Numerical Results
The simulated Gaussian field and squared difference between
the original and reconstructed fields where
  [8,4,4]T ,ˆ  [7.90,3.88,3.88]T
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EM - convergence
• SNRo=SNRc=15dB.
• Number of sensors K=20.
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Box-plot of Square Error
• 1000 Monte Carlo realizations.
• SNRo=SNRc=15dB.
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Square Error SE  
  ˆ
2
Box-plot of Integrated Square Error
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1000 Monte Carlo realizations.
2
SNRo=SNRc=15dB.
IntegratedSquare ErrorISE   G( x, y : ˆ)  G( x, y :  ) dxdy
A
Number of quantization levels
M=8
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Probability of Outliers
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•
•
1000 Monte Carlo realizations.
SNRo=SNRc=15dB.
Number of quantization levels M=8.
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Poutliers ( )  P[ SE   ],
  T hreshold.
Effect of Quantization Levels
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•
1000 Monte Carlo realizations.
SNRo=SNRc=15dB.
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Number of sensors K=20.
Summary
• An iterative linearized EM solution to distributed field estimation is
presented and numerically evaluated.
• SNRo dominates SNRc in terms of its effect on the performance of the
estimator.
• Increasing the number of sensors results in fewer outliers and thus in
increased quality of the estimated values.
• At small number of sensors
number of outliers.
the EM algorithm produces a substantial
• More number of quantization levels makes the EM algorithm takes fewer
iterations to converge.
• For large K, increasing the number of sensors does not have a notable effect
on the performance of the algorithms.
November 1, 2012
Contact Information
• Natalia A. Schmid
e-mail: [email protected]
• Marwan Alkhweldi
e-mail: [email protected]
• Matthew C. Valenti
e-mail: [email protected]
November 1, 2012