tFPR

A

Fuzzy

and

Structural

pattern

recognition

System

of

Multi-Variate

Time-Dependent

Pattern

Classes

based

on

Sigmoidal

Functions

John

A.

Drakopoulos

and

Barbara

Hayes-Roth

Stanford

University

Department

of

Computer

Science

Knowledge

Systems

Laboratory

Welch

Road

Palo

Alto

CA

Abstract

tFPR

is

a

hybrid

fuzzy

and

structural

pattern

recognition

system

that

uses

fuzzy

sets

to

represent

multi-variate

pattern

classes

that

can

be

either

static

or

dynamic

depending

on

time

or

some

other

parameter

space

The

membership

functions

of

the

fuzzy

sets

that

represent

pattern

classes

are

modeled

by

sigmoidal

functions

The

choice

of

sigmoidal

functions

was

motivated

by

their

ability

to

represent

efficiently

and

concisely

different

multi-variate

pattern

classes

Given

a

set

of

input

data

and

a

pattern

class

specification

which

is

defined

as

a

fuzzy

set

tFPR

estimates

how

closely

the

data

match

the

pattern

class

specification

using

the

grade

of

membership

of

the

data

in

the

fuzzy

set

The

input

data

may

be

a

number

of

time-dependent

signals

whose

past

values

may

influence

the

evaluation

of

the

pattern

class

In

that

case

structural

pattern

recognition

methods

in

addition

to

fuzzy

ones

are

employed

in

order

to

match

multi-dimensional

curves

-LRB-

rather

than

points

-RRB-

in

the

input

domain

Two

different

structural

-LRB-

segmentation

-RRB-

methods

are

presented

The

first

is

an

efficient

greedy

algorithm

that

is

linear

to

the

number

of

input

data

points

while

the

second

is

an

algorithm

that

is

optimal

under

some

conditions

but

is

slower

than

the

first

method

Although

efficiency

is

a

very

important

consideration

in

tFPR

the

main

issues

are

knowledge

acquisition

and

knowledge

representation

-LRB-

in

terms

of

pattern

classes

-RRB-

tFPR

has

been

implemented

in

the

blackboard

architecture

It

is

currently

being

applied

in

a

system

for

medical

monitoring

Potential

applications

in

other

domains

are

under

investigation

Introduction

pattern

recognition

is

defined

in

-LSB-

-RSB-

as

the

many-to-one

mapping

from

a

very

large

set

of

possible

pattern

instances

to

a

much

smaller

set

of

pattern

names

It

is

a

process

of

information

reduction

that

results

in

a

concise

description

of

a

state

or

situation

that

represents

a

large

number

of

possible

data

values

which

are

often

only

partially

available

That

description

can

be

useful

in

many

ways

It

may

signal

the

satisfaction

of

some

conditions

or

the

existence

of

some

features

in

a

state

of

a

system

that

require

a

change

in

the

system

's

behavior

As

a

result

pattern

recognition

is

an

essential

ingredient

of

tasks

such

as

perception

decision

making

reaction

design

and

planning

Currently

there

are

so

many

pattern

recognition

approaches

and

methods

that

their

complete

taxonomy

is

a

difficult

and

arduous

task

In

most

of

those

approaches

the

input

data

can

be

described

or

be

transformed

into

a

number

n

of

computable

properties

-LRB-

often

called

features

-RRB-

and

as

such

can

be

represented

as

a

point

in

an

n-dimensional

space

-LRB-

which

is

often

called

feature

space

-RRB-

Subsequently

pattern

classes

correspond

to

sets

of

input

data

that

is

sets

of

point

in

feature

space

which

are

usually

contiguous

regions

and

as

such

can

be

described

by

their

boundaries

Those

boundaries

are

usually

called

decision

boundaries

As

a

result

pattern

recognition

is

in

essence

the

computation

and

consice

representation

of

the

decision

boundaries

of

a

number

of

pattern

classes

so

that

it

would

be

efficient

to

compute

whether

an

arbitrary

point

in

feature

space

lies

within

the

decision

boundaries

or

not

Following

the

above

interpretation

a

classification

of

pattern

recognition

methods

and

techniques

appears

in

-LSB-

-RSB-

based

on

the

way

decision

regions

are

formed

-LRB-

e.g.

probabilistic

receptive

fields

exemplars

etc.

-RRB-

and

the

form

of

the

decision

boundaries

-LRB-

i.e.

hyperplanes

hyper-spheres

quadratic

surfaces

etc.

