Efficient

Bayesian

Hierarchical

User

Modeling

for

Recommendation

Systems

Yi

Zhang,

Jonathan

Koren

School

of

Engineering

University

of

California

Santa

Cruz

Santa

Cruz,

CA,

USA

{yiz,

jonathan}@soe.ucsc.edu

ABSTRACT

A

content-based

personalized

recommendation

system

learns

user

specific

profiles

from

user

feedback

so

that

it

can

deliver

information

tailored

to

each

individual

user"s

interest.

A

system

serving

millions

of

users

can

learn

a

better

user

profile

for

a

new

user,

or

a

user

with

little

feedback,

by

borrowing

information

from

other

users

through

the

use

of

a

Bayesian

hierarchical

model.

Learning

the

model

parameters

to

optimize

the

joint

data

likelihood

from

millions

of

users

is

very

computationally

expensive.

The

commonly

used

EM

algorithm

converges

very

slowly

due

to

the

sparseness

of

the

data

in

IR

applications.

This

paper

proposes

a

new

fast

learning

technique

to

learn

a

large

number

of

individual

user

profiles.

The

efficacy

and

efficiency

of

the

proposed

algorithm

are

justified

by

theory

and

demonstrated

on

actual

user

data

from

Netflix

and

MovieLens.

Categories

and

Subject

Descriptors:

[Information

Search

and

Retrieval]:

Information

filtering

General

Terms:

Algorithms

INTRODUCTION

Personalization

is

the

future

of

the

Web,

and

it

has

achieved

great

success

in

industrial

applications.

For

example,

online

stores,

such

as

Amazon

and

Netflix,

provide

customized

recommendations

for

additional

products

or

services

based

on

a

user"s

history.

Recent

offerings

such

as

My

MSN,

My

Yahoo!,

My

Google,

and

Google

News

have

attracted

much

attention

due

to

their

potential

ability

to

infer

a

user"s

interests

from

his/her

history.

One

major

personalization

topic

studied

in

the

information

retrieval

community

is

content-based

personal

recommendation

These

systems

learn

user-specific

profiles

from

user

feedback

so

that

they

can

recommend

information

tailored

to

each

individual

user"s

interest

without

requiring

the

user

to

make

an

explicit

query.

Learning

the

user

profiles

is

the

core

problem

for

these

systems.

A

user

profile

is

usually

a

classifier

that

can

identify

whether

a

document

is

relevant

to

the

user

or

not,

or

a

regression

model

that

tells

how

relevant

a

document

is

to

the

user.

One

major

challenge

of

building

a

recommendation

or

personalization

system

is

that

the

profile

learned

for

a

particular

user

is

usually

of

low

quality

when

the

amount

of

data

from

that

particular

user

is

small.

This

is

known

as

the

cold

start

problem.

This

means

that

any

new

user

must

endure

poor

initial

performance

until

sufficient

feedback

from

that

user

is

provided

to

learn

a

reliable

user

profile.

There

has

been

much

research

on

improving

classification

accuracy

when

the

amount

of

labeled

training

data

is

small.

The

semi-supervised

learning

approach

combines

unlabeled

and

labeled

data

together

to

achieve

this

goal

Another

approach

is

using

domain

knowledge.

Researchers

have

modified

different

learning

algorithms,

such

as

Na¨ıveBayes

logistic

regression

and

SVMs

to

integrate

domain

knowledge

into

a

text

classifier.

The

third

approach

is

borrowing

training

data

from

other

resources

The

effectiveness

of

these

different

approaches

is

mixed,

due

to

how

well

the

underlying

model

assumption

fits

the

data.

One

well-received

approach

to

improve

recommendation

system

performance

for

a

particular

user

is

borrowing

information

from

other

users

through

a

Bayesian

hierarchical

modeling

approach.

Several

researchers

have

demonstrated

that

this

approach

effectively

trades

off

between

shared

and

user-specific

information,

thus

alleviating

poor

initial

performance

for

each

In

order

to

learn

a

Bayesian

hierarchical

model,

the

system

usually

tries

to

find

the

most

likely

model

parameters

for

the

given

data.

A

mature

recommendation

system

usually

works

for

millions

of

users.

It

is

well

known

that

learning

the

optimal

parameters

of

a

Bayesian

hierarchical

model

is

computationally

expensive

when

there

are

thousands

or

millions

of

users.

The

EM

algorithm

is

a

commonly

used

technique

for

parameter

learning

due

to

its

simplicity

and

convergence

guarantee.

However,

a

content

based

recommendation

system

often

handles

documents

in

a

very

high

dimensional

space,

in

which

each

document

is

represented

by

a

very

sparse

vector.

With

careful

analysis

of

the

EM

algorithm

in

this

scenario

(Section

we

find

that

the

EM

tering,

or

item-based

collaborative

filtering.

In

this

paper,

the

words

filtering

and

recommendation

are

used

interchangeably.

algorithm

converges

very

slowly

due

to

the

sparseness

of

the

input

variables.

We

also

find

that

updating

the

model

parameter

at

each

EM

iteration

is

also

expensive

with

computational

complexity

of

O(MK),

where

M

is

the

number

of

users

and

K

is

the

number

of

dimensions.

This

paper

modifies

the

standard

EM

algorithm

to

create

an

improved

learning

algorithm,

which

we

call

the

Modified

EM

algorithm.

The

basic

idea

is

that

instead

of

calculating

the

numerical

solution

for

all

the

user

profile

parameters,

we

derive

the

analytical

solution

of

the

parameters

for

some

feature

dimensions,

and

at

the

M

step

use

the

analytical

solution

instead

of

the

numerical

solution

estimated

at

E

step

for

those

parameters.

This

greatly

reduces

the

computation

at

a

single

EM

iteration,

and

also

has

the

benefit

of

increasing

the

convergence

speed

of

the

learning

algorithm.

The

proposed

technique

is

not

only

well

supported

by

theory,

but

also

by

experimental

results.

The

organization

of

the

remaining

parts

of

this

paper

is

as

follows:

Section

describes

the

Bayesian

hierarchical

linear

regression

modeling

framework

used

for

content-based

recommendations.

Section

describes

how

to

learn

the

model

parameters

using

the

standard

EM

algorithm,

along

with

using

the

new

technique

proposed

in

this

paper.

The

experimental

setting

and

results

used

to

validate

the

proposed

learning

technique

are

reported

in

Sections

and

Section

summarizes

and

offers

concluding

remarks.

RELATED

WORK

Providing

personalized

recommendations

to

users

has

been

identified

as

a

very

important

problem

in

the

IR

community

since

the

The

approaches

that

have

been

used

to

solve

this

problem

can

be

roughly

classified

into

two

major

categories:

content

based

filtering

versus

collaborative

filtering.

Content-based

filtering

studies

the

scenario

where

a

recommendation

system

monitors

a

document

stream

and

pushes

documents

that

match

a

user

profile

to

the

corresponding

user.

The

user

may

read

the

delivered

documents

and

provide

explicit

relevance

feedback,

which

the

filtering

system

then

uses

to

update

the

user"s

profile

using

relevance

feedback

retrieval

models

(e.g.

Boolean

models,

vector

space

models,

traditional

probabilistic

models

inference

networks

and

language

models

or

machine

learning

algorithms

(e.g.

Support

Vector

Machines

(SVM),

K

nearest

neighbors

(K-NN)

clustering,

neural

networks,

logistic

regression,

or

Winnow

Collaborative

filtering

goes

beyond

merely

using

document

content

to

recommend

items

to

a

user

by

leveraging

information

from

other

users

with

similar

tastes

and

preferences

in

the

past.

Memorybased

heuristics

and

model

based

approaches

have

been

used

in

collaborative

filtering

task

This

paper

contributes

to

the

content-based

recommendation

research

by

improving

the

efficiency

and

effectiveness

of

Bayesian

hierarchical

linear

models,

which

have

a

strong

theoretical

basis

and

good

empirical

performance

on

recommendation

This

paper

does

not

intend

to

compare

content-based

filtering

with

collaborative

filtering

or

claim

which

one

is

a

better.

We

think

each

complements

the

other,

and

that

content-based

filtering

is

extremely

useful

for

handling

new

documents/items

with

little

or

no

user

feedback.

Similar

to

some

other

we

found

that

a

recommendation

system

will

be

more

effective

when

both

techniques

are

combined.

However,

this

is

beyond

the

scope

of

this

paper

and

thus

not

discussed

here.

BAYESIAN

HIERARCHICAL

LINEAR

REGRESSION

Assume

there

are

M

users

in

the

system.

The

task

of

the

system

is

to

recommend

documents

that

are

relevant

to

each

user.

