Entropy-based

Sensor

Selection

Heuristic

for

Target

Localization

We

propose

an

entropy-based

sensor

selection

heuristic

for

localization

Given

-RRB-

a

prior

probability

distribution

of

the

target

location

and

-RRB-

the

locations

and

the

sensing

models

of

a

set

of

candidate

sensors

for

selection

the

heuristic

selects

an

informative

sensor

such

that

the

fusion

of

the

selected

sensor

observation

with

the

prior

target

location

distribution

would

yield

on

average

the

greatest

or

nearly

the

greatest

reduction

in

the

entropy

of

the

target

location

distribution

The

heuristic

greedily

selects

one

sensor

in

each

step

without

retrieving

any

actual

sensor

observations

The

heuristic

is

also

computationally

much

simpler

than

the

mutual-information-based

approaches

The

effectiveness

of

the

heuristic

is

evaluated

using

localization

simulations

in

which

Gaussian

sensing

models

are

assumed

for

simplicity

The

heuristic

is

more

effective

when

the

optimal

candidate

sensor

is

more

informative

H.

-LSB-

MODELS

AND

PRINCIPLES

-RSB-

Systems

and

Information

Theory

Information

theory

C.

-LSB-

COMPUTER-COMMUNICATION

NETWORKS

-RSB-

Distributed

Systems

Distributed

applications

Algorithms

Management

Theory

Permission

to

make

digital

or

hard

copies

of

all

or

part

of

this

work

for

personal

or

classroom

use

is

granted

without

fee

provided

that

copies

are

not

made

or

distributed

for

profit

or

commercial

advantage

and

that

copies

bear

this

notice

and

the

full

citation

on

the

first

page

To

copy

otherwise

to

republish

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requires

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specific

permission

and/or

a

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IPSN

April

Berkeley

California

USA

Copyright

ACM

The

recent

convergence

of

micro-electro-mechanical

systems

-LRB-

MEMS

-RRB-

technology

wireless

communication

and

networking

technology

and

low-cost

low-power

miniature

digital

hardware

design

technology

has

made

the

concept

of

wireless

sensor

networks

viable

and

a

new

frontier

of

research

-LSB-

-RSB-

The

limited

on-board

energy

storage

and

the

limited

wireless

channel

capacity

are

the

major

constraints

of

wireless

sensor

networks

In

order

to

save

precious

resources

a

sensing

task

should

not

involve

more

sensors

than

necessary

From

the

information-theoretic

point

of

view

sensors

are

tasked

to

observe

the

target

in

order

to

increase

the

information

-LRB-

or

to

reduce

the

uncertainty

-RRB-

about

the

target

state

The

information

gain

attributable

to

one

sensor

may

be

very

different

from

that

attributable

to

another

when

sensors

have

different

observation

perspectives

and

sensing

uncertainties

Selective

use

of

informative

sensors

reduces

the

number

of

sensors

needed

to

obtain

information

about

the

target

state

and

therefore

prolongs

the

system

lifetime

In

the

scenario

of

localization

or

tracking

using

wireless

sensor

networks

the

belief

state

of

the

target

location

can

be

gradually

improved

by

repeatedly

selecting

the

most

informative

unused

sensor

until

the

required

accuracy

-LRB-

or

uncertainty

-RRB-

level

of

the

target

state

is

achieved

There

have

been

several

investigations

into

information-theoretic

approaches

to

sensor

fusion

and

management

The

idea

of

using

information

theory

in

sensor

management

was

first

proposed

in

-LSB-

-RSB-

Sensor

selection

based

on

expected

information

gain

was

introduced

for

decentralized

sensing

systems

in

-LSB-

-RSB-

The

mutual

information

between

the

predicted

sensor

observation

and

the

current

target

location

distribution

was

proposed

to

evaluate

the

expected

information

gain

about

the

target

location

attributable

to

a

sensor

in

-LSB-

-RSB-

On

the

other

hand

without

using

information

theory

Yao

et

al.