-RRB-

On

the

other

hand

a

number

of

other

properties

of

classifiers

which

could

be

taken

into

account

in

classifying

pattern

recognition

methods

are

mentioned

in

-LSB-

-RSB-

For

example

a

classifier

can

be

statistical

or

structural

parametric

or

non-parametric

and

supervised

or

unsupervised

Finally

a

more

appealing

taxonomy

of

pattern

recognition

methods

appears

in

-LSB-

-RSB-

where

classifiers

are

divided

into

categories

based

on

their

computational

nature

That

taxonomy

consists

of

three

different

classes

of

pattern

recognition

methods

statistical

structural

and

neural

pattern

recognition

methods

Here

we

describe

a

taxonomy

that

is

primarily

based

on

the

form

of

the

measures

used

internally

by

the

classifier

and

the

nature

of

their

elementary

computations

more

or

less

along

the

lines

of

the

taxonomy

presented

in

-LSB-

-RSB-

Subsequently

within

each

category

classifiers

can

be

partitioned

into

classes

based

on

other

properties

such

as

the

form

of

their

decision

regions

their

parametric

or

non-parametric

formalization

and

their

supervised

or

unsupervised

learning

characteristics

Our

taxonomy

consists

of

four

categories

Statistical

pattern

recognition

methods

-LRB-

also

known

as

stochastic

decision-theoretic

or

probabilistic

-RRB-

Structural

or

syntactic

pattern

recognition

methods

Fuzzy

or

possibilistic

pattern

recognition

methods

Neural

network

or

connectionist

pattern

recognition

methods

The

first

category

contains

the

most

conventional

classifiers

that

are

based

on

some

statistical

or

decision-theoretic

measures

to

classify

their

input

data

A

good

description

and

introduction

to

most

of

those

techniques

appear

in

early

classic

textbooks

on

pattern

recognition

-LSB-

-RSB-

More

recent

textbooks

that

cover

statistical

pattern

recognition

methods

are

-LSB-

-RSB-

Structural

pattern

recognition

methods

differ

from

all

other

pattern

class

recognition

methods

for

they

do

not

directly

form

decision

regions

in

some

feature

space

Instead

they

try

to

decompose

their

input

data

into

a

number

of

smaller

pieces

that

match

some

primitive

pattern

class

elements

or

subpattern

classes

Those

subpattern

classes

are

then

combined

hierarchically

to

form

other

higher

level

subpattern

classes

of

the

overall

pattern

class

Most

of

structural

pattern

recognition

methods

are

presented

in

-LSB-

-RSB-

Despite

the

conceptual

differences

of

statistical

and

structural

pattern

recognition

methods

a

unifying

view

is

given

by

Fu

in

-LSB-

-RSB-

Fuzzy

or

possibilistic

pattern

recognition

methods

are

based

on

possibilistic

measures

to

define

pattern

classes

Fuzzy

sets

which

were

invented

in

-LSB-

-RSB-

are

such

measures

Fuzzy

sets

naturally

fit

pattern

recognition

problems

In

the

most

general

case

those

problems

involve

the

computation

of

the

degree

to

which

a

data

object

x

matches

a

pattern

class

description

p

In

the

fuzzy

set

interpretation

of

those

problems

p

is

usually

represented

by

a

fuzzy

set

Ap

which

in

general

models

a

cluster

class

or

pattern

object

The

grade

of

membership

of

x

in

Ap

represents

the

degree

to

which

x

matches

pattern

class

p

An

original

study

of

fuzzy

sets

in

the

context

of

pattern

recognition

problems

was

made

in

-LSB-

-RSB-

An

early

and

inspiring

application

of

fuzzy

sets

to

pattern

recognition

problems

and

clustering

in

particular

was

developed

in

-LSB-

-RSB-

A

subsequent

and

very

influential

method

was

the

fuzzy

ISODATA

-LRB-

or

fuzzy

c-means

-RRB-

algorithm

-LSB-

-RSB-

Since

then

a

number

of

other

methods

and

techniques

have

been

developed

and

used

in

applications

A

good

description

of

most

of

those

techniques

appears

in

-LSB-

-RSB-

-LSB-

-RSB-

-LSB-

ch

-RSB-

More

recent

collections

of

fuzzy

pattern

recognition

methods

appear

in

-LSB-

ch

-RSB-

and

-LSB-

-RSB-

Although

the

first

and

very

influential

works

on

neural

network

research

appeared

about

fifty

years

ago

-LSB-

-RSB-

and

some

early

and

very

important

developments

took

place

in

late

and

early

-LSB-

-RSB-

the

field

of

neural

networks

as

a

computational

paradigm

has

grown

rapidly

only

in

the

last

few

years

-LRB-

see

-LSB-

chapter

-RSB-

for

an

overview

of

the

history

of

neurocomputing

-RRB-

Neural

network

pattern

recognition

methods

which

are

based

on

neural

networks

to

define

and

describe

pattern

classes

in

feature

space

have

received

a

proportional

amount

of

interest

in

the

last

few

years

The

measures

used

here

are

distance

or

classification

error

metrics

sometimes

coupled

with

measures

of

generalization

or

network

size

-LRB-

such

as

Rissanen

's

minimum

descriptive

length

-RRB-

-LSB-

-RSB-

The

basic

computational

element

is

a

unit

or

node

that

performs

a

simple

function

and

communicates

with

other

nodes

through

connections

of

variable

strength

-LRB-

or

weights

-RRB-

Knowledge

is

usually

stored

in

the

weights

while

the

overall

computation

emerges

through

the

interactions

of

the

units

Neural

network

classifiers

may

use

hyper-planes

or

hyperspheres

or

hyper-ellipses

to

form

regions

in

feature

space

Early

neural

network

approaches

were

focused

in

linear

discriminating

functions

and

in

particular

perceptrons

-LRB-

-LSB-

ch

-RSB-

-LSB-

-RSB-

-RRB-

Now

there

is

a

wide

range

of

neural

network

architectures

and

learning

algorithms

that

find

applications

in

many

different

domains

and

disciplines

Most

of

those

algorithms

are

described

in

neural

network

textbooks

-LSB-

-RSB-

A

good

introduction

to

pattern

recognition

techniques

and

algorithms

of

all

of

the

above

categories

except

fuzzy

methods

is

given

in

-LSB-

-RSB-

Of

course

there

are

hybrid

methods

that

could

belong

to

more

than

one

of

those

categories

For

example

concepts

originated

in

fuzzy

set

theory

have

been

incorporated

into

neural

network

methods

and

neural

networks

have

been

used

in

fuzzy

set

implementation

and

fuzzy

reasoning

-LSB-

-RSB-

Fur

thermore

fuzzy

neural

networks

and

hybrid

fuzzy

and

neural

classification

systems

have

been

developed

Those

include

the

nested

generalized

exemplar

-LRB-

NGE

-RRB-

algorithm

-LSB-

-RSB-

the

fuzzy

min-max

classifier

-LRB-

FMMC

-RRB-

-LSB-

-RSB-

the

fuzzy

ARTMAP

neural

network

architecture

-LSB-

-RSB-

and

fuzzy

multi-layer

perceptrons

and

neurons

-LSB-

-RSB-

Furthermore

hybrid

probabilistic

and

neural

network

methods

exist

such

as

neural

implementations

of

the

k-nearest

neighbor

classifier

and

perceptron

trees

-LSB-

-RSB-

In

addition

a

pattern

recognition

system

or

architecture

may

use

many

different

pattern

recognition

methods

and

so

be

a

hybrid

system

In

this

paper

we

introduce

a

new

hybrid

fuzzy

and

structural

pattern

recognition

architecture

that

uses

sigmoidal

functions

to

model

membership

functions

of

fuzzy

sets

that

represent

pattern

classes

Our

methods

are

fuzzy

for

they

use

fuzzy

sets

to

represent

pattern

classes

However

they

are

also

structural

for

they

decompose

a

non-static

parameter-dependent

pattern

class

into

a

number

of

primitive

pattern

class

elements

or

segments

and

compute

the

evaluation

of

the

overall

pattern

class

by

evaluating

the

segments

and

combining

their

evaluations

into

a

single

evaluation

In

our

approach

a

pattern

class

evaluation

is

an

assessment

of

the

degree

to

which

the

available

input

data

match

a

pattern

class

of

interest

The

input

data

correspond

to

a

set

of

values

of

a

number

of

variables

which

are

called

pattern

measures

A

pattern

class

evaluation

may

use

the

values

of

the

corresponding

pattern

measures

at

some

point

in

time

-LRB-

usually

the

current

time

-RRB-

or

may

use

many

other

values

and

match

those

with

multi-dimensional

curves

or

surfaces

that

represent

the

desired

behavior

of

pattern

measures

through

time

or

some

other

parameter

space

Furthermore

arbitrary

combinations

of

such

multi-dimensional

objects

with

points

or

other

objects

of

different

dimensionality

are

also

allowed

In

our

system

the

above

computations

are

based

on

fuzzy

membership

functions

A

method

for

generating

those

functions

is

thus

necessary

Although

a

wide

range

of

membership

functions

have

been

used

in

the

literature

-LRB-

-LSB-

-RSB-

-LSB-

ch.III

section

-RSB-

-LSB-

-RSB-

-RRB-

we

choose

to

use

a

particular

class

of

functions

derived

from

the

sigmoidal

family

of

functions

-LSB-

-RSB-

Arguments

for

the

suitability

of

sigmoidal

functions

in

modeling

fuzzy

set

membership

have

been

given

in

-LSB-

-RSB-

A

brief

presentation

of

those

arguments

as

well

as

the

basic

model

used

in

tFPR

and

its

class

of

sigmoidal

functions

are

given

in

the

next

section

In

section

an

extension

to

the

basic

model

to

include

time

or

any

parameter-space

dependent

signals

and

pattern

classes

is

presented

Algorithms

that

can

support

this

model

are

presented

in

section

In

section

a

further

extension

of

the

method

to

include

a

wide

range

of

operators

as

well

as

its

particular

implementation

-LRB-

tFPR

-RRB-

in

the

blackboard

architecture

is

presented

In

section

example

applications

are

presented

and

in

section

specifications

of

a

graphical

interface

for

defining

pattern

classes

are

given

Finally

in

section

future

extensions

and

enhancements

of

the

system

are

discussed

The

Underlying

Model

The

basic

building

block

of

our

system

and

essential

ingredient

of

its

methods

is

the

pattern

object

i.e.