For

each

user,

the

system

learns

a

user

model

from

the

user"s

history.

In

the

rest

of

this

paper,

we

will

use

the

following

notations

to

represent

the

variables

in

the

system.

m

M:

The

index

for

each

individual

user.

M

is

the

total

number

of

users.

wm:

The

user

model

parameter

associated

with

user

m.

wm

is

a

K

dimensional

vector.

j

Jm:

The

index

for

a

set

of

data

for

user

m.

Jm

is

the

number

of

training

data

for

user

m.

Dm

{(xm,j,

ym,j)}:

A

set

of

data

associated

with

user

m.

xm,j

is

a

K

dimensional

vector

that

represents

the

mth

user"s

jth

training

ym,j

is

a

scalar

that

represents

the

label

of

document

xm,j.

k

K:

The

dimensional

index

of

input

variable

x.

The

Bayesian

hierarchical

modeling

approach

has

been

widely

used

in

real-world

information

retrieval

applications.

Generalized

Bayesian

hierarchical

linear

models,

one

of

the

simplest

Bayesian

hierarchical

models,

are

commonly

used

and

have

achieved

good

performance

on

collaborative

filtering

and

content-based

adaptive

filtering

tasks.

Figure

shows

the

graphical

representation

of

a

Bayesian

hierarchical

model.

In

this

graph,

each

user

model

is

represented

by

a

random

vector

wm.

We

assume

a

user

model

is

sampled

randomly

from

a

prior

distribution

P(w|Φ).

The

system

can

predict

the

user

label

y

of

a

document

x

given

an

estimation

of

wm

(or

wm"s

distribution)

using

a

function

y

f(x,

w).

The

model

is

called

generalized

Bayesian

hierarchical

linear

model

when

y

f(wT

x)

is

any

generalized

linear

model

such

as

logistic

regression,

SVM,

and

linear

regression.

To

reliably

estimate

the

user

model

wm,

the

system

can

borrow

information

from

other

users

through

the

prior

Φ

(µ,

Σ).

Now

we

look

at

one

commonly

used

model

where

y

wT

x

where

σ2

is

a

random

noise

Assume

that

each

user

model

wm

is

an

independent

draw

from

a

population

distribution

P(w|Φ),

which

is

governed

by

some

unknown

hyperparameter

Φ.

Let

the

prior

distribution

of

user

model

w

be

a

Gaussian

distribution

with

parameter

Φ

(µ,

Σ),

which

is

the

commonly

used

prior

for

linear

models.

µ

µK

is

a

K

dimensional

vector

that

represents

the

mean

of

the

Gaussian

distribution,

and

Σ

is

the

covariance

matrix

of

the

Gaussian.

Usually,

a

Normal

distribution

aI)

and

an

Inverse

Wishart

distribution

P(Σ)

|Σ|−

b

are

used

as

hyperprior

to

model

the

prior

distribution

of

µ

and

Σ

respectively.

I

is

the

K

dimensional

identity

matrix,

and

a,

b,

and

c

are

real

numbers.

With

these

settings,

we

have

the

following

model

for

the

system:

µ

and

Σ

are

sampled

from

aI)

and

IWν

(aI),

respectively.

The

first

dimension

of

x

is

a

dummy

variable

that

always

equals

to

Figure

Illustration

of

dependencies

of

variables

in

the

hierarchical

model.

The

rating,

y,

for

a

document,

x,

is

conditioned

on

the

document

and

the

user

model,

wm,

associated

with

the

user

m.

Users

share

information

about

their

models

through

the

prior,

Φ

(µ,

Σ).

For

each

user

m,

wm

is

sampled

randomly

from

a

Normal

distribution:

wm

N(µ,

Σ2

For

each

item

xm,j,

ym,j

is

sampled

randomly

from

a

Normal

distribution:

ym,j

N(wT

mxm,j,

σ2

Let

θ

(Φ,

wM

represent

the

parameters

of

this

system

that

needs

to

be

estimated.

The

joint

likelihood

for

all

the

variables

in

the

probabilistic

model,

which

includes

the

data

and

the

parameters,

is:

P(D,

θ)

P(Φ)

m

P(wm|Φ)

j

P(ym,j|xm,j,

wm)

For

simplicity,

we

assume

a,

b,

c,

and

σ

are

provided

to

the

system.

MODEL

PARAMETER

LEARNING

If

the

prior

Φ

is

known,

finding

the

optimal

wm

is

straightforward:

it

is

a

simple

linear

regression.

Therefore,

we

will

focus

on

estimating

Φ.

The

maximum

a

priori

solution

of

Φ

is

given

by

ΦMAP

arg

max

Φ

P(Φ|D)

arg

max

Φ

P(Φ,

D)

P(D)

arg

max

Φ

P(D|Φ)P(Φ)

arg

max

Φ

w

P(D|w,

Φ)P(w|Φ)P(Φ)dw

Finding

the

optimal

solution

for

the

above

problem

is

challenging,

since

we

need

to

integrate

over

all

w

wM

which

are

unobserved

hidden

variables.

EM

Algorithm

for

Bayesian

Hierarchical

Linear

Models

In

Equation

Φ

is

the

parameter

needs

to

be

estimated,

and

the

result

depends

on

unobserved

latent

variables

w.

This

kind

of

optimization

problem

is

usually

solved

by

the

EM

algorithm.

Applying

EM

to

the

above

problem,

the

set

of

user

models

w

are

the

unobservable

hidden

variables

and

we

have:

Q

w

P(w|µ,

Σ2

Dm)

log

P(µ,

Σ2

w,

D)dw

Based

on

the

derivation

of

the

EM

formulas

presented

in

we

have

the

following

Expectation-Maximization

steps

for

finding

the

optimal

hyperparameters.

For

space

considerations,

we

omit

the

derivation

in

this

paper

since

it

is

not

the

focus

of

our

work.

E

step:

For

each

user

m,

estimate

the

user

model

distribution

P(wm|Dm,

Φ)

N(wm;

¯wm,

Σ2

m)

based

on

the

current

estimation

of

the

prior

Φ

(µ,

Σ2

¯wm

Sxx,m

σ2

Sxy,m

σ2

µ)

Σ2

m

Sxx,m

σ2

where

Sxx,m

j

xm,jxT

m,j

Sxy,m

j

xm,jym,j

M

step:

Optimize

the

prior

Φ

(µ,

Σ2

based

on

the

estimation

from

the

last

E

step.

µ

M

m

¯wm

Σ2

M

m

Σ2

m

¯wm

µ)(

¯wm

µ)T

Many

machine

learning

driven

IR

systems

use

a

point

estimate

of

the

parameters

at

different

stages

in

the

system.

However,

we

are

estimating

the

posterior

distribution

of

the

variables

at

the

E

step.

This

avoids

overfitting

wm

to

a

particular

user"s

data,

which

may

be

small

and

noisy.

A

detailed

discussion

about

this

subject

appears

in

New

Algorithm:

Modified

EM

Although

the

EM

algorithm

is

widely

studied

and

used

in

machine

learning

applications,

using

the

above

EM

process

to

solve

Bayesian

hierarchical

linear

models

in

large-scale

information

retrieval

systems

is

still

too

computationally

expensive.

In

this

section,

we

describe

why

the

learning

rate

of

the

EM

algorithm

is

slow

in

our

application

and

introduce

a

new

technique

to

make

the

learning

of

the

Bayesian

hierarchical

linear

model

scalable.

The

derivation

of

the

new

learning

algorithm

will

be

based

on

the

EM

algorithm

described

in

the

previous

section.

First,

the

covariance

matrices

Σ2

Σ2

m

are

usually

too

large

to

be

computationally

feasible.

For

simplicity,

and

as

a

common

practice

in

IR,

we

do

not

model

the

correlation

between

features.

Thus

we

approximate

these

matrices

with

K

dimensional

diagonal

matrices.

In

the

rest

of

the

paper,

we

use

these

symbols

to

represent

their

diagonal

approximations:

Σ2

σ2

σ2

σ2

K

Σ2

m

σ2

σ2

σ2

m,K

Secondly,

and

most

importantly,

the

input

space

is

very

sparse

and

there

are

many

dimensions

that

are

not

related

to

a

particular

user

in

a

real

IR

application.

For

example,

let

us

consider

a

movie

recommendation

system,

with

the

input

variable

x

representing

a

particular

movie.

For

the

jth

movie

that

the

user

m

has

seen,

let

xm,j,k

if

the

director

of

the

movie

is

Jean-Pierre

Jeunet

(indexed

by

k).

Here

we

assume

that

whether

or

not

that

this

director

directed

a

specific

movie

is

represented

by

the

kth

dimension.