-LSB-

-RSB-

found

that

the

overall

localization

accuracy

depends

on

not

only

the

accuracy

of

individual

sensors

but

also

the

sensor

locations

relative

to

the

target

location

during

the

development

of

localization

algorithms

We

propose

a

novel

entropy-based

heuristic

for

sensor

selection

based

on

our

experiences

with

target

localization

It

is

computationally

more

efficient

than

mutual-information-based

methods

proposed

in

-LSB-

-RSB-

We

use

the

following

notations

throughout

this

paper

S

is

the

set

of

candidate

sensors

for

selection

i

S

is

the

sensor

index

x

is

the

realization

of

the

random

vector

that

denotes

the

target

location

x

t

is

the

actual

target

location

x

is

the

maximum

likelihood

estimate

of

the

target

location

x

i

is

the

deterministic

location

of

sensor

i

z

i

is

the

realization

of

the

random

variable

that

denotes

the

observation

of

sensor

i

about

the

target

location

z

ti

is

the

actual

observation

of

sensor

i

about

the

target

location

z

v

i

is

the

realization

of

the

random

variable

that

denotes

the

view

of

sensor

i

about

the

target

location

The

rest

of

this

paper

is

organized

as

follows

Section

describes

the

heuristic

in

detail

Section

evaluates

the

heuristic

using

simulations

Section

discusses

the

discrepancy

between

the

heuristic

and

the

mutual

information

based

approaches

Section

outlines

future

work

Section

concludes

the

paper

Section

acknowledges

the

sponsors

This

Sect

formulates

the

sensor

selection

problem

in

localization

presents

the

details

of

the

entropy-based

sensor

selection

heuristic

and

discusses

the

relation

between

the

entropy

difference

proposed

in

this

paper

and

mutual

information

used

in

previous

work

about

sensor

selection

Sensor

Selection

Problem

in

Localization

There

are

several

information

measures

In

this

paper

we

use

Shannon

entropy

-LSB-

-RSB-

to

quantify

the

information

gain

-LRB-

or

uncertainty

reduction

-RRB-

about

the

target

location

due

to

sensor

observation

We

adopt

the

greedy

sensor

selection

strategy

used

in

mutual-information-based

approaches

-LSB-

-RSB-

The

greedy

strategy

gradually

reduces

the

uncertainty

of

the

target

location

distribution

by

repeatedly

selecting

the

currently

unused

sensor

with

maximal

expected

information

gain

The

observation

of

the

selected

sensor

is

incorporated

into

the

target

location

distribution

using

sequential

Bayesian

filtering

-LSB-

-RSB-

The

greedy

sensor

selection

and

the

sequential

information

fusion

continue

until

the

uncertainty

of

the

target

location

distribution

is

less

than

or

equal

to

the

required

level

The

core

problem

of

the

greedy

sensor

selection

approach

is

how

to

efficiently

evaluate

the

expected

information

gain

attributable

to

each

candidate

sensor

without

actually

retrieving

sensor

data

The

sensor

selection

problem

is

formulated

as

follows

Given

the

prior

target

location

distribution

p

-LRB-

x

-RRB-

the

locations

of

candidate

sensors

for

selection

x

i

i

S

the

sensing

models

of

candidate

sensors

for

selection

p

-LRB-

z

i

x

-RRB-

i

S

the

objective

is

to

find

the

sensor

i

whose

observation

z

i

minimizes

the

expected

conditional

entropy

of

the

posterior

target

location

distribution

i

arg

min

iS

H

-LRB-

x

z

i

-RRB-

-LRB-

-RRB-

Equivalently

the

observation

of

sensor

i

maximizes

the

expected

target

location

entropy

reduction

i

arg

max

iS

-LRB-

H

-LRB-

x

-RRB-

H

-LRB-

x

z

i

-RRB-

-RRB-

-LRB-

-RRB-

H

-LRB-

x

-RRB-

H

-LRB-

x

z

i

-RRB-

is

one

expression

of

I

-LRB-

x

z

i

-RRB-

the

mutual

information

between

the

target

location

x

and

the

predicted

sensor

observation

z

i

I

-LRB-

x

z

i

-RRB-

p

-LRB-

x

z

i

-RRB-

log

p

-LRB-

x

z

i

-RRB-

p

-LRB-

x

-RRB-

p

-LRB-

z

i

-RRB-

dxdz

i

-LRB-

-RRB-

where

p

-LRB-

x

z

i

-RRB-

p

-LRB-

z

i

x

-RRB-

p

-LRB-

x

-RRB-

and

p

-LRB-

z

i

-RRB-

p

-LRB-

x

z

i

-RRB-

dx

Thus

the

observation

of

sensor

i

maximizes

the

mutual

information

I

-LRB-

x

z

i

-RRB-

i

arg

max

iS

I

-LRB-

x

z

i

-RRB-

-LRB-

-RRB-

Sensor

selection

based

on

-LRB-

-RRB-

is

the

maximal

mutual

information

criterion

proposed

in

-LSB-

-RSB-

The

target

location

x

could

be

of

up

to

three

dimensions

The

sensor

observation

z

i

-LRB-

e.g.

the

direction

to

a

target

in

a

three-dimensional

space

-RRB-

could

be

of

up

to

two

dimensions

Therefore

I

-LRB-

x

z

i

-RRB-

is

a

complex

integral

in

the

joint

state

space

-LRB-

x

z

i

-RRB-

of

up

to

five

dimensions

The

complexity

of

computing

I

-LRB-

x

z

i

-RRB-

could

be

more

than

that

low-end

sensor

nodes

are

capable

of

If

the

observation

z

i

is

related

to

the

target

location

x

only

through

the

sufficient

statistics

z

-LRB-

x

-RRB-

then

I

-LRB-

x

z

i

-RRB-

I

-LRB-

z

-LRB-

x

-RRB-

z

i

-RRB-

-LRB-

-RRB-

If

z

-LRB-

x

-RRB-

has

fewer

dimensions

than

x

then

I

-LRB-

z

-LRB-

x

-RRB-

z

i

-RRB-

is

less

complex

to

compute

than

I

-LRB-

x

z

i

-RRB-

In

the

above

special

scenario

I

-LRB-

z

-LRB-

x

-RRB-

z

i

-RRB-

has

been

proposed

to

replace

I

-LRB-

x

z

i

-RRB-

to

reduce

the

complexity

of

computing

mutual

information

in

-LSB-

-RSB-

In

this

paper

we

propose

an

alternative

entropy-based

sensor

selection

heuristic

In

general

the

entropy-based

sensor

selection

heuristic

is

computationally

much

simpler

than

the

mutual

information

based

approaches

However

the

observation

of

the

sensor

selected

by

the

heuristic

would

still

yield

on

average

the

greatest

or

nearly

the

greatest

entropy

reduction

of

the

target

location

distribution

Entropy-based

Sensor

Selection

Heuristic

During

the

development

of

wireless

sensor

networks

for

localization

we

have

observed

that

the

localization

uncertainty

reduction

attributable

to

a

sensor

is

greatly

effected

by

the

difference

of

two

quantities

namely

the

entropy

of

the

distribution

of

that

sensor

's

view

about

the

target

location

and

the

entropy

of

that

sensor

's

sensing

model

for

the

actual

target

location

A

sensor

's

view

about

the

target

location

is

the

geometric

projection

of

the

target

location

onto

that

sensor

's

observation

perspective

For

example

a

direction-of-arrival

-LRB-

DOA

-RRB-

sensor

's

view

of

the

target

location

is

the

direction

from

the

sensor

to

the

target

The

view

of

sensor

i

about

the

target

location

is

denoted

as

z

v

i

which

is

a

function

of

the

target

location

x

and

the

sensor

location

x

i

z

v

i

f

-LRB-

x

x

i

-RRB-

-LRB-

-RRB-

z

v

i

usually

has

less

dimensions

than

x

The

probability

distribution

of

the

view

of

sensor

i

about

the

target

location

p

-LRB-

z

v

i

-RRB-

is

the

projection

of

the

target

location

distribution

p

-LRB-

x

-RRB-

onto

the

observation

perspective

of

sensor

i

p

-LRB-

z

v

i

-RRB-

dz

v

i

z

v

i

f

-LRB-

x

x

i

-RRB-

z

v

i

dz

v

i

p

-LRB-

x

-RRB-

dx

-LRB-

-RRB-

Alternatively

p

-LRB-

z

v

i

-RRB-

can

be

regarded

as

the

noise

free

prediction

of

the

sensor

observation

distribution

p

-LRB-

z

i

-RRB-

based

on

the

target

location

distribution

p

-LRB-

x

-RRB-

x

East

North

o

o

o

Figure

A

DOA

sensor

's

view

about

the

target

location

Thestatespaceof

thetarge

t

location

is

gridded

in

cells

Theimagedepicts

theprobability

distribution

of

thetarget

location

Theactual

target

location

is

-LRB-

-RRB-

denoted

by

marker

From

the

perspective

of

the

DOA

sensor

denoted

by

the

square

only

the

direction

to

the

target

is

observable

The

view

of

the

DOA

sensor

about

the

target

is

in

the

interval

-LSB-

o

o

-RSB-

if

and

only

if

the

target

is

inside

the

sector

delimited

by

o

lineand

o

line

In

practice

the

state

space

of

the

target

location

and

the

sensor

view

can

be

discretized

by

griding

for

numerical

analysis

The

discrete

representation

of

p

-LRB-

z

v

i

-RRB-

can

be

computed

as

follows

Let

X

be

the

grid

set

of

the

target

location

x

Let

Z

be

the

grid

set

of

the

sensor

view

z

v

i

For

each

grid

point

z

v

i

Z

initialize

p

-LRB-

z

v

i

-RRB-

to

zero

For

each

grid

point

x

X

determine

the

corresponding

grid

point

z

v

i

Z

using

equation

-LRB-

-RRB-

and

update

its

probability

as

p

-LRB-

z

v

i

-RRB-

p

-LRB-

z

v

i

-RRB-

p

-LRB-

x

-RRB-

Normalize

p

-LRB-

z

v

i

-RRB-

to

make

the

total

probability

of

the

sensor

view

be

The

numerical

computation

of

p

-LRB-

z

v

i

-RRB-

for

a

DOA

sensor

is

illustrated

in

Fig.

and

Fig.

The

entropy

of

the

probability

distribution

of

the

view

of

sensor

i

H

v

i

is

H

v

i

p

-LRB-

z

v

i

-RRB-

log

p

-LRB-

z

v

i

-RRB-

dz

v

i

-LRB-

-RRB-

Given

the

discrete

representation

of

p

-LRB-

z

v

i

-RRB-

with

a

grid

size

of

z

v

i

H

v

i

can

be

numerically

computed

as

H

v

i

p

-LRB-

z

v

i

-RRB-

log

p

-LRB-

z

v

i

-RRB-

z

v

i

-LRB-

-RRB-

The

sensing

model

of

sensor

i

for

the

actual

target

location

x

t

is

p

-LRB-

z

i

x

t

-RRB-

which

describes

the

probability

distribution

of

the

observation

of

sensor

i

given

that

the

target

is

at

x

t

The

sensing

model

incorporates

observation

uncertainty

from

all

sources

including

the

noise

corruption

to

the

signal

the

signal

modeling

error

of

the

sensor

estimation

algorithm

and

the

inaccuracy

of

the

sensor

hardware

For

a

single-modal

target

location

distribution

p

-LRB-

x

-RRB-

we

can

use

the

maximum

DOA

-LRB-

degree

-RRB-

Probability

Thediscreteprobability

distribution

of

a

DOA

se

nsor

's

vie

w.