an

object

that

describes

a

pattern

class

of

interest

The

difference

between

a

pattern

object

and

its

pattern

class

is

that

the

first

is

a

declarative

implementation

of

the

second

Having

a

collection

of

such

pattern

objects

we

can

build

a

pattern

manager

that

evaluates

a

pattern

object

on

the

current

input

whenever

input

related

to

that

pattern

object

is

presented

to

the

system

Each

such

evaluation

results

in

a

value

in

-LSB-

-RSB-

that

is

the

value

of

a

membership

function

of

a

fuzzy

set

that

is

associated

with

the

current

pattern

object

Thus

we

have

to

define

a

model

for

pattern

objects

and

in

particular

for

the

membership

functions

of

their

corresponding

fuzzy

sets

In

the

most

general

case

a

pattern

object

would

be

associated

with

many

pattern

measures

-LRB-

i.e.

input

variables

or

sequences

-RRB-

that

have

different

and

maybe

non-numeric

ranges

A

series

of

transformations

may

be

required

to

transform

the

pattern

measure

values

into

a

domain

which

would

be

the

domain

of

a

fuzzy

set

Let

as

call

this

the

x-domain

Then

an

application

of

the

membership

function

of

the

fuzzy

set

would

result

in

the

desired

membership

value

in

the

interval

-LSB-

-RSB-

Let

as

call

this

interval

the

y-domain

Now

in

order

to

define

formally

pattern

objects

assume

the

following

p

is

a

pattern

class

Mp

is

the

set

of

pattern

measures

associated

with

p

R

-LSB-

Mp

-RSB-

is

the

Cartesian

product

of

the

ranges

of

the

measures

in

Mp

jMpj

is

the

size

of

the

set

Mp

R

is

the

set

of

real

numbers

Then

a

pattern

object

can

be

defined

as

follows

Definition

A

pattern

object

corresponding

to

a

pattern

class

p

is

a

tuple

-LRB-

Mp

Ap

vp

np

ap

dp

Cp

-RRB-

where

Mp

is

the

set

of

pattern

measures

associated

with

p

ap

dp

are

integers

Ap

is

a

function

from

R

-LSB-

Mp

-RSB-

to

Rap

vp

is

a

function

from

Rap

to

Rdp

np

is

a

function

from

Rdp

to

-LSB-

-RSB-

Cp

is

a

set

of

conditions

Cp

is

a

set

of

conditions

that

determine

the

applicability

of

the

pattern

object

and

can

be

used

to

enforce

or

inhibit

an

evaluation

of

pattern

object

at

some

point

in

time

There

is

not

anything

special

about

the

number

of

mapping

functions

used

in

a

pattern

object

If

one

would

replace

Ap

vp

and

np

by

a

function

fp

that

is

the

composition

of

those

three

functions

he

should

get

a

functionally

equivalent

pattern

class

definition

that

would

involve

only

a

single

mapping

function

Our

dividing

the

mapping

function

process

into

three

parts

was

primarily

motivated

by

the

difference

in

functionality

of

the

tasks

these

three

functions

perform

but

is

not

otherwise

required

Furthermore

putting

the

whole

mapping

into

a

single

function

would

result

in

complicated

and

hard-to-understand

functions

Of

course

the

most

general

solution

would

allow

the

user

to

define

an

arbitrary

sequence

of

functions

to

be

applied

to

the

input

data

and

the

most

restrictive

one

would

allow

only

one

such

function

Our

choice

stands

in

the

middle

The

transformations

Ap

and

vp

are

arbitrary

For

example

when

the

input

is

a

non-stationary

signal

like

a

sound

and

the

pattern

class

is

defined

as

a

number

of

relations

among

some

harmonics

present

in

the

sound

Ap

could

compute

the

spectrogram

of

the

signal

in

a

time

window

of

interest

and

vp

could

compute

the

set

of

values

of

the

harmonics

the

pattern

class

in

question

examines

in

that

time

window

As

we

shall

see

in

that

case

np

could

be

a

combination

of

sigmoidal

functions

that

encode

the

relations

among

the

harmonics

Then

ap

should

be

equal

to

-LRB-

since

the

spectrogram

measures

energy

over

time

and

frequency

-RRB-

and

dp

should

be

the

number

of

harmonics

involved

in

the

pattern

class

In

tFPR

the

specification

of

membership

functions

np

that

map

from

Rdp

to

-LSB-

-RSB-

is

always

based

on

sigmoidal

functions

The

choice

of

sigmoidal

functions

was

motivated

by

the

sigmoidal

bubble

theorem

-LRB-

see

-LSB-

-RSB-

for

more

details

-RRB-

For

purposes

of

completeness

we

present

here

a

modified

version

of

the

theorem

that

makes

direct

references

to

fuzzy

sets

To

this

purpose

let

S

indicate

a

basic

unit

sigmoidal

function

and

AS

indicate

the

affine

sigmoidal

class

of

S

i.e.

AS

fSa

c

a

c

R

a

g

where

Sa

c

-LRB-

x

-RRB-

S

-LRB-

ax

c

-RRB-

Now

observe

that

if

fBi

is

the

membership

function

of

the

fuzzy

set

Bi

and

fBi

AS

-LRB-

i.e.

fBi

Sai

ci

for

some

ai

ci

R

-RRB-

then

fBi

-LSB-

Bj

-LRB-

x

-RRB-

supffBi

-LRB-

x

-RRB-

fBj

-LRB-

x

-RRB-

g

S

-LRB-

supfaix

ci

ajx

cjg

-RRB-

fBi

Bj

-LRB-

x

-RRB-

infffBi

-LRB-

x

-RRB-

fBj

-LRB-

x

-RRB-

g

S

-LRB-

inffaix

ci

ajx

cjg

-RRB-

f

Bi

-LRB-

x

-RRB-

fBi

-LRB-

x

-RRB-

S

-LRB-

aix

ci

-RRB-

Therefore

we

can

always

interchange

sigmoidal

functions

with

sup

inf

and

complementation

operations

As

a

result

if

all

the

constituent

subexpressions

of

a

fuzzy

set

theoretic

expression

are

evaluated

through

sigmoidal

functions

of

a

single

class

only

a

single

sigmoidal

computation

is

required

to

evaluate

the

overall

expression

no

matter

how

complicated

the

expression

is

Furthermore

each

sigmoidal

function

that

belongs

to

a

particular

affine

class

can

now

be

represented

by

a

pair

of

numbers

-LRB-

a

c

-RRB-

This

reduces

the

storage

requirements

of

sigmoidal

functions

to

a

level

comparable

to

that

of

linear

segments

and

makes

them

very

attractive

as

representation

functions

As

reported

in

-LSB-

-RSB-

some

preliminary

experimentation

with

the

tFPR

system

showed

that

sigmoidal

functions

outperformed

linear

segments

and

lower

order

polynomials

Finally

it

has

been

shown

in

-LRB-

-LSB-

-RSB-

-RRB-

that

sigmoidal

functions

are

the

only

functions

that

posses

the

above

properties

Figure

shows

the

functions

Sa

c

for

various

values

of

a

c

and

figure

shows

fuzzy

set

operations

on

sigmoidal

functions

Now

sigmoidal

functions

can

be

used

to

approximate

arbitrary

membership

functions

As

an

example

of

membership

function

approximation

consider

the

situation

depicted

in

figure

where

the

membership

function

y

-LRB-

x

-RRB-

is

defined

in

the

following

way

y

-LRB-

x

-RRB-

if

x

if

x

if

x

if

x

if

x

a

c

a

c

Figure

Sigmoidal

functions

for

various

values

of

a

c.