If

the

user

m

has

never

seen

a

movie

directed

by

Jean-Pierre

Jeunet,

then

the

corresponding

dimension

is

always

zero

(xm,j,k

for

all

j)

One

major

drawback

of

the

EM

algorithm

is

that

the

importance

of

a

feature,

µk,

may

be

greatly

dominated

by

users

who

have

never

encountered

this

feature

(i.e.

j

xm,j,k

at

the

M

step

(Equation

Assume

that

out

of

million

users

have

viewed

the

movie

directed

by

Jean-Pierre

Jeunet,

and

that

the

viewers

have

rated

all

of

his

movies

as

excellent.

Intuitively,

he

is

a

good

director

and

the

weight

for

him

(µk)

should

be

high.

Before

the

EM

iteration,

the

initial

value

of

µ

is

usually

set

to

Since

the

other

users

have

not

seen

this

movie,

their

corresponding

weights

wm,k...,

for

that

director

would

be

very

small

initially.

Thus

the

corresponding

weight

of

the

director

in

the

prior

µk

at

the

first

M

step

would

be

very

low

and

the

variance

σm,k

will

be

large

(Equations

and

It

is

undesirable

that

users

who

have

never

seen

any

movie

produced

by

the

director

influence

the

importance

of

the

director

so

much.

This

makes

the

convergence

of

the

standard

EM

algorithm

very

slow.

Now

let"s

look

at

whether

we

can

improve

the

learning

speed

of

the

algorithm.

Without

a

loss

of

generality,

let

us

assume

that

the

kth

dimension

of

the

input

variable

x

is

not

related

to

a

particular

user

m.

By

which

we

mean,

xm,j,k

for

all

j

Jm.

It

is

straightforward

to

prove

that

the

kth

row

and

kth

column

of

Sxx,m

are

completely

filled

with

zeros,

and

that

the

kth

dimension

of

Sxy,m

is

zeroed

as

well.

Thus

the

corresponding

kth

dimension

of

the

user

model"s

mean,

¯wm,

should

be

equal

to

that

of

the

prior:

¯wm,k

µk,

with

the

corresponding

covariance

of

σm,k

σk.

At

the

M

step,

the

standard

EM

algorithm

uses

the

numerical

solution

of

the

distribution

P(wm|Dm,

Φ)

estimated

at

E

step

(Equation

and

Equation

However,

the

numerical

solutions

are

very

unreliable

for

¯wm,k

and

σm,k

when

the

kth

dimension

is

not

related

to

the

mth

user.

A

better

approach

is

using

the

analytical

solutions

¯wm,k

µk,

and

σm,k

σk

for

the

unrelated

(m,

k)

pairs,

along

with

the

numerical

solution

estimated

at

E

step

for

the

other

(m,

k)

pairs.

Thus

we

get

the

following

new

EM-like

algorithm:

Modified

E

step:

For

each

user

m,

estimate

the

user

model

distribution

P(wm|Dm,

Φ)

N(wm;

¯wm,

Σ2

m)

based

on

the

current

estimation

of

σ

µ,

Σ2

¯wm

Sxx,m

σ2

Sxy,m

σ2

σ2

m,k

sxx,m,k

σ2

where

sxx,m,k

j

m,j,k

and

sxy,m,k

j

xm,j,kym,j

Modified

M

Step

Optimize

the

prior

Φ

(µ,

Σ2

based

on

the

estimation

from

the

last

E

step

for

related

userfeature

pairs.

The

M

step

implicitly

uses

the

analytical

solution

for

unrelated

user-feature

pairs.

µk

Mk

m:related

¯wm,k

σ2

k

Mk

m:related

σ2

m,k

¯wm,k

µk)(

¯wm,k

µk)T

where

Mk

is

the

number

of

users

that

are

related

to

feature

k

We

only

estimate

the

diagonal

of

Σ2

m

and

Σ

since

we

are

using

the

diagonal

approximation

of

the

covariance

matrices.

To

estimate

¯wm,

we

only

need

to

calculate

the

numerical

solutions

for

dimensions

that

are

related

to

user

m.

To

estimate

σ2

k

and

µk,

we

only

sum

over

users

that

are

related

to

the

kth

feature.

There

are

two

major

benefits

of

the

new

algorithm.

First,

because

only

the

related

(m,

k)

pairs

are

needed

at

the

modified

M

step,

the

computational

complexity

in

a

single

EM

iteration

is

much

smaller

when

the

data

is

sparse,

and

many

of

(m,

k)

pairs

are

unrelated.

Second,

the

parameters

estimated

at

the

modified

M

step

(Equations

are

more

accurate

than

the

standard

M

step

described

in

Section

because

the

exact

analytical

solutions

¯wm,k

µk

and

σm,k

σk

for

the

unrelated

(m,

k)

pairs

were

used

in

the

new

algorithm

instead

of

an

approximate

solution

as

in

the

standard

algorithm.

EXPERIMENTAL

METHODOLOGY

Evaluation

Data

Set

To

evaluate

the

proposed

technique,

we

used

the

following

three

major

data

sets

(Table

MovieLens

Data:

This

data

set

was

created

by

combining

the

relevance

judgments

from

the

data

set

with

documents

from

the

Internet

Movie

Database

(IMDB).

MovieLens

allows

users

to

rank

how

much

he/she

enjoyed

a

specific

movie

on

a

scale

from

to

This

likeability

rating

was

used

as

a

measurement

of

how

relevant

the

document

representing

the

corresponding

movie

is

to

the

user.

We

considered

documents

with

likeability

scores

of

or

as

relevant,

and

documents

with

a

score

of

to

as

irrelevant

to

the

user.

MovieLens

provided

relevance

judgments

on

documents

from

separate

users.

On

average,

each

user

rated

movies,

of

these

were

judged

to

be

relevant.

The

average

score

for

a

document

was

Documents

representing

each

movie

were

constructed

from

the

portion

of

the

IMDB

database

that

is

available

for

public

Based

on

this

database,

we

created

one

document

per

movie

that

contained

the

relevant

information

about

it

(e.g.

directors,

actors,

etc.).

Table

Data

Set

Statistics.

On

Reuters,

the

number

of

rating

for

a

simulated

user

is

the

number

of

documents

relevant

to

the

corresponding

topic.

Data

Users

Docs

Ratings

per

User

MovieLens

Netflix-all

Reuters-C

Reuters-E

Reuters-G

Reuters-M

Netflix

Data:

This

data

set

was

constructed

by

combining

documents

about

movies

crawled

from

the

web

with

a

set

of

actual

movie

rental

customer

relevance

judgments

from

Netflix

publicly

provides

the

relevance

judgments

of

anonymous

customers.

There

are

around

million

rating

on

a

scale

of

to

for

documents.

Similar

to

MovieLens,

we

considered

documents

with

likeability

scores

of

or

as

relevant.

This

number

was

reduced

to

customers

through

random

sampling.

The

average

customer

on

the

reduced

data

set

provided

judgments,

with

being

deemed

relevant.

The

average

score

for

documents

is

Reuters

Data:

This

is

the

Reuters

Corpus,

Volume

It

covers

Reuters

English

language

news

stories

from

August

to

August

Only

the

first

news

were

used

in

our

experiments.

The

Reuters

corpus

comes

with

a

topic

hierarchy.

Each

document

is

assigned

to

one

of

several

locations

on

the

hierarchical

tree.

The

first

level

of

the

tree

contains

four

topics,

denoted

as

C,

E,

M,

and

G.

For

the

experiments

in

this

paper,

the

tree

was

cut

at

level

to

create

four

smaller

trees,

each

of

which

corresponds

to

one

smaller

data

set:

Reuters-E

Reuters-C,

ReutersM

and

Reuters-G.

For

each

small

data

set,

we

created

several

profiles,

one

profile

for

each

node

in

a

sub-tree,

to

simulate

multiple

users,

each

with

a

related,

yet

separate

definition

of

relevance.

All

the

user

profiles

on

a

sub-tree

are

supposed

to

share

the

same

prior

model

distribution.

Since

this

corpus

explicitly

indicates

only

the

relevant

documents

for

a

topic(user),

all

other

documents

are

considered

irrelevant.

Evaluation

We

designed

the

experiments

to

answer

the

following

three

questions:

Do

we

need

to

take

the

effort

to

use

a

Bayesian

approach

and

learn

a

prior

from

other

users?

Does

the

new

algorithm

work

better

than

the

standard

EM

algorithm

for

learning

the

Bayesian

hierarchical

linear

model?

Can

the

new

algorithm

quickly

learn

many

user

models?

To

answer

the

first

question,

we

compared

the

Bayesian

hierarchical

models

with

commonly

used

regularized

linear

regression

models.