Thestatespaceof

theDOA

sensor

view

is

gridded

in

o

intervals

The

target

location

distribution

and

theDOA

sensor

location

are

illustrated

in

Fig.

Marker

X

denotes

the

probability

of

the

DOA

view

interval

-LSB-

o

o

-RSB-

which

is

the

summation

of

theprobability

of

all

target

locations

inside

the

sector

delimited

by

o

lineand

o

line

in

Fig.

Please

note

that

the

sensor

view

distribution

does

not

depends

on

the

sensing

uncertainty

characteristics

at

all

likelihood

estimate

x

of

the

target

location

to

approximate

the

actual

target

location

x

t

Thus

the

entropy

of

the

sensing

model

of

sensor

i

for

the

actual

target

location

x

t

is

approximated

as

H

si

p

-LRB-

z

i

x

-RRB-

log

p

-LRB-

z

i

x

-RRB-

dz

i

-LRB-

-RRB-

For

a

multi-modal

target

location

distribution

p

-LRB-

x

-RRB-

with

M

peaks

x

-LRB-

m

-RRB-

where

m

M

the

entropy

of

the

sensing

model

of

sensor

i

for

the

actual

target

location

x

t

can

be

approximated

as

a

weighted

average

of

the

entropy

of

the

sensing

model

for

all

modes

H

si

M

m

p

-LRB-

x

-LRB-

m

-RRB-

-RRB-

p

-LRB-

z

i

x

-LRB-

m

-RRB-

-RRB-

log

p

-LRB-

z

i

x

-LRB-

m

-RRB-

-RRB-

dz

i

-LRB-

-RRB-

Given

a

target

location

distribution

p

-LRB-

x

-RRB-

the

target

location

with

maximum

likelihood

or

local

maximum

likelihood

can

be

found

using

standard

search

algorithms

We

have

repeatedly

observed

that

the

incorporation

of

the

observation

of

sensor

i

with

larger

entropy

difference

H

v

i

H

si

yields

on

average

larger

reduction

in

the

uncertainty

of

the

posterior

target

location

distribution

p

-LRB-

x

z

i

-RRB-

Therefore

given

a

prior

target

location

distribution

and

the

location

and

the

sensing

uncertainty

model

of

a

set

of

candidate

sensors

for

selection

the

entropy

difference

H

v

i

H

si

can

sort

candidate

sensors

into

nearly

the

same

order

as

mutual

information

I

-LRB-

x

z

i

-RRB-

does

Specifically

the

sensor

with

the

maximal

entropy

difference

H

v

i

H

si

also

has

the

maximum

or

nearly

the

maximal

mutual

information

I

-LRB-

x

z

i

-RRB-

Hence

we

propose

to

use

the

entropy

difference

H

v

i

H

si

as

an

alternative

to

mutual

information

I

-LRB-

x

z

i

-RRB-

for

selecting

the

most

informative

sensor

The

entropy-based

heuristic

is

to

compute

H

v

i

H

si

for

every

candidate

sensor

i

S

and

then

to

select

sensor

i

such

that

i

arg

max

iS

-LRB-

H

v

i

H

si

-RRB-

-LRB-

-RRB-

In

Sect.

the

validity

of

the

heuristic

is

evaluated

using

simulations

and

the

complexity

of

the

heuristic

is

analyzed

for

two-dimensional

localization

The

entropy-based

sensor

selection

heuristic

works

nearly

as

well

as

the

mutual-information

based

approaches

In

addition

the

heuristic

is

computationally

much

simpler

than

mutual

information

Relation

of

Entropy

Difference

and

Mutual

Information

A

brief

analysis

of

the

relation

between

entropy

difference

H

v

i

H

si

and

mutual

information

I

-LRB-

x

z

i

-RRB-

helps

to

reveal

fundamental

properties

of

our

sensor

selection

heuristic

Mutual

information

I

-LRB-

x

z

i

-RRB-

has

another

expression

namely

H

-LRB-

z

i

-RRB-

H

-LRB-

z

i

x

-RRB-

The

entropy

difference

H

v

i

H

si

is

closely

related

to

H

-LRB-

z

i

-RRB-

H

-LRB-

z

i

x

-RRB-

H

-LRB-

z

i

-RRB-

is

the

entropy

of

the

predicted

sensor

observation

distribution

p

-LRB-

z

i

-RRB-

H

-LRB-

z

i

-RRB-

p

-LRB-

z

i

-RRB-

log

p

-LRB-

z

i

-RRB-

dz

i

-LRB-

-RRB-

The

predicted

sensor

observation

distribution

p

-LRB-

z

i

-RRB-

becomes

the

sensor

's

view

distribution

p

-LRB-

z

v

i

-RRB-

when

the

sensing

model

p

-LRB-

z

i

x

-RRB-

is

deterministic

without

uncertainty

The

uncertainty

in

the

sensing

model

p

-LRB-

z

i

x

-RRB-

makes

H

-LRB-

z

i

-RRB-

larger

than

the

sensor

's

view

entropy

H

v

i

defined

in

-LRB-

-RRB-

H

v

i

closely

approximates

H

-LRB-

z

i

-RRB-

when

the

entropy

of

the

sensing

model

p

-LRB-

z

i

x

-RRB-

is

small

relative

to

H

v

i

H

-LRB-

z

i

x

-RRB-

is

actually

the

expected

entropy

of

the

sensing

model

p

-LRB-

x

-RRB-

averaged

for

all

possible

target

locations

H

-LRB-

z

i

x

-RRB-

p

-LRB-

x

z

i

-RRB-

log

p

-LRB-

z

i

x

-RRB-

dxdz

i

p

-LRB-

x

-RRB-

-LCB-

p

-LRB-

z

i

x

-RRB-

log

p

-LRB-

z

i

x

-RRB-

dz

i

-RCB-

dx

-LRB-

-RRB-

When

p

-LRB-

x

-RRB-

is

a

single-modal

distribution

H

si

is

defined

in

-LRB-

-RRB-

which

is

the

entropy

of

the

sensing

model

for

the

most

likely

target

location

estimate

x

When

p

-LRB-

x

-RRB-

is

a

multi-modal

distribution

H

si

is

defined

in

-LRB-

-RRB-

which

is

the

average

entropy

of

the

sensing

model

for

all

target

locations

with

local

maximal

likelihood

When

the

entropy

of

the

sensing

model

p

-LRB-

z

i

x

-RRB-

log

p

-LRB-

z

i

x

-RRB-

dz

i

changes

gradually

with

x

H

si

can

reasonably

approximate

H

-LRB-

z

i

x

-RRB-

The

entropy

difference

H

v

i

H

si

reasonably

approximates

the

mutual

information

H

-LRB-

z

i

-RRB-

H

-LRB-

z

i

x

-RRB-

when

H

si

is

small

relative

to

H

v

i

and

the

entropy

of

the

sensing

model

changes

gradually

with

x.

