Using Sparse Noisy Data

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November 1, 2012

Presented by Marwan M.
Alkhweldi


Co
-
authors Natalia A.
Schmid

and Matthew C.
Valenti


Distributed Estimation of a Parametric Field

Using Sparse Noisy Data

This work was sponsored by the Office of Naval Research under Award No. N00014
-
09
-
1
-
1189.


Overview and Motivation


Assumptions


Problem Statement


Proposed
Solution


Numerical Results


Summary


Outline

November 1, 2012


WSNs

have

been

used

for

area

monitoring,

surveillance,

target

recognition

and

other

inference

problems

since

1980
s

[
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
.




Overview and Motivation

November 1, 2012

[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
.


Assumptions

November 1, 2012

Z1

Z2

.

ZK

Fusion Center











.
,
0
~

where
,
*
.





.
*
.
,
0
~

where
,
,
R

*
A.

area
over

placed
randomly

sensors

*
2
2
i


N
N
N
R
Q
Z
quantizer
Level
M
an
is
Q
N
W
W
y
x
G
K
i
i
i
i
i
i
i
i




http://www.classictruckposters.com/wp
-
content/uploads/2011/03/dream
-
truck.png

A

Transmission Channel

Observation Model


i
R

)
,
(
c
c
y
x
The object generates

fumes that
are modeled as

a Gaussian
shaped field.

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 be the log
-
likelihood function of the observations

at
the Fusion Center. Then the ML estimates solve:


Problem Statement

November 1, 2012



.
:
max
arg
ˆ
θ
Z
θ
Θ
θ
l




θ
Z
:
l
Proposed Solution

November 1, 2012


The

log
-
likelihood

function

of

is
:








The

necessary

condition

to

find

the

maximum

is
:

K
Z
Z
Z
,...,
,
2
1












Q(.).
quantizer

the
of

points
on
reproducti

are

,...,


,
2
exp
2
1

where
,
2
log
2
2
exp
log

1
2
2
2
2
1
1
2
2
1
M
k
j
k
K
k
M
j
j
k
j
k
v
v
and
dt
G
t
v
p
K
v
z
v
p
l
j
j











































z


.
0
:
ˆ


ML
Z
l



Iterative Solution

November 1, 2012

A. P. Dempster, N. M. Laird, and D. B. Rubin, “Maximum likelihood from incomplete data via the em algorithm," J. of the Royal

St
at. Soc.
Series B, vol. 39, no. 1, pp. 1
-
38, 1977.


Incomplete

data
:



Complete

data
:

,


where

,

and

.


Mapping
:

.


where

.



The

complete

data

log
-
likelihood
:















K
i
i
i
i
cd
y
x
G
R
l
1
2
2
.

of

function
not

terms
,
2
1


k
Z


k
k
N
R
,




2
,
:
,
~


k
k
k
y
x
G
N
R


2
,
0
~

N
N
k


k
k
k
n
R
q
Z


K
k
,...,
1


Expectation Step
:




Maximization Step:



E
-

and

M
-

steps

November 1, 2012







.
ˆ
,
2
1
1
2
2
1












k
K
k
k
k
k
z
G
R
E
Q




























.
in
nonlinear

are


and


where
L.
1,2,...,
t
,
0
L.
1,...,
t
,
0
ˆ
,
2
1
k
)
(
K
1
i
1
1
1
1
ˆ
1
2
2
1
1








k
i
k
i
K
i
k
i
t
k
i
k
i
k
i
t
k
i
k
K
i
t
i
i
i
t
k
G
B
G
A
G
B
d
dG
G
G
A
d
dG
z
d
dG
G
R
E
d
dQ
k


























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:




SNR in transmission channel is defined as:


Experimental Set Up

November 1, 2012



.
:
,
2
2


A
dxdy
y
x
G
SNR
A
O








.
,
2
2

A
dxdy
y
x
R
q
E
SNR
A
C


Performance Measures

November 1, 2012









]
[
)
(

outliers

of

Occurrence
*
]
[
Error

Square
Mean

Integrated
*
)
:
,
(
)
ˆ
:
,
(
Error

Square

Integrated
*
]
[
Error

Square
Mean
*
ˆ
Error

Square
*
2
2

























SE
P
P
ISE
E
IMSE
dxdy
y
x
G
y
x
G
ISE
SE
E
MSE
SE
outliers
A
Target Localization

Shape Reconstruction

The

simulated

Gaussian

field

and

squared

difference

between

the

original

and

reconstructed

fields

where





Numerical Results

November 1, 2012

T
3.88]
7.90,3.88,
[
ˆ
,
]
4
,
4
,
8
[




T
EM
-

convergence

November 1, 2012


SNRo
=
SNRc
=
15
dB
.



Number

of

sensors

K=
20
.



Box
-
plot of Square Error

November 1, 2012


1000

Monte

Carlo

realizations
.


SNRo
=
SNRc
=
15
dB
.




2
ˆ
Error

Square




SE

Box
-
plot of Integrated Square Error

November 1, 2012


1000

Monte

Carlo

realizations
.


SNRo
=
SNRc
=
15
dB
.


Number

of

quantization

levels

M=
8







A
dxdy
y
x
G
y
x
G
ISE
2
)
:
,
(
)
ˆ
:
,
(
Error

Square

Integrated


Probability of Outliers

November 1, 2012


1000

Monte

Carlo

realizations
.


SNRo
=
SNRc
=
15
dB
.


Number

of

quantization

levels

M=
8
.


Threshold.

],
[
)
(






SE
P
P
outliers
Effect of Quantization Levels

November 1, 2012


1000

Monte

Carlo

realizations
.


SNRo
=
SNRc
=
15
dB
.


Number

of

sensors

K=
20
.


Summary

November 1, 2012


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

the

EM

algorithm

produces

a

substantial

number

of

outliers
.


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
.




Natalia A.
Schmid



e
-
mail: Natalia.Schmid@mail.wvu.edu



Marwan
Alkhweldi




e
-
mail: malkhwel@mix.wvu.edu





Matthew C.
Valenti




e
-
mail: Matthew.Valenti@mail.wvu.edu

Contact Information

November 1, 2012