In

fact,

the

commonly

used

approach

is

equivalent

to

the

model

learned

at

the

end

of

the

first

EM

iteration.

To

answer

the

second

question,

we

compared

the

proposed

new

algorithm

with

the

standard

EM

algorithm

to

see

whether

the

new

learning

algorithm

is

better.

To

answer

the

third

question,

we

tested

the

efficiency

of

the

new

algorithm

on

the

entire

Netflix

data

set

where

about

half

a

million

user

models

need

to

be

learned

together.

For

the

MovieLens

and

Netflix

data

sets,

algorithm

effectiveness

was

measured

by

mean

square

error,

while

on

the

Reuters

data

set

classification

error

was

used

because

it

was

more

informative.

We

first

evaluated

the

performance

on

each

individual

user,

and

then

estimated

the

macro

average

over

all

users.

Statistical

tests

(t-tests)

were

carried

out

to

see

whether

the

results

are

significant.

For

the

experiments

on

the

MovieLens

and

Netflix

data

sets,

we

used

a

random

sample

of

of

each

user

for

training,

and

the

rest

for

testing.

On

Reuters"

data

set,

because

there

are

too

many

relevant

documents

for

each

topic

in

the

corpus,

we

used

a

random

sample

of

of

each

topic

for

training,

and

of

the

remaining

documents

for

testing.

For

all

runs,

we

set

(a,

b,

c,

Σ

manually.

EXPERIMENTAL

RESULTS

Figure

Figure

and

Figure

show

that

on

all

data

sets,

the

Bayesian

hierarchical

modeling

approach

has

a

statistical

significant

improvement

over

the

regularized

linear

regression

model,

which

is

equivalent

to

the

Bayesian

hierarchical

models

learned

at

the

first

iteration.

Further

analysis

shows

a

negative

correlation

between

the

number

of

training

data

for

a

user

and

the

improvement

the

system

gets.

This

suggests

that

the

borrowing

information

from

other

users

has

more

significant

improvements

for

users

with

less

training

data,

which

is

as

expected.

However,

the

strength

of

the

correlation

differs

over

data

sets,

and

the

amount

of

training

data

is

not

the

only

characteristics

that

will

influence

the

final

performance.

Figure

and

Figure

show

that

the

proposed

new

algorithm

works

better

than

the

standard

EM

algorithm

on

the

Netflix

and

MovieLens

data

sets.

This

is

not

surprising

since

the

number

of

related

feature-users

pairs

is

much

smaller

than

the

number

of

unrelated

feature-user

pairs

on

these

two

data

sets,

and

thus

the

proposed

new

algorithm

is

expected

to

work

better.

Figure

shows

that

the

two

algorithms

work

similarly

on

the

Reuters-E

data

set.

The

accuracy

of

the

new

algorithm

is

similar

to

that

of

the

standard

EM

algorithm

at

each

iteration.

The

general

patterns

are

very

similar

on

other

Reuters"

subsets.

Further

analysis

shows

that

only

of

the

user-feature

pairs

are

unrelated

on

this

data

set.

Since

the

number

of

unrelated

user-feature

pairs

is

not

extremely

large,

the

sparseness

is

not

a

serious

problem

on

the

Reuters

data

set.

Thus

the

two

learning

algorithms

perform

similarly.

The

results

suggest

that

only

on

a

corpus

where

the

number

of

unrelated

user-feature

pairs

is

much

larger

than

the

number

of

related

pairs,

such

as

on

the

Netflix

data

set,

the

proposed

technique

will

get

a

significant

improvement

over

standard

EM.

However,

the

experiments

also

show

that

when

the

assumption

does

not

hold,

the

new

algorithm

does

not

hurt

performance.

Although

the

proposed

technique

is

faster

than

standard

Figure

Performance

on

a

Netflix

subset

with

users.

The

new

algorithm

is

statistical

significantly

better

than

EM

algorithm

at

iterations

regularized

linear

models

are

equivalent

to

the

Bayesian

hierarchical

models

learned

at

the

first

iteration,

and

are

statistical

significantly

worse

than

the

Bayesian

hierarchical

models.

Iterations

MeanSquareError

New

Algorithm

Traditional

EM

Iterations

ClassificationError

New

Algorithm

Traditional

EM

Figure

Performance

on

a

MovieLens

subset

with

users.

The

new

algorithm

is

statistical

significantly

better

than

EM

algorithm

at

iteration

to

(evaluated

with

mean

square

error).

Iterations

MeanSquareError

New

Algorithm

Traditional

EM

Iterations

ClassificationError

New

Algorithm

Traditional

EM

Figure

Performance

on

a

Reuters-E

subset

with

profiles.

Performances

on

Reuters-C,

Reuters-M,

Reuters-G

are

similar.

Iterations

MeanSquareError

New

Algorithm

Traditional

EM

Iterations

ClassificationError

New

Algorithm

Traditional

EM

EM,

can

it

really

learn

millions

of

user

models

quickly?

Our

results

show

that

the

modified

EM

algorithm

converges

quickly,

and

modified

EM

iterations

would

result

in

a

reliable

estimation.

We

evaluated

the

algorithm

on

the

whole

Netflix

data

set

users,

features,

and

million

ratings)

running

on

a

single

CPU

PC

memory,

The

system

finished

one

modified

EM

iteration

in

about

hours.

This

demonstrates

that

the

proposed

technique

can

efficiently

handle

large-scale

system

like

Netflix.

CONCLUSION

Content-based

user

profile

learning

is

an

important

problem

and

is

the

key

to

providing

personal

recommendations

to

a

user,

especially

for

recommending

new

items

with

a

small

number

of

ratings.

The

Bayesian

hierarchical

modeling

approach

is

becoming

an

important

user

profile

learning

approach

due

to

its

theoretically

justified

ability

to

help

one

user

through

information

transfer

from

the

other

users

by

way

of

hyperpriors.

This

paper

examined

the

weakness

of

the

popular

EM

based

learning

approach

for

Bayesian

hierarchical

linear

models

and

proposed

a

better

learning

technique

called

Modified

EM.

We

showed

that

the

new

technique

is

theoretically

more

computationally

efficient

than

the

standard

EM

algorithm.

Evaluation

on

the

MovieLens

and

Netflix

data

sets

demonstrated

the

effectiveness

of

the

new

technique

when

the

data

is

sparse,

by

which

we

mean

the

ratio

of

related

user-feature

pairs

to

unrelated

pairs

is

small.

Evaluation

on

the

Reuters

data

set

showed

that

the

new

technique

performed

similar

to

the

standard

EM

algorithm

when

the

sparseness

condition

does

not

hold.

In

general,

it

is

better

to

use

the

new

algorithm

since

it

is

as

simple

as

standard

EM,

the

performance

is

either

better

or

similar

to

EM,

and

the

computation

complexity

is

lower

at

each

iteration.

It

is

worth

mentioning

that

even

if

the

original

problem

space

is

not

sparse,

sparseness

can

be

created

artificially

when

a

recommendation

system

uses

user-specific

feature

selection

techniques

to

reduce

the

noise

and

user

model

complexity.

The

proposed

technique

can

also

be

adapted

to

improve

the

learning

in

such

a

scenario.

We

also

demonstrated

that

the

proposed

technique

can

learn

half

a

million

user

profiles

from

million

ratings

in

a

few

hours

with

a

single

CPU.

The

research

is

important

because

scalability

is

a

major

concern

for

researchers

when

using

the

Bayesian

hierarchical

linear

modeling

approach

to

build

a

practical

large

scale

system,

even

though

the

literature

have

demonstrated

the

effectiveness

of

the

models

in

many

applications.

Our

work

is

one

major

step

on

the

road

to

make

Bayesian

hierarchical

linear

models

more

practical.

The

proposed

new

technique

can

be

easily

adapted

to

run

on

a

cluster

of

machines,

and

thus

further

speed

up

the

learning

process

to

handle

a

larger

scale

system

with

hundreds

of

millions

of

users.

The

research

has

much

potential

to

benefit

people

using

EM

algorithm

on

many

other

IR

problems

as

well

as

machine

learning

problems.

EM

algorithm

is

a

commonly

used

machine

learning

technique.

It

is

used

to

find

model

parameters

in

many

IR

problems

where

the

training

data

is

very

sparse.

Although

we

are

focusing

on

the

Bayesian

hierarchical

linear

models

for

recommendation

and

filtering,

the

new

idea

of

using

analytical

solution

instead

of

numerical

solution

for

unrelated

user-feature

pairs

at

the

M

step

could

be

adapted

to

many

other

problems.