-LRB-

x

-RRB-

-LRB-

x

-RRB-

-LRB-

x

-RRB-

-LRB-

x

-RRB-

-LRB-

x

-RRB-

-LRB-

x

-RRB-

g

-LRB-

x

-RRB-

-LRB-

x

-RRB-

g

Figure

Fuzzy

set

operations

on

sigmoidal

functions

Now

let

S

be

the

sigmoidal

function

that

passes

through

the

points

-LRB-

ffl

-RRB-

and

-LRB-

ffl

-RRB-

and

be

the

sigmoidal

that

passes

through

-LRB-

ffl

-RRB-

and

-LRB-

ffl

-RRB-

where

ffl

is

a

small

positive

constant

Then

the

combination

min

-LRB-

S

-RRB-

is

a

good

approximation

of

y

as

shown

in

figure

Now

let

us

call

set

operations

the

traditional

fuzzy

set

operations

-LRB-

e.g.

fuzzy

set

union

intersection

and

complementation

-RRB-

We

have

shown

that

sigmoidal

functions

combine

very

nicely

with

set

operations

However

it

is

not

possible

to

approximate

any

arbitrary

function

using

only

set

operations

on

sigmoidal

functions

To

solve

this

problem

the

domain

of

the

fuzzy

sets

-LRB-

x-domain

-RRB-

was

divided

into

appropriate

intervals

and

set

operations

were

applied

within

each

interval

to

approximate

a

given

function

The

extra

conditions

that

define

the

set

of

intervals

of

a

pattern

class

are

called

interval

conditions

The

resulting

functions

are

piece-wise

sigmoidal

functions

Using

an

appropriately

large

number

of

intervals

a

sigmoidal

approximation

of

an

arbitrary

function

-LRB-

similar

to

its

polygonal

approximation

-RRB-

can

be

found

The

use

of

interval

conditions

extends

the

representational

spectrum

of

set

operations

to

include

new

functions

Furthermore

since

only

set

operations

are

performed

within

each

interval

only

a

single

sigmoidal

computation

per

pattern

object

is

required

However

the

emphasis

in

tFPR

is

on

knowledge

acquisition

and

knowledge

representation

An

extended

set

of

tFPR

operators

would

conform

to

that

principle

Furthermore

even

in

cases

where

new

tFPR

operators

would

require

more

than

one

sigmoidal

computation

per

pattern

object

effi

Figure

A

pattern

class

and

its

sigmoidal

approximation

ciency

would

not

be

sacrificed

or

compromised

in

any

way

On

the

contrary

sometimes

it

would

be

nearly

intractable

to

approximate

a

given

function

using

only

set

operations

and

interval

conditions

for

the

number

of

intervals

may

grow

enormously

high

In

those

cases

extra

tFPR

operators

-LRB-

such

as

summations

or

weighted

summations

-RRB-

become

not

only

useful

but

also

resourceful

In

short

an

extended

tFPR

operator

set

not

only

would

provide

an

agent

with

higher

flexibility

in

defining

pattern

classes

and

reduce

the

overall

knowledge

acquisition

effort

but

it

also

should

result

in

more

effective

pattern

class

specifications

The

set

of

operators

currently

used

in

tFPR

is

described

in

-LSB-

-RSB-

and

is

presented

in

section

In

the

next

section

we

present

an

extension

of

the

basic

model

presented

above

that

would

deal

with

dynamic

pattern

classes

that

depend

on

time

or

any

other

parameter

space

The

Extended

Model

An

extension

of

the

basic

model

allows

pattern

measures

to

be

arbitrary

functions

instead

of

arbitrary

values

in

a

domain

This

extension

requires

the

system

to

deal

with

multi-dimensional

curves

-LRB-

e.g.

parametric

functions

-RRB-

instead

of

points

-LRB-

e.g.

values

-RRB-

To

this

purpose

pattern

objects

are

defined

over

some

parameter

space

The

extension

to

parameter-space-dependent

pattern

objects

is

very

useful

in

cases

where

as

an

example

the

definition

of

a

pattern

class

requires

access

not

only

to

current

but

also

to

past

values

of

pattern

measures

In

that

case

the

paramenter

space

is

time

and

pattern

measures

are

functions

of

time

In

the

model

below

we

use

a

single

one-dimensional

parameter

space

not

only

because

this

simplifies

our

presentation

but

also

because

the

extension

of

our

model

to

other

multi-dimensional

parameter

spaces

is

straightforward

In

tFPR

curves

and

surfaces

are

broken

into

segments

transforming

the

problem

of

representing

arbitrary

shapes

into

a

problem

of

segmentation

and

segment

specification

Each

segment

consists

of

a

basic

ideal

curve

which

is

called

the

ridge

of

the

segment

and

which

can

be

described

by

a

small

amount

of

information

in

terms

of

conditions

upon

the

literal

values

of

the

curve

and

the

values

of

its

k-order

derivatives

with

respect

to

the

parameter

space

-LRB-

for

k

:-RRB-

A

number

of

sigmoidal

functions

are

used

to

evaluate

the

degree

of

similarity

between

a

portion

of

a

curve

in

the

input

data

and

a

segment

's

ridge

by

matching

the

values

and

the

derivatives

of

the

curve

against

those

of

the

ridge

Those

functions

produce

an

evaluation

-LRB-

y-value

-RRB-

for

each

segment

All

the

evaluations

are

eventually

combined

to

produce

an

overall

evaluation

of

the

whole

curve

This

way

both

description

-LRB-

representation

-RRB-

and

recognition

-LRB-

matching

-RRB-

of

arbitrary

curves

can

be

accomplished

using

exactly

the

same

primitives

and

formalism

In

tFPR

we

use

only

two

primitives

for

describing

segments

and

a

single

one

for

combining

them

Uniformity

in

the

representation

and

evaluation

of

functions

has

not

been

our

primary

motive

in

choosing

to

segment

curves

In

fact

our

choice

was

initially

based

on

a

very

basic

observation

When

people

try

to

describe

a

trajectory

a

curve

or

a

surface

most

of

the

time

they

describe

it

as

a

collection

of

segments

over

a

set

of

regions

in

which

some

very

basic

properties

of

the

curve

or

the

surface

-LRB-

such

as

curvature

rate

of

growth/decline

range

etc.