However

selection

of

the

most

informative

sensor

does

not

require

an

exact

evaluation

of

sensor

information

utility

Instead

an

order

of

sensors

in

terms

of

information

utility

is

needed

H

v

i

H

si

could

sort

sensors

into

approximately

the

same

order

as

mutual

information

does

Therefore

a

sensor

with

the

maximal

entropy

difference

H

v

i

H

si

also

has

the

maximal

or

nearly

the

maximal

mutual

information

The

correlation

between

the

entropy

difference

H

v

i

H

si

and

mutual

information

I

-LRB-

x

z

i

-RRB-

is

analyzed

using

simulations

in

Sect.

Section

discusses

the

discrepancy

between

the

heuristic

and

the

mutual

information

based

approaches

This

Sect

presents

the

evaluation

of

the

entropy-based

sensor

selection

heuristic

using

simulations

The

computational

complexity

of

the

heuristic

is

also

analyzed

The

Gaussian

noise

model

has

been

widely

assumed

for

sensor

observations

in

many

localization

and

tracking

algorithms

e.g.

the

Kalman

filter

-LSB-

-RSB-

Successes

of

these

algorithms

indicate

that

the

Gaussian

sensing

model

is

a

reasonable

first-order-approximation

of

the

reality

As

a

starting

point

we

assume

Gaussian

sensing

models

in

the

evaluative

simulations

for

simplicity

The

simple

Gaussian

sensing

models

assumed

here

are

not

accurate

especially

when

sensors

are

very

close

to

the

target

To

avoid

the

problem

of

over-simplified

sensing

models

in

the

simulations

we

only

analyze

sensors

with

some

middle

distance

range

to

the

target

The

heuristic

will

be

evaluated

further

under

more

realistic

sensing

models

in

the

future

Four

scenarios

of

sensor

selection

for

localization

have

been

studied

Three

of

them

involve

DOA

sensors

range

sensors

or

time-difference-of-arrival

-LRB-

TDOA

-RRB-

sensors

respectively

One

of

them

involves

all

of

the

above

sensors

mixed

together

In

every

sensor

selection

scenario

both

the

entropy

difference

H

v

i

H

si

and

mutual

information

I

-LRB-

x

z

i

-RRB-

are

evaluated

and

compared

for

all

candidate

sensors

In

all

sensor

selection

scenarios

the

entropy

difference

H

v

i

H

si

can

sort

all

candidate

sensors

into

nearly

the

same

order

as

mutual

information

I

-LRB-

x

z

i

-RRB-

does

Therefore

the

sensor

with

the

maximal

entropy

difference

H

v

i

H

si

selected

by

the

heuristic

always

has

the

maximum

or

nearly

the

maximal

mutual

information

I

-LRB-

x

z

i

-RRB-

The

larger

the

entropy

difference

H

v

i

H

si

and

mutual

information

I

-LRB-

x

z

i

-RRB-

are

the

more

consistent

their

sensor

selection

decisions

are

Selection

of

DOA

Sensors

Consider

now

entropy-based

sensor

selection

when

all

candidate

sensors

are

DOA

sensors

as

depicted

in

Fig.

The

prior

probability

distribution

p

-LRB-

x

-RRB-

of

the

target

location

x

is

non-zero

in

a

limited

area

We

assume

the

unbiased

Gaussian

sensing

models

for

DOA

sensors

in

some

middle

distance

range

to

the

target

Specifically

given

a

target

location

such

that

x

x

i

the

probability

distribution

of

DOA

observation

z

i

is

assumed

to

be

p

-LRB-

z

i

x

-RRB-

e

-LRB-

z

i

z

v

i

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

where

z

v

i

f

-LRB-

x

x

i

-RRB-

is

the

direction

from

sensor

i

to

the

target

location

x.

For

many

DOA

estimation

algorithms

like

the

approximate

maximum

likelihood

-LRB-

AML

-RRB-

algorithm

-LSB-

-RSB-

DOA

estimation

usually

becomes

much

more

uncertain

when

the

candidate

sensor

is

either

very

near

or

very

far

from

the

target

In

this

scenario

we

exclude

sensors

that

are

either

outside

the

study

area

or

within

a

distance

of

to

the

area

of

non-zero

p

-LRB-

x

-RRB-

The

entropy

difference

H

v

i

H

si

and

mutual

information

I

-LRB-

x

z

i

-RRB-

of

DOA

sensors

are

evaluated

and

compared

in

five

cases

In

each

case

Gaussian

sensing

models

of

the

same

standard

deviation

are

assumed

for

all

candidate

sensors

However

the

standard

deviation

varies

with

the

case

As

shown

in

fig.

mutual

information

I

-LRB-

x

z

i

-RRB-

vs

the

entropy

difference

H

v

i

H

si

is

plotted

for

all

candidate

sensors

in

all

cases

Mutual

information

I

-LRB-

x

z

i

-RRB-

increases

nearly

monotonically

with

the

entropy

difference

H

v

i

H

si

The

larger

the

entropy

difference

H

v

i

H

si

and

mutual

information

I

-LRB-

x

z

i

-RRB-

are

the

more

correlated

they

are

Therefore

x

Figure

Scenario

of

sensor

selection

for

localization

using

DOA

sensors

exclusively

The

image

depicts

theprior

probability

distribution

p

-LRB-

x

-RRB-

of

thetarget

location

x.

p

-LRB-

x

-RRB-

is

zero

outside

the

solid

rectangle

The

actual

target

location

is

-LRB-

-RRB-

denoted

by

marker

The

squares

denote

candidate

DOA

sensors

for

selection

DOA

sensors

are

uniformly

randomly

placed

outside

the

dotted

rectangle

The

gap

between

the

solid

rectangle

and

the

dotted

rect-angleis

the

entropy

difference

H

v

i

H

si

sorts

DOA

sensors

in

nearly

the

same

order

as

mutual

information

I

-LRB-

x

z

i

-RRB-

does

especially

when

the

entropy

difference

H

v

i

H

si

is

large

The

candidate

DOA

sensor

selected

by

the

proposed

heuristic

has

the

maximal

entropy

difference

H

v

i

H

si

and

also

has

the

maximal

mutual

information

I

-LRB-

x

z

i

-RRB-

Selection

of

Range

Sensors

and

TDOA

Sensors

This

Subsect

evaluates

the

entropy-based

sensor

selection

heuristic

for

range

sensors

and

TDOA

sensors

respectively

Fig.

shows

the

sensor

selection

scenario

in

which

all

candidate

sensors

can

only

measure

the

range

to

the

target

The

prior

probability

distribution

p

-LRB-

x

-RRB-

of

the

target

location

x

is

non-zero

in

a

limited

area

We

assume

the

unbiased

Gaussian

sensing

models

p

-LRB-

z

i

x

-RRB-

for

range

sensors

used

in

-LSB-

-RSB-

When

the

actual

range

is

small

relative

to

the

standard

deviation

of

the

Gaussian

sensing

model

p

-LRB-

z

i

x

-RRB-

is

significantly

greater

than

zero

even

for

negative

values

of

range

observation

z

i

Because

a

range

of

negative

value

has

no

physical

meaning

the

above

Gaussian

sensing

model

is

not

valid

for

short

ranges

To

avoid

the

above

difficulty

of

the

Gaussian

sensing

model

we

only

consider

candidate

sensors

in

some

middle

distance

range

to

the

target

Specifically

in

this

range

sensor

selection

scenario

we

exclude

sensors

that

are

either

outside

the

study

area

or

within

a

distance

of

to

the

area

of

non-zero

p

-LRB-

x

-RRB-

Fig.

shows

the

sensor

selection

scenario

in

which

only

TDOA

sensors

are

used

The

prior

probability

distribution

p

-LRB-

x

-RRB-

of

the

target

location

x

is

non-zero

in

a

limited

area

As

in

-LSB-

-RSB-

the

signal

arrival

time

difference

observed

by

every

TDOA

sensor

is

relative

to

a

common

reference

sensor

We

Entropy

difference

-LRB-

bit

-RRB-

Mutual

information

-LRB-

bit

-RRB-

Mutual

information

I

-LRB-

x

z

i

-RRB-

vs

entropy

difference

H

v

i

H

si

of

DOA

sensors

Each

symbol

denotes

-LRB-

H

v

i

H

si

I

-LRB-

x

z

i

-RRB-

-RRB-

pair

evaluated

for

one

candidatese

nsor

Theprior

targe

t

location

distribution

and

the

candidate

sensor

placements

are

shown

in

Fig.