ACKNOWLEDGMENTS

We

thank

Wei

Xu,

David

Lewis

and

anonymous

reviewers

for

valuable

feedback

on

the

work

described

in

this

paper.

Part

of

the

work

was

supported

by

Yahoo,

Google,

the

Petascale

Data

Storage

Institute

and

the

Institute

for

Scalable

Scientific

Data

Management.

Any

opinions,

findings,

conclusions,

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expressed

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material

are

those

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reflect

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of

the

sponsors.

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O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O B O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O B I O O O O O O B O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I B O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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Sensor

Deployment

Strategy

for

Target

Detection

Thomas

Clouqueur,

Veradej

Phipatanasuphorn,

Parameswaran

Ramanathan

and

Kewal

K.

Saluja

University

of

Wisconsin-Madison

Engineering

Drive

Madison,

WI

clouqueu@ece.wisc.edu

ABSTRACT

In

order

to

monitor

a

region

for

traffic

traversal,

sensors

can

be

deployed

to

perform

collaborative

target

detection.

Such

a

sensor

network

achieves

a

certain

level

of

detection

performance

with

an

associated

cost

of

deployment.

This

paper

addresses

this

problem

by

proposing

path

exposure

as

a

measure

of

the

goodness

of

a

deployment

and

presents

an

approach

for

sequential

deployment

in

steps.

It

illustrates

that

the

cost

of

deployment

can

be

minimized

to

achieve

the

desired

detection

performance

by

appropriately

choosing

the

number

of

sensors

deployed

in

each

step.

Categories

and

Subject

Descriptors

[Computer-Communication

Networks]:

Distributed

Systems-distributed

applications;

[Special-purpose

and

Application-based

Systems]:

[signal

processing

systems]

General

Terms

Algorithms,

Design,

Performance

INTRODUCTION

Recent

advances

in

computing

hardware

and

software

are

responsible

for

the

emergence

of

sensor

networks

capable

of

observing

the

environment,

processing

the

data

and

making

decisions

based

on

the

observations.

Such

a

network

can

be

used

to

monitor

the

environment,

detect,

classify

and

locate

specific

events,

and

track

targets

over

a

specific

region.

Examples

of

such

systems

are

in

surveillance,

monitoring

of

pollution,

traffic,

agriculture

or

civil

infrastructures

The

deployment

of

sensor

networks

varies

with

the

application

considered.

It

can

be

predetermined

when

the

environment

is

sufficiently

known

and

under

control,

in

which

case

the

sensors

can

be

strategically

hand

placed.

The

deployment

can

also

be

a

priori

undetermined

when

the

environment

is

unknown

or

hostile

in

which

case

the

sensors

may

be

airdropped

from

an

aircraft

or

deployed

by

other

means,

generally

resulting

in

a

random

placement.

This

paper

investigates

deployment

strategies

for

sensor

networks

performing

target

detection

over

a

region

of

interest.

In

order

to

detect

a

target

moving

through

the

region,

sensors

have

to

make

local

observations

of

the

environment

and

collaborate

to

produce

a

global

decision

that

reflects

the

status

of

the

region

covered

This

collaboration

requires

local

processing

of

the

observations,

communication

between

different

nodes,

and

information

fusion

Since

the

local

observations

made

by

the

sensors

depend

on

their

position,

the

performance

of

the

detection

algorithm

is

a

function

of

the

deployment.

One

possible

measure

of

the

goodness

of

deployment

for

target

detection

is

called

path

exposure.

It

is

a

measure

of

the

likelihood

of

detecting

a

target

traversing

the

region

using

a

given

path.

The

higher

the

path

exposure,

the

better

the

deployment.

The

set

of

paths

to

be

considered

may

be

constrained

by

the

environment.

For

example,

if

the

target

is

expected

to

be

following

a

road,

only

the

paths

consisting

of

the

roads

need

to

be

considered.

In

this

study,

the

deployment

is

assumed

to

be

random

which

corresponds

to

many

practical

applications

where

the

region

to

be

monitored

is

not

accessible

for

precise

placement

of

sensors.

The

focus

of

this

paper

is

to

determine

the

number

of

sensors

to

be

deployed

to

carry

out

target

detection

in

a

region

of

interest.

The

tradeoffs

lie

between

the

network

performance,

the

cost

of

the

sensors

deployed,

and

the

cost

of

deploying

the

sensors.

This

paper

is

organized

as

follows.

In

section

a

definition

for

path

exposure

is

proposed

and

a

method

to

evaluate

the

exposure

of

a

given

path

is

developed.

In

section

the

problem

of

random

deployment

is

formulated

and

several

solutions

are

presented.

An

analytical

study

of

these

solutions

is

given

in

section

and

section

presents

simulation

results

that

are

used

in

section

to

determine

the

optimum

solution

for

a

given

environment.

The

paper

concludes

with

section

PATH

EXPOSURE

In

this

section,

a

model

for

sensor

network

target

detection

is

proposed,

a

definition

of

path

exposure

is

presented

and

expressions

for

evaluating

this

path

exposure

are

developed.

Model

Consider

a

rectangular

sensor

field

with

n

sensors

ployed

at

locations

si,

i

n.

A

target

at

location

u

emits

a

signal

which

is

measured

by

the

sensors.

The

signal

from

the

target

decays

as

a

polynomial

of

the

distance.

If

the

decay

coefficient

is

k,

the

signal

energy

of

a

target

at

location

u

measured

by

the

sensor

at

si

is

given

by

Si(u)

K

||u

si||k

where

K

is

the

energy

emitted

by

the

target

and

||u

si||

is

the

geometric

distance

between

the

target

and

the

sensor.

Depending

on

the

environment

the

value

k

typically

ranges

from

to

Energy

measurements

at

a

sensor

are

usually

corrupted

by

noise.

If

Ni

denotes

the

noise

energy

at

sensor

i

during

a

particular

measurement,

then

the

total

energy

measured

at

sensor

i,

when

the

target

is

at

location

u,

is

Ei(u)

Si(u)

Ni

K

||u

si||k

Ni.

The

sensors

collaborate

to

arrive

at

a

consensus

decision

as

to

whether

a

target

is

present

in

the

region.

We

consider

two

basic

approaches

for

reaching

this

consensus:

Value

fusion

and

Decision

fusion

In

value

fusion,

one

of

the

sensors

gathers

the

energy

measurements

from

the

other

sensors,

totals

up

the

energy

and

compares

the

sum

to

a

threshold

to

decide

whether

a

target

is

present.

If

the

sum

exceeds

the

threshold,

then

the

consensus

decision

is

that

a

target

is

present.

In

contrast,

in

decision

fusion,

each

individual

sensor

compares

its

energy

measurement

to

a

threshold

to

arrive

at

a

local

decision

as

to

whether

a

target

is

present.

The

local

decisions

for

target

present

and

otherwise)

from

the

sensors

are

totaled

at

a

sensor

and

the

sum

is

compared

to

another

threshold

to

arrive

at

the

consensus

decision.

In

some

situations,

value

fusion

outperforms

decision

fusion

and

vice

versa.

Value

Fusion.

The

probability

of

consensus

target

detection

when

the

target

is

at

location

u

is

Dv(u)

Prob

n

K

||u

si||k

Ni

η

Prob

n

Ni

η

n

K

||u

si||k

where

η

is

the

value

fusion

threshold.

If

the

noise

processes

at

the

sensors

are

independent,

then

the

probability

density

function

of

n

Ni

equals

the

convolution

of

the

probability

density

function

of

Ni,

i

n.

In

particular,

if

the

noise

process

at

each

sensor

is

Additive

White

Gaussian

Noise

(AWGN),

then

n

Ni

has

a

Chi-square

distribution

of

degree

n.

Due

to

the

presence

of

noise,

the

sensors

may

incorrectly

decide

that

a

target

is

present

even

though

there

is

no

target

in

the

field.

The

probability

of

a

consensus

false

target

detection

is

Fv

Prob

n

Ni

η

As

above,

if

the

noise

processes

at

the

sensors

are

independent

and

AWGN,

then

the

false

alarm

probability

can

be

computed

from

the

Chi-square

distribution

of

degree

n.

Decision

Fusion.

For

decision

fusion,

the

probability

of

consensus

target

detection

when

the

target

is

located

at

u

is

Dd(u)

Prob

n

hd,i(u)

η2

n

n

j

j

(n−j)

where

Prob

[hd,i(u)

Prob

Ni

η1

K

||u

si||k

and

Prob

[hd,i(u)

Prob

[hd,i(u)

can

be

computed

from

Chi-square

distribution

of

degree

for

AWGN

noise

process.