-RRB-

are

more

or

less

within

some

specifiable

range

Thus

segmentation

seems

to

be

the

natural

approach

to

the

problem

In

tFPR

in

addition

to

segmentation

functions

we

use

some

basic

functions

to

extract

some

features

of

the

actual

values

of

a

segment

-LRB-

such

as

maximal

or

minimal

value

duration

etc.

-RRB-

whenever

the

pattern

class

evaluation

depends

on

those

features

too

The

subsequent

evaluations

of

those

features

are

used

in

the

overall

evaluation

of

the

pattern

class

and

often

help

in

determining

the

segments

themselves

For

example

consider

a

pattern

class

that

requires

its

input

to

stay

at

zero

for

some

time

then

start

increasing

with

a

bounded

range

of

rates

up

to

a

value

then

stay

constant

for

a

while

and

then

fall

back

to

zero

-LRB-

again

with

bounded

rate

-RRB-

and

stay

there

This

pattern

class

could

consist

of

segments

segment

constant

at

-LRB-

with

some

tolerance

-RRB-

segment

going

from

to

with

rate

bounded

by

and

segment

constant

at

-LRB-

with

tolerance

-RRB-

segment

going

from

to

with

rate

bounded

by

and

segment

constant

at

-LRB-

with

tolerance

-RRB-

We

can

also

add

another

kind

of

constraints

that

would

relate

properties

of

different

segments

For

example

we

may

require

the

duration

of

segment

to

be

greater

than

the

duration

of

each

of

segments

and

but

less

than

minutes

In

addition

we

may

require

the

maximum

value

of

the

input

in

segment

to

be

less

than

the

corresponding

maximum

value

in

segment

In

the

next

section

we

present

algorithms

-LRB-

segmentation

algorithms

-RRB-

that

combine

segments

together

in

order

to

match

-LRB-

evaluate

-RRB-

arbitrary

curves

Segmentation

Algorithms

During

the

development

of

tFPR

many

segmentation

algorithms

were

developed

each

having

its

own

characteristics

precision

noise

tolerance

complexity

and

performance

In

this

paper

only

two

of

them

are

presented

First

a

greedy

non-optimal

algorithm

that

is

linear

with

the

data

size

is

given

This

algorithm

is

of

practical

interest

due

to

the

high

requirements

for

speed

in

a

system

that

deals

with

a

large

number

of

pattern

classes

The

second

algorithm

is

polynomial

yet

optimal

when

the

cost

function

is

additive

and

there

are

no

constraints

among

features

of

different

segments

similar

to

those

described

in

the

previous

section

The

above

requirements

as

well

as

the

higher

time

complexity

of

the

second

algorithm

severely

limit

its

practicality

and

applicability

in

any

multi-pattern

class

system

We

attribute

to

it

only

theoretical

importance

for

it

shows

that

the

segmentation

problem

is

polynomial

in

cases

where

the

algorithm

applies

Both

of

these

algorithms

scan

the

input

in

reverse

temporal

order

-LRB-

right

to

left

-RRB-

and

dynamically

collect

new

data

as

needed

to

match

the

pattern

class

The

implicit

assumption

is

that

the

pattern

class

ends

at

a

given

point

in

time

which

is

used

as

the

starting

point

for

a

right-to-left

trip

The

choice

of

right

to

left

evaluation

was

initially

motivated

by

the

fact

that

the

most

recent

data

are

most

probably

the

most

important

ones

In

addition

the

opposite

-LRB-

i.e.

left

to

right

-RRB-

evaluation

would

require

some

state

bookkeeping

in

order

to

have

the

same

computational

complexity

as

the

right

to

left

evaluation

This

state-related

information

can

grow

linearly

with

the

input

data

set

size

a

fact

that

forbids

its

use

on

systems

with

large

input

data

sets

Furthermore

that

information

even

in

cases

where

it

would

not

grow

prohibitively

large

would

degrade

the

time

performance

of

the

pattern

class

evaluation

functions

for

it

enlarges

the

active

data

set

of

the

running

code

-LRB-

see

-LSB-

section

-RSB-

-RRB-

The

greedy

algorithm

developed

is

called

the

RL-algorithm

and

is

very

simple

Given

a

paprameter-dependent

pattern

object

consisting

of

P

segments

-LRB-

SP

-RRB-

their

weights

-LRB-

wP

-RRB-

a

number

of

input

signals

-LRB-

dI

-RRB-

and

the

current

time

t

-LRB-

ending

point

-RRB-

it

computes

a

sequence

of

times

-LRB-

TP

-RRB-

that

define

the

segments

-LRB-

segment

i

lasts

from

Ti

to

Ti

-RRB-

as

well

as

an

evaluation

of

that

segmentation

The

algorithm

works

from

present

to

past

keeping

track

of

the

number

of

sampled

data-points

ni

and

the

current

evaluation

for

each

segment

Si

Initially

the

last

point

belongs

to

the

last

segment

SP

which

is

the

current

segment

At

each

step

the

algorithm

decides

whether

the

next

point

-LRB-

previous

in

time

-RRB-

should

belong

to

the

current

segment

-LRB-

Si

-RRB-

or

to

the

next

one

-LRB-

Si

-RRB-

The

criterion

used

to

make

that

decision

is

the

local

maximization

of

the

evaluation

function

The

algorithm

also

computes

the

earliest

and

latest

end

-LRB-

Li

and

Ri

respectively

-RRB-

of

each

segment

in

order

to

compute

durations

and

other

segment

features

that

may

be

required

in

order

to

evaluate

constraints

among

segments

during

the

segmentation

process

Termination

occurs

whenever

the

algorithm

has

matched

all

the

segments

or

when

the

matching

process

is

at

an

intermediate

segment

and

the

evaluations

of

the

remaining

segments

to

be

matched

can

not

change

significantly

the

matching

process

outcome

either

by

perfect

match

or

mismatch

Eventually

for

some

p

-LRB-

p

P

-RRB-

it

would

be

fp

p

Pg

-LRB-

Li

Ti

Ri

Ti

-RRB-

where

Sp

is

the

segment

where

the

matching

process

was

terminated

These

times

are

also

used

to

compute

durations

and

can

help

in

constraint

evaluation

during

the

process

of

segmentation

The

RL

algorithm

runs

in

O

-LRB-

n

-RRB-

time

-LRB-

where

n

is

the

number

of

data

points

used

-RRB-

since

it

spends

O

-LRB-

-RRB-

time

at

each

point

to

compare

two

evaluations

of

two

competing

-LRB-

neighboring

-RRB-

segments

and

choose

the

greater

one

That

choice

may

lead

to

suboptimal

solution

especially

in

cases

where

there

is

noise

or

neighboring

segments

whose

definitions

overlap

As

a

result

the

RL

algorithm

is

not

optimal

The

second

algorithm

is

optimal

in

cases

where

there

are

no

special

constraints

among

features

of

segments

and

the

segmentation

cost

function

is

additive

This

algorithm

essentially

is

a

modification

of

Dijkstra

's

shortest

path

algorithm

-LRB-

-LSB-

-RSB-

-RRB-

applied

to

a

graph

of

nP

nodes

Each

node

in

that

graph

is

associated

with

a

pair

of

numbers

-LRB-

i

j

-RRB-

-LRB-

for

i

P

and

j

n

-RRB-

and

is

connected

to

the

nodes

of

the

graph

that

correspond

to

pairs

-LRB-

i

j

-RRB-

and

-LRB-

i

j

-RRB-

-LRB-

figure

-RRB-

Intuitively

going

from

-LRB-

i

j

-RRB-

to

-LRB-

i

j

-RRB-

means

that

data

D

collected

at

time

tj

belong

to

segment

Si

On

the

other

hand

going

from

-LRB-

i

j

-RRB-

to

-LRB-

i

j

-RRB-

means

that

the

matching

of

segment

Si

ends

n

P

n

sgm

no

P

P

time

i

j

i

j

Figure

Search

graph

for

the

optimal

algorithm

and

the

matching

of

segment

Si

starts

at

tj

-LRB-

i.e.