Five

cases

with

different

standard

deviation

of

Gaussian

sensing

models

are

studied

In

each

case

all

candidate

sensors

are

assumed

to

have

the

same

value

also

assume

the

unbiased

Gaussian

sensing

models

p

-LRB-

z

i

x

-RRB-

for

TDOA

sensors

In

order

to

be

comparable

with

scenarios

of

DOA

sensors

and

range

sensors

we

only

consider

TDOA

sensors

in

middle

range

distance

to

the

target

Specifically

we

exclude

TDOA

sensors

that

are

either

outside

the

study

area

or

within

a

distance

of

to

the

area

of

non-zero

p

-LRB-

x

-RRB-

Following

the

same

approach

to

the

heuristic

evaluation

for

DOA

sensors

the

entropy

difference

H

v

i

H

si

and

mutual

information

I

-LRB-

x

z

i

-RRB-

of

every

candidate

sensor

are

evaluated

and

compared

for

range

sensor

selection

scenario

in

Fig.

and

for

TDOA

sensor

selection

scenario

in

Fig.

respectively

Mutual

information

I

-LRB-

x

z

i

-RRB-

vs

the

entropy

difference

H

v

i

H

si

is

plotted

in

Fig.

for

all

range

sensors

and

in

Fig.

for

all

TDOA

sensors

In

both

scenarios

mutual

information

I

-LRB-

x

z

i

-RRB-

increases

nearly

monotonically

with

the

entropy

difference

H

v

i

H

si

The

larger

the

entropy

difference

H

v

i

H

si

and

mutual

information

I

-LRB-

x

z

i

-RRB-

are

the

more

correlated

they

are

Using

the

proposed

heuristic

both

the

selected

range

sensor

and

the

selected

TDOA

sensor

have

the

maximal

entropy

difference

H

v

i

H

si

and

also

have

nearly

the

maximal

mutual

information

I

-LRB-

x

z

i

-RRB-

Selection

of

Mixed

Sensors

In

order

to

evaluate

the

entropy-based

sensor

selection

heuristic

across

different

sensing

modalities

this

Subsect

is

devoted

to

the

sensor

selection

scenario

in

which

candidate

sensors

are

a

mixture

of

DOA

sensors

range

sensors

and

TDOA

sensors

Fig.

shows

the

sensor

selection

scenario

for

mixed

candidate

sensors

Each

candidate

sensor

is

randomly

assigned

one

of

three

sensing

modalities

namely

DOA

range

and

TDOA

Gaussian

sensing

models

are

assumed

for

all

candidate

sensors

with

middle

range

distance

to

the

target

Each

x

Figure

Scenario

of

sensor

selection

for

localization

using

rangese

nsors

Theimagede

picts

theprior

probability

distribution

p

-LRB-

x

-RRB-

of

thetarget

location

x.

p

-LRB-

x

-RRB-

is

zero

outside

the

solid

rectangle

The

actual

target

location

is

-LRB-

-RRB-

denoted

by

marker

The

circles

denote

candidate

range

sensors

for

selection

range

sensors

are

uniformly

randomly

placed

outside

the

dotted

rectangle

The

gap

between

the

solid

rectangle

and

the

dotted

rectangle

is

candidate

sensor

is

also

randomly

assigned

one

of

five

values

of

the

standard

deviation

of

the

sensing

model

namely

and

candidate

sensors

are

uniformly

randomly

placed

in

the

vicinity

of

the

prior

target

location

estimation

In

order

to

avoid

the

difficulties

of

Gaussian

sensing

models

for

DOA

sensors

and

range

sensors

close

to

the

target

we

exclude

sensors

either

outside

the

study

area

or

within

a

distance

of

to

the

non-zero

area

of

the

prior

target

location

distribution

p

-LRB-

x

-RRB-

The

entropy

difference

H

v

i

H

si

and

mutual

information

I

-LRB-

x

z

i

-RRB-

of

every

candidate

sensor

are

evaluated

and

plotted

in

Fig.

The

correlation

between

H

v

i

H

si

and

I

-LRB-

x

z

i

-RRB-

of

mixed

sensors

is

very

similar

to

the

correlation

between

H

v

i

H

si

and

I

-LRB-

x

z

i

-RRB-

of

sensors

with

single

modality

Across

various

sensing

modalities

mutual

information

I

-LRB-

x

z

i

-RRB-

increases

nearly

monotonically

with

the

entropy

difference

H

v

i

H

si

Therefore

across

various

sensing

modalities

the

candidate

sensor

with

the

maximal

entropy

difference

H

v

i

H

si

selected

by

the

proposed

heuristic

has

the

maximal

mutual

information

I

-LRB-

x

z

i

-RRB-

Computational

Complexity

Computational

complexity

analysis

is

an

important

part

of

the

evaluation

of

the

heuristic

We

will

analyze

the

complexity

of

the

heuristic

and

compare

it

to

the

complexity

of

the

mutual-information-based

approaches

For

two-dimensional

localization

the

target

location

x

is

two-dimensional

The

sensor

's

view

z

v

i

of

the

target

location

x

is

one-dimensional

The

sensor

observation

z

i

is

one-dimensional

We

assume

that

all

random

variables

are

gridded

for

numerical

computation

Specifically

the

area

with

non-trivial

p

-LRB-

x

-RRB-

is

gridded

into

n

n

The

interval

with

x

Figure

Scenario

of

sensor

selection

for

localization

using

TDOA

sensors

The

image

depicts

the

prior

probability

distribution

p

-LRB-

x

-RRB-

of

thetarget

location

x.