The

probability

of

false

target

detection

at

sensor

i

is

Prob[gd,i

Prob[Ni

and

Prob[gd,i

Prob[gd,i

Therefore,

the

probability

of

consensus

false

target

detection

is

Fd

Prob

n

gd,i

η2

n

n

j

(Prob

[gd,i

(Prob

[gd,i

The

above

equations

serve

as

an

analytic

basis

for

evaluating

exposure

as

defined

in

the

following

subsection.

Note

that

in

value

and

decision

fusion

the

knowledge

of

the

sensors

location

can

be

used

to

make

the

detection

decision.

For

example,

a

sensor

can

report

values

that

differ

substantially

from

its

neighbors

values.

This

discrepancy

can

be

analyzed

to

avoid

false

alarms

or

misses

and

therefore

improve

the

detection

performance.

However,

such

algorithms

are

not

considered

in

this

paper.

De£nition

of

exposure

We

define

exposure

to

be

the

probability

of

detecting

the

target

or

an

intruder

carrying

out

the

unauthorized

activity,

where

the

activity

depends

on

the

problem

under

consideration.

In

this

paper,

the

activity

considered

is

the

Unauthorized

Traversal

(UT)

as

defined

below.

Unauthorized

Traversal

(UT)

Problem:

We

are

given

a

sensor

field

with

n

sensors

at

locations

sn

(see

Figure

We

are

also

given

the

stochastic

characterization

of

the

noise

at

each

sensor

and

a

tolerable

bound,

α,

on

the

false

alarm

probability.

Let

P

denote

a

path

from

the

west

to

the

east

periphery

of

the

sensor

field.

A

target

traversing

the

sensor

field

using

path

P

is

detected

if

it

is

detected

at

some

point

u

P.

The

exposure

of

path

P

is

the

net

probability

of

detecting

a

target

that

traverses

the

field

using

P.

The

target

is

assumed

to

be

able

to

follow

any

path

through

the

field

and

the

problem

is

to

find

the

path

P

with

the

least

exposure.

Sensor

Figure

Example

sensor

fields

for

UT

problem.

Solution

to

the

UT

problem

Let

P

denote

a

path

from

the

west

to

the

east

periphery

through

the

sensor

field.

A

target

that

traverses

the

field

using

P

is

not

detected

if

and

only

if

it

is

not

detected

at

any

time

while

it

is

on

that

path.

Since

detection

attempts

by

the

sensor

network

occur

at

a

fixed

frequency,

we

can

associate

each

detection

attempt

with

a

point

u

P

when

assuming

that

the

target

traverses

the

field

at

a

constant

speed.

The

detection

attempts

are

based

on

energy

measured

over

a

period

of

time

T

during

which

the

target

is

moving.

Therefore,

the

detection

probability

associated

with

each

point

u

reflects

the

measurements

performed

during

time

T.

Considering

the

path,

the

net

probability

of

not

detecting

a

target

traversing

the

field

using

P

is

the

product

of

the

probabilities

of

no

detection

at

each

point

u

P.

That

is,

if

G(P)

denotes

the

net

probability

of

not

detecting

a

target

as

it

traverses

over

path

P,

then,

log

G(P)

u∈P

D(u))du,

where

D(u)

is

either

Dv(u)

of

Dd(u)

depending

on

whether

the

sensors

use

value

or

decision

fusion

to

arrive

at

a

consensus

decision.

Since

the

exposure

of

P

is

G(P)),

the

problem

is

to

find

the

path

which

minimizes

G(P))

or

equivalently

the

path

that

minimizes

log

In

general,

the

path

P

that

minimizes

log

G(P)|

can

be

fairly

arbitrary

in

shape.

The

proposed

solution

does

not

exactly

compute

this

path.

Instead,

we

rely

on

the

following

approximation.

We

first

divide

the

sensor

field

into

a

fine

grid

and

then

assume

that

the

target

only

moves

along

this

grid.

The

problem

then

is

to

find

the

path

P

on

this

grid

that

minimizes

log

G(P)|.

Note

that,

the

finer

the

grid

the

closer

the

approximation.

Also,

one

can

use

higher

order

grids

such

as

in

instead

of

the

rectangular

grid

we

use

in

this

paper.

The

higher

order

grids

change

the

runtime

of

the

algorithm

but

the

approach

is

the

same

as

with

the

rectangular

grid.

For

the

target

not

to

be

detected

at

any

point

u

P,

Note

that,

G(P)

lies

between

and

and

thus

log

G(P)

is

negative.

Generate

a

suitably

fine

rectangular

grid.

For

each

line

segment

l

between

adjacent

grid

points

Compute

log

ml|

using

Equation

Assign

l

a

weight

equal

to

log

ml|

Endfor

Add

a

link

from

virtual

point

a

to

each

grid

point

on

the

west

Add

a

link

from

virtual

point

b

to

each

grid

point

on

the

east

Assign

a

weight

of

to

all

the

line

segments

from

a

and

b

Compute

the

least

weight

path

P

from

a

to

b

using

Dijkstra"s

algorithm

Let

w

equal

the

total

weight

of

P.

Return

P

as

the

least

exposure

path

with

an

exposure

equal

to

Figure

Pseudo-code

of

the

proposed

solution

for

the

UT

problem.

it

need

not

be

detected

at

any

point

u

lying

between

any

two

adjacent

grid

points

of

P.

We

therefore

subdivide

any

path

P

as

a

chain

of

grid

segments.

Let

us

consider

two

adjacent

points,

say

and

on

the

grid.

Let

l

denote

the

line

segment

between

and

Also,

let

ml

denote

the

probability

of

not

detecting

a

target

traveling

between

and

on

the

line

segment

l.

Then,

from

the

discussion

above,

log

ml

u∈l

D(u))du

The

probability

ml

can

be

evaluated

by

finding

the

detection

probability

D(u)

at

each

point

u

l.

Note

that,

ml

lies

between

and

and,

therefore,

log

ml

is

negative.

To

find

the

least

exposed

path,

a

non-negative

weight

equal

to

log

ml|

is

assigned

to

each

segment

l

on

this

grid.

Also,

a

fictitious

point

a

is

created

and

a

line

segment

is

added

from

a

to

each

grid

point

on

the

west

periphery

of

the

sensor

field.

A

weight

equal

to

is

assigned

to

each

of

these

line

segments.

Similarly,

a

fictitious

point

b

is

created

and

a

line

segment

is

added

from

b

to

each

grid

point

on

the

east

periphery

of

the

sensor

field.

A

weight

equal

to

is

assigned

to

each

of

these

line

segments.

The

problem

of

finding

the

least

exposed

path

from

west

periphery

to

east

periphery

is

then

equivalent

to

the

problem

of

finding

the

least

weight

path

from

a

to

b

on

this

grid.

Such

a

path

can

be

efficiently

determined

using

the

Dijkstra"s

shortest

path

algorithm

A

pseudo-code

of

the

overall

algorithm

is

shown

in

Figure

Example:

Figure

shows

a

sensor

field

with

eight

sensors

at

locations

marked

by

dark

circles.

Assume

the

noise

process

at

each

sensor

is

Additive

White

Gaussian

with

mean

and

variance

Further

assume

that

the

sensors

use

value

fusion

to

arrive

at

a

consensus

decision.

Then,

from

Equation

we

chose

a

threshold

η

to

achieve

a

false

alarm

probability

of

The

field

has

been

divided

into

a

grid.

The

target

emits

an

energy

K

and

the

energy

decay

factor

is

The

figure

shows

the

weight

assigned

to

each

line

segment

in

the

grid

as

described

above.

The

least

exposure

path

found

by

the

Dijkstra"s

algorithm

for

this

weighted

grid

is

highlighted.

The

probability

of

Fictitious

Fictitious

Threshold

Detection

Probability

of

the

Path

Point

A

Point

0.030.030.030.050.060.070.040.020.100.03

0.000.010.010.220.420.440.240.020.000.00

0.010.010.010.050.480.490.060.010.010.01

0.850.390.190.120.361.361.210.270.610.53

Sensor

Edge

Weight

Figure

Illustration

of

the

proposed

solution

for

an

example

UT

problem.

tecting

the

target

traversing

the

field

using

the

highlighted

path

is

DEPLOYMENT

In

this

section,

the

problem

of

sensor

deployment

for

unauthorized

traversal

detection

is

formulated

and

solutions

are

identified.

Problem

formulation

Consider

a

region

to

be

monitored

for

unauthorized

traversal

using

a

sensor

network.

The

target

to

traverse

the

sensor

field

emits

a

given

energy

level

K

and

the

stochastic

of

the

noise

in

the

region

is

known.

The

sensors

are

to

be

deployed

over

the

region

in

a

random

fashion

where

the

sensors

locations

in

the

region

are

a

priori

undetermined

and

only

the

number

or

density

of

sensors

can

be

chosen.