Ti

tj

-RRB-

In

short

matching

consequent

data

to

the

same

segment

corresponds

to

horizontal

transitions

while

switching

segments

corresponds

to

diagonal

transitions

in

the

graph

of

figure

Now

if

we

associate

costs

with

the

graph

's

links

to

reflect

changes

in

the

evaluation

of

a

pattern

object

as

we

follow

different

paths

in

this

graph

we

can

solve

the

input

segmentation

problem

by

finding

a

maximum

cost

path

that

goes

from

-LRB-

p

t

-RRB-

to

-LRB-

-RRB-

for

some

t

and

p

P

Obviously

a

path

denotes

a

particular

segmentation

of

the

data

while

its

cost

measures

the

degree

in

which

this

segmentation

matches

the

current

pattern

class

We

can

further

simplify

this

optimal

algorithm

by

observing

that

each

node

in

the

graph

has

exactly

two

immediate

successors

Assuming

that

MC

-LRB-

i

j

-RRB-

indicates

the

maximum

cost

path

to

node

-LRB-

i

j

-RRB-

we

can

solve

the

maximum

cost

path

problem

by

traversing

the

graph

from

right

to

left

-LRB-

i.e

from

j

n

down

to

j

-RRB-

and

solving

the

following

recurrent

equation

MC

-LRB-

i

j

-RRB-

maxfMC

-LRB-

i

j

-RRB-

MC

-LRB-

i

j

-RRB-

g

v

-LRB-

Si

tj

-RRB-

with

initial

conditions

MC

-LRB-

i

n

-RRB-

v

-LRB-

Si

tn

-RRB-

Pg

where

v

-LRB-

S

t

-RRB-

denotes

the

evaluation

of

segment

S

at

time

t

The

complexity

of

this

algorithm

is

O

-LRB-

-RRB-

The

optimal

algorithm

presented

above

has

two

major

limitations

First

the

cost

function

v

must

be

additive

along

paths

in

order

for

the

algorithm

to

apply

Unfortunately

this

is

not

true

in

the

current

formulation

of

the

problem

where

we

take

into

account

the

number

of

sampled

points

in

order

to

normalize

each

segment

's

evaluation

and

its

contribution

to

the

overall

evaluation

of

the

pattern

class

Second

the

algorithm

can

not

be

extended

to

deal

with

constraints

among

different

segments

without

sacrificing

its

optimality

The

above

two

limitations

and

the

higher

time

complexity

of

the

optimal

algorithm

render

it

inappropriate

for

applications

where

some

strict

time

limitations

apply

Yet

the

algorithm

is

of

theoretical

importance

for

it

proves

that

the

segmentation

problem

has

polynomial

time

complexity

when

the

cost

function

is

additive

and

there

are

no

constraints

among

different

segments

Implementation

tFPR

has

been

implemented

as

a

software

library

that

provides

users

with

functions

that

create

and

evaluate

pattern

objects

In

addition

it

has

been

embedded

in

the

blackboard

architecture

-LSB-

-RSB-

where

it

acts

as

a

frontend

signal

interpretation

module

For

our

own

purposes

we

have

implemented

the

tFPR

pattern

manager

as

a

knowledge

source

in

However

the

organization

of

the

pattern

manager

is

architecturally

independent

We

describe

it

here

in

very

general

terms

It

can

be

reimplemented

easily

in

other

software

architectures

The

pattern

manager

is

triggered

by

satisfaction

of

its

trigger-condition

which

is

the

addition

of

a

new

interval

of

data

to

a

temporal

database

which

in

is

called

the

timeline

Triggering

binds

the

pattern

manager

's

local

variables

to

particular

attributes

of

the

trigger

ing

data

These

values

indicate

which

measure

was

updated

and

the

time

at

which

it

occurred

Next

pattern

objects

from

the

blackboard

database

that

are

related

to

the

triggering

data

are

retrieved

using

the

context

mechanism

of

knowledge

sources

Finally

the

pattern

manager

creates

a

unique

instance

of

its

action

which

reflects

the

established

bindings

of

its

local

variables

and

a

unique

binding

for

its

context

variables

that

correspond

to

pattern

objects

Thus

each

new

interval

of

data

added

to

the

timeline

can

trigger

one

or

more

instances

of

the

pattern

manager

's

action

each

one

prepared

to

evaluate

a

particular

relevant

pattern

object

in

a

right-to-left

fashion

starting

with

the

new

data

interval

We

refer

to

each

such

instance

as

an

executable

pattern

class

matching

action

Any

executable

pattern

class

matching

action

may

or

may

not

actually

be

executed

In

this

is

determined

by

the

larger

application

system

's

current

strategy

A

strategy

might

differentially

weight

pattern

recognition

versus

other

reasoning

tasks

And

it

might

differentially

weight

pattern

recognition

activities

for

different

data

measures

or

different

pattern

objects

For

example

the

Guardian

application

for

monitoring

medical

patients

dynamically

adopts

context-dependent

strategies

that

differentially

weight

pattern

class

matching

actions

versus

actions

involved

in

diagnosis

prediction

planning

and

explanation

And

it

dynamically

adjusts

its

pattern

class

matching

activities

to

favor

the

most

important

patient

data

and

the

most

important

pattern

objects

in

the

current

context

Any

non-trivial

application

system

-LRB-

in

terms

of

number

of

observed

data

measures

rate

of

adding

new

data

intervals

to

the

timeline

and

number

of

pattern

objects

in

the

knowledge

base

-RRB-

will

need

to

control

its

pattern

recognition

activities

with

some

such

mechanism

for

selective

attention

Executing

a

pattern

class

matching

action

involves

calling

an

evaluation

function

with

the

instantiated

variable

bindings

Then

this

function

enters

a

cycle

in

which

iteratively

collects

timeline

data

for

the

related

pattern

measures

and

updates

its

current

pattern

class

evaluation

based

on

the

values

of

the

collected

data

Termination

occurs

whenever

the

process

reaches

a

state

where

no

significant

changes

can

occur

to

the

final

outcome

either

by

perfect

match

or

mismatch

of

the

remaining

data

and

segments

In