p

-LRB-

x

-RRB-

is

zero

outside

the

solid

rectangle

The

actual

target

location

is

-LRB-

-RRB-

denoted

by

marker

The

triangles

denote

candidate

TDOA

sensors

for

selection

Every

TDOA

observation

is

relative

to

a

common

reference

sensor

denoted

by

marker

TDOA

sensors

are

uniformly

randomly

placed

outside

the

dotted

rectangle

The

gap

between

the

solid

rectangle

and

the

dotted

rectangle

is

non-trivial

p

-LRB-

z

i

-RRB-

or

p

-LRB-

z

v

i

-RRB-

is

also

gridded

into

n

We

assume

there

are

K

candidate

sensors

for

selection

K

is

usually

a

small

number

The

proposed

heuristic

evaluates

the

entropy

difference

H

v

i

H

si

of

all

sensors

and

then

selects

the

one

with

the

maximal

H

v

i

H

si

As

shown

in

-LRB-

-RRB-

p

-LRB-

z

v

i

-RRB-

can

be

computed

from

p

-LRB-

x

-RRB-

with

cost

O

-LRB-

n

-RRB-

As

shown

in

-LRB-

-RRB-

H

v

i

can

be

computed

from

p

-LRB-

z

v

i

-RRB-

with

cost

O

-LRB-

n

-RRB-

As

shown

in

-LRB-

-RRB-

and

-LRB-

-RRB-

H

si

can

be

computed

from

p

-LRB-

z

i

x

-RRB-

with

cost

O

-LRB-

n

-RRB-

The

cost

to

compute

H

v

i

H

si

for

one

candidate

sensor

is

O

-LRB-

n

-RRB-

Therefore

the

total

cost

for

the

heuristic

to

select

one

out

of

K

candidate

sensors

is

O

-LRB-

n

-RRB-

The

mutual-information-based

approaches

evaluate

the

mutual

information

I

-LRB-

x

z

i

-RRB-

of

all

sensors

and

then

select

the

one

with

the

maximal

I

-LRB-

x

z

i

-RRB-

As

shown

in

-LRB-

-RRB-

I

-LRB-

x

z

i

-RRB-

can

be

directly

computed

from

p

-LRB-

x

-RRB-

and

p

-LRB-

z

i

x

-RRB-

with

cost

of

O

-LRB-

n

-RRB-

Therefore

the

total

cost

to

select

one

out

of

K

candidate

sensors

is

O

-LRB-

n

-RRB-

As

we

mentioned

early

in

Subsect

the

computational

cost

of

mutual

information

I

-LRB-

x

z

i

-RRB-

could

be

reduced

in

some

special

scenarios

In

general

however

the

heuristic

is

computationally

much

simpler

than

the

mutual-information-based

approaches

As

shown

in

Sect.

when

the

mutual

information

I

-LRB-

x

z

i

-RRB-

is

close

to

bit

the

entropy

difference

H

v

i

H

si

might

not

sort

candidate

sensors

into

exactly

the

same

order

as

the

mutual

information

does

Such

discrepancy

is

caused

by

the

dispersion

of

the

correlation

between

the

entropy

difference

Entropy

difference

-LRB-

bit

-RRB-

Mutual

information

-LRB-

bit

-RRB-

Mutual

information

I

-LRB-

x

z

i

-RRB-

vs

entropy

difference

H

v

i

H

si

of

range

senors

Each

symbol

denotes

-LRB-

H

v

i

H

si

I

-LRB-

x

z

i

-RRB-

-RRB-

pair

evaluated

for

one

candidate

sensor

The

prior

target

location

distribution

and

the

candidate

sensor

placements

are

shown

in

Fig.

Five

cases

with

different

standard

deviation

of

Gaussian

sensing

models

are

studied

In

each

case

all

candidate

sensors

are

assumed

to

have

the

same

value

H

v

i

H

si

and

the

mutual

information

I

-LRB-

x

z

i

-RRB-

when

the

mutual

information

is

small

In

this

Sect.

we

examine

such

correlation

dispersion

and

evaluate

its

impact

on

the

discrepancy

of

sensor

selection

decisions

of

the

entropy-based

heuristic

and

the

mutual

information

based

approaches

Dispersion

In

this

Subsect.

we

describe

the

dispersion

of

the

correlation

between

the

entropy

difference

H

v

i

H

si

and

the

mutual

information

I

-LRB-

x

z

i

-RRB-

when

the

mutual

information

is

small

We

also

examine

possible

sources

for

such

correlation

dispersion

Close

examination

on

the

convex

part

of

the

mutual

information

vs.

entropy

difference

curve

in

Fig.

and

Fig.

reveals

that

the

correlation

between

the

mutual

information

I

-LRB-

x

z

i

-RRB-

and

the

entropy

difference

H

v

i

H

si

is

not

strictly

monotonic

Instead

there

is

obvious

dispersion

of

the

correlation

The

convex

part

corresponds

to

the

situation

in

which

candidate

sensors

are

not

very

informative

because

the

mutual

information

between

the

target

location

distribution

and

the

sensor

observation

is

close

to

bit

In

another

words

when

candidate

sensors

are

not

very

informative

the

entropy

difference

H

v

i

H

si

might

not

sort

candidate

sensors

into

the

same

order

as

the

mutual

information

I

-LRB-

x

z

i

-RRB-

does

Given

a

set

of

candidate

sensors

whose

observation

could

only

reduce

a

little

amount

of

uncertainty

of

the

target

location

distribution

the

sensor

selected

on

the

basis

of

the

maximum

entropy

difference

H

v

i

H

si

might

not

have

the

maximum

mutual

information

I

-LRB-

x

z

i

-RRB-

Thus

there

might

be

discrepancy

between

the

sensor

selection

decision

of

the

entropy-based

heuristic

and

that

of

the

mutual

information

based

approaches

if

no

candidate

sensor

is

very

informative

Entropy

difference

-LRB-

bit

-RRB-

Mutual

information

-LRB-

bit

-RRB-

Mutual

information

I

-LRB-

x

z

i

-RRB-

vs

entropy

difference

H

v

i

H

si

of

TDOA

senors

Each

symbol

denotes

-LRB-

H

v

i

H

si

I

-LRB-

x

z

i

-RRB-

-RRB-

pair

evaluated

for

one

candidatese

nsor

Theprior

targe

t

location

distribution

and

the

candidate

sensor

placements

are

shown

in

Fig.

Five

cases

with

different

standard

deviation

of

Gaussian

sensing

models

are

studied

In

each

case

all

candidate

sensors

are

assumed

to

have

the

same

value

There

might

be

multiple

causes

of

such

correlation

dispersion

between

the

entropy

difference

H

v

i

H

si

and

the

mutual

information

I

-LRB-

x

z

i

-RRB-

As

pointed

out

in

Subsect

the

entropy

difference

H

v

i

H

si

can

be

viewed

as

an

approximation

of

the

mutual

information

I

-LRB-

x

z

i

-RRB-

Thus

the

order

of

sensors

sorted

by

the

entropy

difference

H

v

i

H

si

is

intrinsically

an

approximation

of

that

by

the

mutual

information

I

-LRB-

x

z

i

-RRB-

In

practice

the

discretization

of

the

state

space

of

the

target

location

random

variable

and

the

sensor

view

random

variable

might

also

introduce

inaccuracy

into

the

evaluation

of

H

v

i

Besides

as

shown

in

-LRB-

-RRB-

and

-LRB-

-RRB-

the

maximum

likelihood

estimate

of

the

target

location

is

used

to

approximate

the

actual

target

location

when

evaluating

the

entropy

of

the

sensing

model

for

the

actual

target

location

Impact

In

this

Subsect.

we

examine

the

impact

of

the

dispersion

of

the

correlation

between

the

entropy

difference

H

v

i

H

si

and

the

mutual

information

I

-LRB-

x

z

i

-RRB-

when

the

mutual

information

is

small

The

analysis

shows

that

such

correlation

dispersion

causes

very

little

degradation

to

the

quality

of

sensor

selection

decision

of

the

entropy-based

heuristic

As

shown

by

the

convex

part

of

the

mutual

information

vs.

entropy

difference

curve

in

Fig.

and

Fig.

there

is

dispersion

of

the

correlation

between

the

entropy

difference

H

v

i

H

si

and

the

mutual

information

I

-LRB-

x

z

i

-RRB-

when

candidate

sensors

are

not

very

informative

We

model

such

dispersion

using

a

uniform

distribution

bounded

by

a

parallelogram

illustrated

in

Fig.