The

problem

is

to

find

a

deployment

strategy

that

results

in

a

desired

performance

level

in

unauthorized

traversal

monitoring

of

the

region.

This

performance

is

measured

by

the

false

alarm

probability

and

the

path

exposure

defined

in

section

The

false

alarm

probability

does

not

depend

on

the

sensor

placement

and

is

only

determined

by

the

number

of

sensors

deployed

and

the

thresholds

used

in

the

detection

algorithms.

It

is

assumed

to

be

fixed

in

this

study

so

that

the

problem

consists

of

maximizing

the

exposure

at

constant

false

alarm

rate.

Since

targets

can

traverse

the

region

through

any

path,

the

goal

of

deployment

is

to

maximize

the

exposure

of

the

least

exposed

path

in

the

region.

Obviously,

the

minimum

exposure

in

the

region

increases

(if

false

alarm

rate

is

kept

constant)

as

more

sensors

are

deployed

in

the

region.

However,

since

the

deployment

is

random,

there

are

no

guarantees

that

the

desired

exposure

level

is

achieved

for

a

given

number

of

sensors.

Indeed

some

sensor

placements

can

result

in

very

poor

detection

ability,

in

particular

when

the

sensors

are

all

deployed

in

the

same

vicinity.

A

study

of

the

statistical

distribution

of

exposure

for

varying

sensor

placement

for

a

given

number

of

sensors

can

provide

a

confidence

level

that

the

desired

detection

level

is

achieved.

In

practical

situations,

only

a

limited

number

of

sensors

are

available

for

deployment

and

only

a

limited

detection

level

with

associated

confidence

level

is

achievable

at

a

given

false

alarm

rate.

Solution

Based

on

the

above

discussion,

we

develop

a

solution

method

to

the

deployment

problem

when

a

maximum

of

M

sensors

can

be

used.

Deploying

the

M

sensors

results

in

the

maximum

achievable

detection

level

but

is

not

optimal

when

considering

the

cost

of

sensors.

To

reduce

the

number

of

sensors

deployed,

only

part

of

the

available

sensors

can

be

deployed

first

and

the

sensors

can

then

report

their

position.

The

random

sensor

placement

obtained

can

be

analyzed

to

determine

if

it

satisfies

the

desired

performance

level.

If

it

does

not,

additional

sensors

can

be

deployed

until

the

desired

exposure

level

is

reached

or

until

the

all

M

available

sensors

are

deployed.

The

number

of

sensors

used

in

this

strategy

can

be

minimized

by

deploying

one

sensor

at

a

time.

However,

a

cost

is

usually

associated

with

each

deployment

of

sensors

and

deploying

one

sensor

at

a

time

may

not

be

cost

effective

if

the

cost

of

deployment

is

sufficiently

large

with

respect

to

the

cost

of

single

sensors.

By

assigning

a

cost

to

both

single

sensors

and

deployment,

the

optimal

number

of

sensors

to

be

deployed

at

first

and

thereafter

can

be

determined.

In

the

next

section,

we

develop

analytical

expressions

for

finding

the

optimal

solution.

In

general,

the

optimal

cost

solution

is

neither

deploying

one

sensor

at

a

time

nor

deploying

all

the

sensors

at

a

time.

ANALYTICAL

SOLUTION

In

this

section,

we

derive

an

analytical

model

for

the

cost

of

deployment.

Let

ed

be

the

desired

minimum

exposure

for

the

sensor

network

to

be

deployed

when

a

maximum

of

M

sensors

are

available

for

deployment.

The

position

of

sensors

are

random

in

the

region

of

interest

R

and

for

a

given

Minimum

exposure

Density

Minimum

exposure

Density

Minimum

exposure

Density

sensors

sensors

sensors

Figure

Probability

density

function

for

the

distribution

of

minimum

exposure

for

deployments

of

and

sensors.

ber

of

sensors

n,

the

least

exposure

e

is

a

random

variable.

Let

Fn(x)

denote

the

cumulative

distribution

function

of

e,

that

is

the

probability

that

e

is

less

than

x,

when

n

sensors

are

deployed.

As

mentioned

in

the

previous

section,

there

is

no

a

priori

guarantee

that

the

exposure

ed

will

be

obtained

when

deploying

the

sensors.

If

M

is

the

maximum

number

of

sensors

available,

the

confidence

of

obtaining

a

least

exposure

of

ed

or

more

is

(ed).

For

the

proposed

solution,

we

assume

that

the

position

of

the

sensors

obtained

after

a

deployment

is

known

so

that

additional

sensors

can

be

deployed

if

the

minimum

exposure

ed

is

not

reached.

To

evaluate

the

probability

that

the

exposure

ed

is

reached

after

additional

sensor

deployment,

we

make

the

following

approximation:

the

distribution

of

exposure

for

n

sensors

is

independent

of

the

exposure

corresponding

to

k

of

these

n

sensors,

k

n

This

is

an

approximation

since

the

exposure

obtained

with

n

sensors

is

always

higher

than

the

exposure

obtained

with

only

k

of

these

n

sensors,

k

n

We

observed

that

the

re-deployment

of

just

a

few

sensors

can

substantially

modify

the

coverage

of

the

region,

for

example

when

filling

empty

spaces.

The

approximation

is

also

justified

by

the

loose

relation

between

exposure

and

sensors

positions.

Indeed,

a

given

minimum

exposure

can

correspond

to

many

different

deployment

positions,

some

of

which

can

be

easily

improved

by

deploying

a

few

additional

sensors

(e.g.

when

there

is

a

empty

space

in

the

region

coverage),

some

of

which

can

only

be

improved

by

deploying

many

additional

sensors

(e.g.

when

the

sensors

are

evenly

distributed

on

the

region).

As

the

minimum

exposure

e

is

a

random

variable,

the

cost

of

deploying

the

sensors

in

steps

until

the

desired

exposure

is

reached

is

also

a

random

variable

C.

We

now

derive

the

expression

for

the

expected

value

of

C.

Let

ni

be

the

total

number

of

sensors

deployed

after

each

step

i

for

a

maximum

number

of

steps

S

so

that

nS

M.

Note

that

ni

is

the

number

of

sensors

deployed

at

step

i.

Also

let

Cd

be

the

cost

of

deploying

the

sensors

at

each

step

and

Cs

be

the

cost

of

each

sensor.

If

the

desired

exposure

is

obtained

after

the

first

step,

the

cost

of

deployment

is

Cd

and

this

Number

of

sensors

Probability

ed

ed

ed

ed

Figure

Probability

that

the

minimum

exposure

is

above

ed

for

varying

number

of

sensors

and

and

event

happens

with

probability

(ed).

Considering

all

the

possible

events,

the

expected

cost

is

given

by

E{C}

(i.Cd

ni.Cs)

Fnj

(ed)

Fni

(ed))

(S.Cd

M.Cs)

Fnj

(ed)

Note

that

a

different

expression

is

needed

for

the

cost

of

step

S

since

no

additional

sensors

are

deployed

after

this

step

even

when

the

desired

exposure

is

not

obtained.

SIMULATION

In

this

section,

we

present

results

of

simulations

that

were

conducted

to

collect

the

exposure

distribution

function

of

the

number

of

sensors

deployed.

Method

The

exposure

distribution

is

obtained

by

collecting

statistics

on

the

exposure

when

deploying

sensors

randomly

in

a

predefined

region.

The

random

deployment

is

assumed

to

be

uniformly

distributed

over

the

region,

which

is

a

local

approximation.

For

every

deployment,

the

minimum

exposure

is

found

using

a

simulator

implementing

the

algorithm

presented

in

section

A

decay

factor

of

k

and

maximum

energy

of

K

are

chosen

to

model

the

energy

emitted

by

targets

(cf

Equation

The

region

monitored

is

of

size

with

a

noise

(AWGN)

of

variance

so

that

the

signal

coming

from

the

target

is

covered

by

noise

when

the

target

is

or

more

units

length

away

from

a

sensor.

The

sensors

use

value

fusion

to

collaborate

when

making

a

common

decision

on

the

presence

of

a

target

in

the

region.

The

threshold

for

detection

is

chosen

as

a

function

of

the

number

of

sensors

to

give

a

constant

false

alarm

probability.

The

false

alarm

probability

for

each

detection

attempt

is

chosen

so

that

the

probability

to

get

one

or

more

false

alarm

along

a

path

of

length

units

(corresponding

to

detection

attempts

by

the

sensors)

is

Cost

for

and

n

Expectedcost

Cost

for

and

n

Expectedcost

Cost

for

and

n

Expectedcost

Figure

Expected

cost

of

achieving

minimum

exposure

of

as

function

of

the

number

of

sensors

for

three

different

cost

assignments.