the

abnormal

case

where

there

would

be

missing

data

values

at

some

point

in

time

the

process

is

terminated

and

its

current

evaluation

is

returned

Eventually

the

returned

evaluation

would

appear

as

a

new

episode

on

the

timeline

tFPR

contains

many

operators

that

are

used

to

represent

and

evaluate

np

functions

Those

operators

are

described

in

detail

in

-LSB-

-RSB-

A

summary

of

them

appears

in

table

The

syntactic

definition

of

np

functions

appears

in

table

In

that

table

np-list

denotes

a

non-null

list

of

nps

and

wp-list

bp-list

denote

lists

of

real

numbers

used

as

weights

or

bounds

respectively

Square

brackets

-LRB-

-LSB-

-RSB-

-RRB-

are

used

to

denote

optional

expressions

The

pair

-LRB-

a

c

-RRB-

denotes

the

sigmoidal

function

Sa

c

The

number

argument

in

rate

is

used

to

denote

the

order

of

the

derivative

Finally

Cp-list

is

a

list

of

constraints

each

of

which

has

the

following

format

-LRB-

op

arg

arg

weight

-RRB-

where

op

f

S-GT

S-GE

S-LT

S-LE

S-EQ

S-NE

g

arg

f

-LRB-

S-DT

segment

-RRB-

-LRB-

S-MAX

segment

measure

-RRB-

number

-LRB-

S-MIN

segment

measure

-RRB-

g

S-GT

stands

for

greater

than

S-GE

for

greater

equal

and

so

on

-LRB-

S-DT

segment

-RRB-

is

used

to

refer

to

the

duration

of

segment

segment

while

-LRB-

S-MAX

segment

measure

-RRB-

is

used

to

refer

to

the

maximum

value

of

measure

in

segment

segment

The

difference

between

an

operator

-LRB-

say

or

-RRB-

and

m

-LRB-

e.g.

mor

-RRB-

is

that

the

former

uses

all

of

its

measures

to

all

of

its

arguments

while

the

later

distributes

its

measures

across

its

arguments

To

illustrate

the

above

let

E

be

the

function

that

would

be

used

to

evaluate

an

np

function

on

some

input

data

x

-LSB-

xk

-RSB-

Then

E

-LRB-

np

x

-RRB-

should

return

the

membership

grade

of

x

in

the

fuzzy

set

associated

with

np

Now

E

can

be

defined

recursively

as

follows

E

-LRB-

x

-RRB-

x

E

-LRB-

-LRB-

-LSB-

npk

-RSB-

-RRB-

x

-RRB-

f

-LRB-

E

-LRB-

x

-RRB-

E

-LRB-

npk

x

-RRB-

-RRB-

E

-LRB-

-LRB-

m

-LSB-

npk

-RSB-

-RRB-

-LSB-

xk

-RSB-

-RRB-

f

-LRB-

E

-LRB-

-RRB-

E

-LRB-

npk

xk

-RRB-

-RRB-

where

denotes

a

tFPR

operator

-LSB-

npk

-RSB-

is

a

list

of

np

functions

denotes

the

empty

list

is

a

list

of

arguments

used

by

the

function

f

Illustrative

Application

We

currently

apply

tFPR

in

a

system

called

Guardian

for

monitoring

intensive

care

patients

-LSB-

-RSB-

Guardian

performs

several

real-time

reasoning

tasks

including

diagnosis

prediction

planning

and

explanation

of

the

patient

's

dynamic

condition

All

of

these

reasoning

activities

are

based

on

the

recognition

of

important

pattern

classes

in

observed

patient

data

Before

we

present

some

examples

of

tFPR

pattern

objects

let

us

define

some

auxiliary

functions

that

could

be

used

to

create

sigmoidal

functions

Assuming

an

affine

sigmoidal

class

AS

where

S

is

a

basic

unit

sigmoidal

function

define

the

following

functions

produce

sigmoidal

-LRB-

-RRB-

It

creates

the

unique

sigmoidal

function

in

AS

that

passes

through

the

points

-LRB-

-RRB-

and

-LRB-

-RRB-

produce

le

-LRB-

a

da

-RRB-

produce

sigmoidal

-LRB-

a

ffl

a

da

ffl

-RRB-

It

creates

a

sigmoidal

function

that

is

nearly

equal

to

in

the

interval

-LRB-

a

-RSB-

produce

ge

-LRB-

a

da

-RRB-

produce

sigmoidal

-LRB-

a

da

ffl

a

ffl

-RRB-

It

creates

a

sigmoidal

function

that

is

nearly

equal

to

in

the

interval

-LSB-

a

-RRB-

produce

gt

-LRB-

a

da

-RRB-

produce

sigmoidal

-LRB-

a

ffl

a

da

ffl

-RRB-

It

creates

a

sigmoidal

function

that

is

nearly

equal

to

in

the

interval

-LRB-

a

-RRB-

produce

ccint

-LRB-

a

b

da

db

-RRB-

min

-LRB-

produce

ge

-LRB-

a

da

-RRB-

produce

le

-LRB-

b

db

-RRB-

-RRB-

It

creates

a

sigmoidal

function

that

is

nearly

equal

to

in

the

interval

-LSB-

a

b

-RSB-

produce

eq

-LRB-

a

da

-RRB-

produce

ccint

-LRB-

a

a

da

da

-RRB-

It

creates

a

sigmoidal

function

that

is

nearly

equal

to

only

at

point

a.

ffl

da

db

are

assumed

to

be

small

positive

constants

Table

illustrates

the

tFPR

pattern

objects

for

three

conditions

of

interest

to

Guardian

Hypoxia

on

top

of

table

exhibits

a

pattern

class

that

is

described

as

a

set

of

constraints

upon

its

measures

as

well

as

a

constraint

between

two

of

its

measures

The

latter

is

a

linear

constraint

and

is

eliminated

in

the

vp

function

which

replaces

its

third

argument

which

is

by

Finally

the

np

function

requires

this

new

value

to

be

less

or

equal

to

Post

bypass

myocardial

depression

with

adequate

volume

in

table

is

a

pattern

class

specification

based

on

many

measures

and

constraints

which

have

been

assigned

relative

importance

with

respect

to

the

pattern

class

evaluation

This

example

illustrates

the

difficulty

in

using

only

set

operations

to

describe

pattern

classes

depending

on

many

conditions/measures

In

this

pattern

class

we

used

a

weighted

sum

to

express

the

vague

interelationship

of

pattern

measures

regarding

the

evaluation

of

the

pattern

class

Trying

to

put

this

pattern

class

into

a

form

of

set

operations

-LRB-

probably

by

using

some

interval

conditions

-RRB-

not

only

would

result

in

a

much

larger

and

more

complicated

definition

but

would

also

require

an

immensely

greater

effort

for

the

pattern

class

to

be

described

at

the

first

place

Finally

sudden

increases

in

chest

tube

output

are

recognized

by

the

pattern

class

at

the

bottom

of

table

That

pattern

class

is

a

time

dependent

pattern

class

consisting

of

two

segments

The

first

segment

requires

the

rate

of

chest

tube

output

to

be

zero

-LRB-

e.g.