A

candidate

sensor

could

assume

any

position

-LRB-

H

v

i

H

si

I

-LRB-

x

z

i

-RRB-

-RRB-

within

the

parallelogram

with

uniform

probability

As

shown

in

Fig.

the

geometry

of

the

parallelogram

is

defined

by

parameters

a

b

and

c.

a

x

Figure

Scenario

of

sensor

selection

for

localization

using

sensors

with

various

modalities

The

imagedepicts

theprior

probability

distribution

p

-LRB-

x

-RRB-

of

thetarge

t

location

x.

p

-LRB-

x

-RRB-

is

zero

outside

the

solid

rectangle

The

actual

target

location

is

-LRB-

-RRB-

denoted

by

marker

The

squares

the

circles

and

the

triangles

denote

DOA

sensors

range

sensors

and

TDOA

sensors

respectively

Every

TDOA

observation

is

relative

to

a

common

reference

sensor

denoted

by

marker

Each

sensor

is

randomly

chosen

to

bea

DOA

se

nsor

a

rangese

nsor

or

a

TDOA

sensor

Each

sensor

is

also

randomly

assigned

one

of

five

values

of

the

standard

deviation

of

Gaussian

sensing

models

namely

and

The

sizeof

a

symbol

indicates

themagnitudeof

sensors

of

various

sensing

modalities

and

values

areuniformly

randomly

place

d

outsidethedotte

d

rectangle

The

gap

between

the

solid

rectangle

and

the

dotted

rectangle

is

is

the

variation

scope

of

entropy

difference

H

v

i

H

si

among

the

set

of

candidate

sensors

c

indicates

the

variation

scope

of

the

mutual

information

I

-LRB-

x

z

i

-RRB-

among

the

set

of

candidate

sensors

b

describes

the

magnitude

of

dispersion

of

the

correlation

between

the

entropy

difference

H

v

i

H

si

and

the

mutual

information

I

-LRB-

x

z

i

-RRB-

Although

the

bounded

uniform

distribution

is

not

accurate

it

captures

the

major

features

of

the

correlation

dispersion

revealed

by

simulations

in

Sect.

We

choose

this

dispersion

model

for

simplicity

As

the

first

order

approximation

the

simple

dispersion

model

does

help

to

reveal

some

major

characteristics

of

the

impact

of

the

correlation

dispersion

on

the

heuristics-based

sensor

selection

A

typical

dispersion

scenario

is

illustrated

in

Fig.

The

mutual

information

I

-LRB-

x

z

i

-RRB-

of

candidate

sensors

varies

from

bit

to

bit

Correspondingly

the

entropy

difference

H

v

i

H

si

of

candidate

sensors

changes

from

bit

to

bit

For

any

value

of

the

entropy

difference

H

v

i

H

si

the

disperse

of

the

mutual

information

I

-LRB-

x

z

i

-RRB-

is

bit

Given

the

above

scenario

we

run

simulations

In

each

simulation

candidate

sensors

randomly

assume

their

-LRB-

H

v

i

H

si

I

-LRB-

x

z

i

-RRB-

-RRB-

pairs

within

the

specified

dispersion

range

In

each

simulation

we

identify

both

the

sensor

with

the

maximum

entropy

Entropy

difference

-LRB-

bit

-RRB-

Mutual

information

-LRB-

bit

-RRB-

TDOA

sensor

DOA

sensor

range

sensor

Mutual

information

I

-LRB-

x

z

i

-RRB-

vs

entropy

difference

H

v

i

H

si

of

mixed

senors

Each

symbol

denotes

-LRB-

H

v

i

H

si

I

-LRB-

x

z

i

-RRB-

-RRB-

pair

evaluated

for

one

candidate

sensor

The

prior

target

location

distribution

and

the

candidate

sensor

placements

are

shown

in

Fig.

difference

H

v

i

H

si

and

the

sensor

with

the

maximum

mutual

information

I

-LRB-

x

z

i

-RRB-

With

chance

the

sensor

selected

by

the

entropy-based

heuristic

also

has

the

maximum

mutual

information

Even

when

the

heuristic

fails

to

select

the

sensor

of

the

maximum

mutual

information

the

mutual

information

of

the

selected

sensor

is

on

average

only

about

bit

less

than

the

maximum

mutual

information

Overall

the

mutual

information

of

the

sensor

selected

by

the

entropy-based

heuristic

is

about

-LRB-

-RRB-

bit

less

than

the

maximum

mutual

information

Therefore

most

of

the

time

the

correlation

dispersion

does

not

cause

discrepancy

of

the

sensor

selection

decisions

between

the

entropy-based

heuristic

and

the

mutual

information

based

approaches

Over

all

the

entropy-based

heuristic

introduces

very

little

degradation

to

the

quality

of

the

sensor

select

decision

even

when

candidate

sensors

are

not

very

informative

We

have

analyzed

the

impact

of

the

correlation

dispersion

for

different

configurations

of

a

b

c

and

the

number

of

candidate

sensors

In

table

a

bit

b

bit

and

c

bit

are

fixed

We

only

change

the

number

of

candidate

sensors

The

chance

for

the

heuristic

to

successfully

select

the

sensor

with

the

maximum

mutual

information

decreases

as

the

number

of

candidate

sensors

increases

When

the

heuristic

fails

to

select

the

sensor

with

the

maximum

mutual

information

the

degradation

of

sensor

selection

decision

based

on

the

heuristic

compared

to

that

based

on

the

mutual

information

does

not

change

with

the

number

of

candidate

sensors

Thus

the

overall

degradation

of

sensor

selection

decision

based

on

the

heuristic

compared

to

that

based

on

mutual

information

also

increases

as

the

number

of

candidate

sensors

increases

In

table

a

bit

and

c

bit

are

fixed

and

the

number

of

candidate

sensors

are

fixed

to

be

We

only

change

the

dispersion

width

b

The

chance

for

the

heuristic

to

successfully

select

the

sensor

with

the

maximum

mutual

Entropy

difference

-LRB-

bit

-RRB-

Mutual

information

-LRB-

bit

-RRB-

b

a

c

Figure

Discrepancy

between

the

entropy-based

sensor

selection

heuristic

and

the

mutual

information

based

approaches

when

candidate

sensors

are

not

very

informative

The

dispersion

of

the

correlation

between

the

entropy

difference

H

v

i

H

si

and

themutual

information

I

-LRB-

x

z

i

-RRB-

is

modeled

by

a

uniform

distribution

bounded

by

a

parallelogram

The

geometry

of

the

parallelogram

is

defined

by

parameters

a

b

and

c.