Distribution

of

minimum

exposure

The

distribution

of

minimum

exposure

were

found

for

the

number

of

sensor

deployed

varying

from

to

To

illustrate

our

results,

the

probability

density

functions

for

and

sensors

are

shown

in

Figure

We

observe

that

for

sensors

deployed,

the

minimum

exposure

has

zero

density

for

values

less

than

the

false

alarm

probability

of

The

highest

density

is

obtained

for

values

around

and

then

drops

exponentially

towards

zero

for

higher

values

of

exposure.

For

deployment

of

sensors,

we

find

again

that

the

minimum

exposure

has

zero

density

for

values

below

then

increases

and

has

about

constant

density

for

values

lying

between

and

We

also

observe

a

peak

of

density

around

For

deployment

of

sensors,

densities

start

at

zero

for

small

values

and

remain

very

small

for

most

values

of

minimum

exposure.

The

density

slowly

increases

and

has

a

large

peak

for

minimum

exposure

of

As

expected,

the

minimum

exposure

increases

on

average

as

the

number

of

sensors

deployed

increases.

When

randomly

deploying

sensors,

it

is

very

unlikely

to

obtain

a

placement

providing

a

desirable

minimum

exposure.

When

deploying

sensors,

most

of

the

exposure

levels

are

equally

likely

and

only

poor

confidence

is

given

to

obtain

a

desirable

exposure

level.

When

deploying

sensors,

it

is

very

likely

that

the

sensor

placement

will

give

good

exposure

and

this

likelihood

keeps

increasing

with

the

number

of

sensors

deployed.

We

use

the

cumulative

distribution

function

obtained

from

the

statistics

collected

to

evaluate

the

likelihood

that

the

desired

level

of

exposure

ed

is

obtained

for

varying

number

of

sensors.

The

graph

of

Figure

shows

the

probability

that

the

minimum

exposure

is

above

ed

as

a

function

of

the

number

of

sensors

deployed

for

ed

and

These

values

can

be

used

to

evaluate

the

cost

expressed

in

Equation

The

graph

shows

that

the

confidence

level

to

obtain

a

given

minimum

exposure

level

ed

increases

with

the

number

of

sensors

deployed.

The

confidence

for

ed

when

deploying

sensors

is

above

which

is

sufficient

for

most

applications,

and

therefore

we

did

not

evaluate

the

distribution

of

exposure

when

deploying

more

than

sensors.

RESULTS

In

this

section,

we

evaluate

the

expected

cost

of

deploying

sensors

using

the

simulation

results.

The

optimal

number

of

sensor

to

deploy

at

first

and

in

the

succeeding

steps

can

be

derived

from

these

results.

For

this

cost

analysis,

the

region

parameters

and

signal

model

are

the

same

as

specified

in

section

We

further

assume

that

the

number

of

sensors

deployed

at

every

step

is

constant

so

that

ni

n

for

all

i

S.

In

this

case,

Equation

reduces

to

E{C}

(Cd

n.Cs)

i.

Fj.n(ed)

Fi.n(ed))

(S.Cd

M.Cs)

Fj.n(ed)

We

evaluated

the

expected

cost

as

a

function

of

n

for

three

different

cost

assignments

with

a

desired

exposure

of

ed

The

three

corresponding

graphs

are

shown

in

Figure

The

first

cost

assignment

is

(Cd

Cs

so

that

the

expected

cost

is

the

expected

number

of

sensors

to

be

used

to

achieve

an

exposure

of

Since

Cd

the

number

of

steps

used

to

deploy

the

sensors

doesn"t

affect

the

cost

and

it

is

therefore

optimal

to

deploy

one

sensor

at

a

time

until

the

minimum

exposure

ed

is

reached,

as

we

observe

on

the

graph.

Overall,

the

expected

number

of

sensor

to

be

deployed

increases

with

n

but

we

observe

a

local

minimum

for

n

that

can

be

explained

by

the

following

analysis.

The

expected

number

of

sensors

is

a

weighted

sum

of

i.n,

i

S

that

are

the

different

number

of

sensors

than

can

be

deployed

at

a

time

when

deploying

n

sensors

at

each

step.

For

n

around

the

probability

that

the

minimum

exposure

is

above

ed

varies

a

lot

as

shown

in

Figure

and

the

weight

associated

with

the

first

term

of

the

sum

(n)

increases

rapidly

while

the

weights

associated

with

higher

number

of

sensors

decrease.

This

is

the

cause

of

the

local

minimum

and

the

cost

starts

to

increase

again

when

the

increase

in

n

compensates

for

the

decrease

in

weights.

The

second

cost

assignment

is

(Cd

Cs

so

that

the

cost

of

a

deployment

is

equal

to

the

cost

of

five

sensors

(note

that

only

the

relative

cost

of

Cd/Cs

determines

the

shape

of

the

graphs).

In

this

case,

deploying

one

sensor

at

a

time

is

prohibited

by

the

cost

of

deployment

and

the

optimal

number

of

sensors

to

deploy

at

every

step

is

Again,

we

find

that

the

curve

presents

a

local

minimum

for

n

that

is

due

to

the

variations

in

weights.

The

last

cost

assignment

is

(Cd

Cs

and

the

minimum

cost

is

achieved

when

deploying

sensors

at

every

step.

These

results

are

specific

to

the

region

and

the

parameters

characterizing

the

signal

emitted

by

the

target

that

were

chosen

for

the

simulation.

Similar

results

can

be

derived

for

other

parameters,

most

of

the

effort

residing

in

finding

the

exposure

distributions

through

simulation.

CONCLUSION

This

paper

addresses

the

problem

of

sensor

deployment

in

a

region

to

be

monitored

for

target

intrusion.

A

mechanism

for

sensor

collaboration

to

perform

target

detection

is

proposed

and

analyzed

to

evaluate

the

exposure

of

paths

through

the

region.

The

minimum

exposure

is

used

as

a

measure

of

the

goodness

of

deployment,

the

goal

being

to

maximize

the

exposure

of

the

least

exposed

path

in

the

region.

In

the

case

where

sensors

are

randomly

placed

in

a

region

to

be

monitored,

a

mechanism

for

sequential

deployment

in

steps

is

developed.

The

strategy

consists

of

deploying

a

limited

number

of

sensors

at

a

time

until

the

desired

minimum

exposure

is

achieved.

The

cost

function

used

in

this

study

depends

on

the

number

of

sensors

deployed

in

each

step

and

the

cost

of

each

deployment.

Through

simulation,

the

distribution

of

minimum

exposure

obtained

by

random

deployment

was

evaluated

for

varying

number

of

sensors

deployed.

These

results

were

used

to

evaluate

the

cost

of

deployment

for

varying

number

of

sensors

deployed

in

each

step.

We

found

that

the

optimal

number

of

sensors

deployed

in

each

step

varies

with

the

relative

cost

assigned

to

deployment

and

sensors.

The

results

of

this

study

can

be

extended

to

larger

regions

with

different

target

parameters.

The

solution

proposed

in

this

paper

can

also

be

improved

by

considering

deploying

variable

number

of

sensors

at

each

step

and

this

multiple

variables

problem

requires

further

investigation.

ACKNOWLEDGMENTS

This

work

was

supported

in

part

by

the

Defense

Advanced

Research

Projects

Agency

(DARPA)

and

the

Air

Force

research

Laboratory,

Air

Force

Material

Command,

USAF,

under

agreement

number

The

U.S.

Government

is

authorized

to

reproduce

and

distribute

reprints

for

Governmental

purposes

notwithstanding

any

copyright

annotation

thereon.

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O B O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O B I O O O O O O O O O O O O B O O O O O O O B I O O O O O O O O B O O O O O B I O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O B O B I O O B I O O O O O O O O O O O O O O O O O O O O O B I O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O B O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O B I B I O O O O O B I O O O O O O O B I O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O B I O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O B I O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O B O O O O B O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O B I O O B I O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O B I O O O O O O O O O O B O O O O O O B O O O O O O O O B O O O O O O O B O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O B I O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O B O O O O O O O O B O O O O O O O O O O B O O O O O O O O O O O O O O O O B O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O B I O O O B I O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O B O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O B I O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O B O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O B O O O O O O O O O O O O O O O O B I O O O O O O O B O O O O O O B I O O O O O O O O O B O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O B I O O O B O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O

sensor number
target decay
collaborative target detection

target detection
sensor network
path exposure
number of sensor
sequential deployment
minimum exposure
random sensor placement
sensor field
deployment
exposure
value fusion