Candidate

sensors

are

denoted

by

marker

whosecoordinates

are

-LRB-

H

v

i

H

si

I

-LRB-

x

z

i

-RRB-

-RRB-

The

entropy-based

heuristic

selects

the

rightmost

sensor

which

has

the

maximum

entropy

difference

H

v

i

H

si

and

is

enclosed

by

a

square

marker

The

mutual

information

based

approaches

selects

the

top

sensor

which

has

the

maximum

mutual

information

I

-LRB-

x

z

i

-RRB-

and

is

enclosed

by

a

diamond-shaped

marker

The

above

two

selected

sensors

might

not

be

the

same

In

the

scenario

of

this

figure

a

bits

b

bit

c

bit

and

candidatesensors

areavailable

for

selection

information

decreases

as

the

dispersion

width

b

increases

When

the

heuristic

fails

to

select

the

sensor

with

the

maximum

mutual

information

the

degradation

of

sensor

selection

decision

based

on

the

heuristic

compared

to

that

based

on

the

mutual

information

increases

as

the

dispersion

width

b

increases

Thus

the

overall

degradation

of

sensor

selection

decision

based

on

the

heuristic

compared

to

that

based

on

mutual

information

also

increases

as

the

dispersion

width

b

increases

In

table

a

bit

and

b

bit

are

fixed

and

the

number

of

candidate

sensors

are

fixed

to

be

We

only

change

the

mutual

information

variation

scope

c

The

chance

for

the

heuristic

to

successfully

select

the

sensor

with

the

maximum

mutual

information

increases

as

the

mutual

information

variation

scope

c

increases

When

the

heuristic

Table

Impact

Change

with

Number

of

Sensors

Number

of

Candidate

Sensors

Chance

of

Success

-LRB-

-RRB-

Degradation

per

Failure

-LRB-

bit

-RRB-

Overall

Degradation

-LRB-

bit

-RRB-

Impact

Changewith

Dispersion

Width

Dispersion

Width

b

-LRB-

bit

-RRB-

Chance

of

Success

-LRB-

-RRB-

Degradation

per

Failure

-LRB-

bit

-RRB-

Overall

Degradation

-LRB-

bit

-RRB-

Impact

Changewith

Mutual

Info

Scope

Mutual

Info

Scope

c

-LRB-

bit

-RRB-

Chance

of

Success

-LRB-

-RRB-

Degradation

per

Failure

-LRB-

bit

-RRB-

Overall

Degradation

-LRB-

bit

-RRB-

fails

to

select

the

sensor

with

the

maximum

mutual

information

the

degradation

of

sensor

selection

decision

based

on

the

heuristic

compared

to

that

based

on

the

mutual

information

does

not

change

much

with

the

mutual

information

variation

scope

c.

Thus

the

overall

degradation

of

sensor

selection

decision

based

on

the

heuristic

compared

to

that

based

on

mutual

information

decreases

as

the

mutual

information

variation

scope

c

increases

In

table

b

bit

is

fixed

and

the

number

of

candidate

sensors

are

fixed

to

be

We

proportionally

change

the

entropy

difference

variation

scope

a

and

the

mutual

information

variation

scope

c

so

that

c/a

is

fixed

The

chance

for

the

heuristic

to

successfully

select

the

sensor

with

the

maximum

mutual

information

increases

as

the

entropy

difference

variation

scope

a

and

the

mutual

information

variation

scope

c

proportionally

increase

When

the

heuristic

fails

to

select

the

sensor

with

the

maximum

mutual

information

the

degradation

of

sensor

selection

decision

based

on

the

heuristic

compared

to

that

based

on

the

mutual

information

does

not

change

Thus

the

overall

degradation

of

sensor

selection

decision

based

on

the

heuristic

compared

to

that

based

on

mutual

information

decreases

as

the

entropy

difference

variation

scope

a

and

the

mutual

information

variation

scope

c

proportionally

increase

When

the

sensors

is

selected

for

tracking

a

temporally

continuous

source

the

prior

target

location

distribution

at

time

t

can

be

obtained

from

the

posterior

target

location

distribution

at

time

t

by

using

the

target

dynamic

model

as

described

in

-LSB-

-RSB-

However

when

the

sensor

selection

heuristic

is

applied

to

locate

a

temporally

discontinuous

source

such

as

a

bird

call

it

is

not

straightforward

to

obtain

the

prior

target

location

distribution

used

in

the

sequential

Bayesian

fusion

One

possible

solution

to

the

above

problem

could

be

as

follows

First

all

sensors

buffer

the

signal

once

an

event

Impact

Changewith

Entropy

Diff

Scopec

and

Mutual

Info

Scope

a

in

Proportion

Entropy

Diff

Scope

a

-LRB-

bit

-RRB-

Mutual

Info

Scope

c

-LRB-

bit

-RRB-

Chance

of

Success

-LRB-

-RRB-

Degradation

per

Failure

-LRB-

bit

-RRB-

Overall

Degradation

-LRB-

bit

-RRB-

such

as

a

bird

call

is

detected

Then

all

triggered

sensors

elect

a

leader

that

received

the

strongest

signal

intensity

using

a

protocol

similar

to

that

described

in

-LSB-

-RSB-

Finally

the

leader

can

pick

a

few

sensors

to

generate

an

initial

prior

target

location

distribution

assuming

a

certain

sensing

model

With

the

initial

prior

target

location

distribution

we

can

apply

the

sensor

selection

heuristic

to

incrementally

reduce

the

uncertainty

of

the

target

location

distribution

We

plan

to

implement

and

test

the

above

mechanism

in

the

future

Discretization

of

State

Space

There

is

a

trade

offof

computational

efficiency

and

numerical

accuracy

in

the

discretization

of

the

state

space

of

random

variables

such

as

the

target

location

and

the

sensor

view

The

bigger

the

grid

size

is

the

fewer

grids

are

involved

in

the

computation

However

a

bigger

grid

size

also

introduces

more

inaccuracy

into

the

evaluation

of

the

entropy

difference

heuristic

In

the

future

we

must

study

more

details

about

the

trade

offin

order

to

choose

a

proper

grid

size

Sensing

Uncertainty

Model

We

have

assumed

Gaussian

sensing

models

in

the

simulations

as

the

first

step

to

evaluate

the

heuristic

Inaccuracy

of

sensing

models

diminishes

the

effectiveness

of

any

sensor

selection

criterion

We

plan

to

construct

a

more

realistic

sensing

model

for

the

AML-based

DOA

estimation

We

have

implemented

AML

algorithm

for

real-time

DOA

estimation

on

a

wireless

sensor

network

testbed

-LSB-

-RSB-

We

will

first

analyze

the

sensing

uncertainty

characteristic

of

the

AML

algorithm

and

then

experimentally

validate

and

refine

it

using

the

testbed

We

will

also

evaluate

the

effectiveness

of

the

entropy-based

sensor

selection

heuristic

using

realistic

sensing

models

and

implement

the

heuristic

on

the

real-time

wireless

sensor

network

testbed

for

localization

We

have

proposed

an

entropy-based

sensor

selection

heuristic

for

localization

The

effectiveness

of

the

heuristic

has

been

evaluated

using

simulations

in

which

Gaussian

sensing

models

are

assumed

for

simplicity

Simulations

have

shown

that

the

heuristic

selects

the

sensor

with

nearly

the

maximal

mutual

information

between

the

target

location

and

the

sensor

observation

Given

the

prior

target

location

distribution

the

sensor

locations

and

the

sensing

models

on

average

the

sensor

selected

by

the

heuristic

would

yield

nearly

the

greatest

reduction

in

the

entropy

of

the

posterior

target

location

distribution

The

heuristic

is

more

effective

when

the

optimal

candidate

sensor

is

more

informative

As

mutual-information

based

sensor

selection

approaches

-LSB-

-RSB-

do

the

heuristic

greedily

selects

one

sensor

in

each

step

without

retrieving

any

actual

sensor

observations

In

addition

in

general

our

heuristic

is

computationally

much

simpler

than

the

mutual-information-based

approaches

This

material

is

based

upon

work

partially

supported

by

the

National

Science

Foundation

-LRB-

NSF

-RRB-

under

Cooperative

Agreement

#CCR

and

DARPA

SensIT

program

under

contract

AFRL/IFG

315 330-1865

and

AROD-MURI

PSU

-LSB-

-RSB-

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F.

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W.

Su

Y.

Sankarasubramaniam

and

E.

Cayirci

Wireless

sensor

networks

a

survey

Computer

Networks

-LRB-

-RRB-

March

-LSB-

-RSB-

G.

Asada

M.

Dong

T.

Lin

F.

Newberg

G.

Pottie

W.

Kaiser

and

H.

Marcy

Wireless

integrated

network

sensors

low

power

systems

on

a

chip

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Proc

the

European

Solid

State

Circuits

Conference

The

Hague

Netherlands

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M.

Bernardo

and

A.

F.

M.

Smith

Bayesian

theory

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New

York

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Chen

R.

Hudson

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Yao

Maximum-likelihood

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target tracking
information fusion
localization
heuristic
entropy
wireless sensor networks
sensor selection
mutual information