chest

tube

output

to

be

constant

-RRB-

while

the

second

segment

requires

the

rate

of

increment

of

chest

tube

output

to

be

greater

than

In

addition

the

pattern

class

puts

a

constraint

on

the

first

segment

requiring

its

duration

to

be

less

than

seconds

The

relative

importance

of

the

two

segments

and

the

constraint

is

Guardian

currently

has

a

knowledge

base

of

about

pattern

objects

We

expect

it

to

grow

to

a

large

database

especially

after

we

would

develop

user-friendly

or

automatic

tools

for

pattern

class

acquisition

Specifications

of

a

Graphical

Interface

The

knowledge

acquisition

bottleneck

-LRB-

-LSB-

-RSB-

-RRB-

is

manifested

in

tFPR

as

the

pattern

class

specification

process

The

current

version

of

tFPR

relies

solely

on

the

user

or

the

expert

to

provide

the

system

with

all

the

necessary

knowledge

that

is

a

definition

of

each

pattern

class

of

interest

This

may

be

a

major

obstacle

in

the

development

of

a

useful

system

based

on

tFPR

when

the

number

of

pattern

classes

would

be

large

As

stated

in

the

next

section

it

is

our

belief

that

the

most

appropriate

solution

to

this

problem

is

the

design

and

integration

of

a

learning

component

into

the

system

However

we

present

here

the

specifications

of

a

graphical

interface

that

would

facilitate

the

pattern

class

specification

process

in

those

cases

where

the

user

himself

would

define

the

pattern

class

Our

solution

requires

that

the

user

provide

the

system

with

n

points

-LRB-

-RRB-

-LRB-

xn

yn

-RRB-

that

specify

a

pattern

class

Then

the

system

tries

to

find

a

minimum

combination

in

terms

of

sigmoidal

functions

that

would

interpolate

the

pattern

class

at

the

n

given

points

with

a

given

tolerance

Though

it

is

very

hard

to

find

an

optimal

algorithm

an

approximate

but

fast

solution

is

the

following

Specify

the

tolerance

-LRB-

-RRB-

in

the

approximation

Sort

the

n

given

points

so

that

xn

For

i

n

let

Si

be

the

sigmoidal

that

passes

through

the

i-th

and

-LRB-

i

-RRB-

th

point

let

LSi

i

RSi

i

While

two

successive

sigmoidal

functions

and

can

be

merged

-LRB-

or

equivalently

when

jSa

c

-LRB-

xj

-RRB-

yjj

where

a

-LRB-

-RRB-

c

-LRB-

-RRB-

-RRB-

merge

them

-LRB-

i.e

replace

them

by

Sa

c

and

let

LSa

c

RSa

c

-RRB-

Return

the

current

collection

of

sigmoidal

functions

The

algorithm

above

can

run

-LRB-

if

implemented

carefully

-RRB-

in

O

-LRB-

-RRB-

Its

average

time

depends

on

the

distribution

of

the

number

of

sigmoidal

mergings

the

larger

the

number

of

mergings

the

slower

the

algorithm

For

a

uniform

distribution

the

average

time

is

O

-LRB-

nlogn

-RRB-

However

the

worst

case

occurs

when

all

the

sigmoidal

functions

merge

into

a

single

one

In

that

case

we

need

-LRB-

-RRB-

sigmoidal

computations

In

the

case

of

time-dependent

pattern

classes

the

user

would

usually

define

the

ridge

of

the

pattern

class

i.e.

the

ideal

variation

of

the

input

to

match

the

pattern

class

perfectly

This

corresponds

to

the

locus

of

points

where

the

membership

function

of

the

fuzzy

set

that

describes

the

pattern

class

in

question

is

maximal

Then

he

can

proceed

by

defining

another

set

of

points

where

the

pattern

class

is

constant

i.e.

the

user

can

use

isometric

curves

to

define

its

pattern

classes

For

example

the

user

may

define

the

set

of

points

where

the

pattern

class

evaluates

to

After

having

defined

at

least

two

such

sets

the

system

can

use

the

original

algorithm

to

interpolate

sigmoidal

functions

across

the

sets

-LRB-

within

a

given

tolerance

of

approximation

-RRB-

Of

course

the

numerical

nature

of

our

computations

require

the

set

of

points

to

be

selected

so

that

they

correspond

to

values

that

are

not

very

close

to

each

other

and

not

too

close

to

either

or

Furthermore

since

the

center

of

symmetry

of

every

sigmoidal

is

its

most

stable

point

-LRB-

i.e.

changes

in

the

y-axis

result

in

the

least

changes

in

the

x-axis

-RRB-

it

is

suggested

to

choose

at

least

one

set

corresponding

to

values

that

are

close

to

Finally

the

set

of

interpolating

sigmoidal

functions

could

be

used

to

form

an

np

function

The

construction

of

the

rest

of

the

elements

of

the

pattern

object

for

the

current

pattern

class

is

straightforward

Extensions

There

are

a

number

of

ways

to

extend

tFPR

and

enhance

its

performance

or

capabilities

As

noted

in

the

last

section

the

pattern

class

specification

task

is

the

most

limiting

factor

in

tFPR

for

it

requires

a

large

amount

of

knowledge

to

be

encoded

into

the

system

Furthermore

as

the

number

of

pattern

objects

would

grow

and

parameter

spaces

of

potentially

higher

dimensionality

would

be

used

-LRB-

such

as

the

Euclidean

space

-RRB-

pattern

class

specification

would

become

a

very

difficult

and

laborious

task

The

solution

to

this

problem

is

to

have

the

system

undertake

the

pattern

class

specification

task

In

that

case

the

system

would

have

to

extract

the

definition

of

a

pattern

class

by

examining

a

set

of

training

input

data

Such

a

pattern

class

learning

capability

would

make

the

system

scale

better

be

more

adaptive

and

even

appear

to

be

more

intelligent

Another

extension/enhancement

of

the

system

results

from

a

basic

observation

made

in

section

Any

non-trivial

application

must

control

and

limit

its

pattern

recognition

activities

with

a

mechanism

for

selective

attention

Although

such

a

mechanism

was

outlined

in

that

section

and

is

partially

implemented

in

the

Guardian

system

-LSB-

-RSB-

alternative

solutions

should

be

studied

as

well

One

such

solution

could

be

to

create

control

pattern

objects

i.e.

pattern

objects

that

would

control

the

applicability

of

other

pattern

objects

based

on

observations

on

the

input

data

These

pattern

objects

serve

no

purpose

but

to

control

the

matching

process

in

order

to

make

it

more

efficient

Alternatively

one

can

allow

the

Cp

conditions

in

pattern

objects

to

be

learnable

and

based

on

experience

These

extensions

should

result

in

a

more

efficient

pattern

recognition

system

Another

problem

tFPR

has

to

deal

with

is

missing

data

In

its

current

state

tFPR

does

not

deal

with

that

problem

at

all

It

aborts

the

current

pattern

object

evaluation

as

soon

as

it

finds

that

some

related

data

are

missing

A

better

solution

would

be

to

include

in

the

pattern

object

information

regarding

the

behavior

of

the

pattern

class

in

cases

where

information

is

missing

For

example

in

order

to

deal

with

missing

data

the

pattern

class

matching

process

may

choose

to

abort

use

default

values

to

anticipate

the

lack

of

some

information

use

expected

values

assign

some

predefined

UN

KNOWN

value

to

anything

that

is

missing

or

use

a

combination

of

these

policies

depending

on

the

definition

of

the

pattern

object

that

is

under

evaluation

Currently

there

are

no

control

pattern

classes

in

tFPR

and

pattern

class

evaluations

with

missing

data

are

discarded

However

a

learning

component

based

on

multi-sigmoidal

neural

networks

-LSB-

-RSB-

is

under

development

and

is

expected

to

be

operational

in

the

next

few

months

The

final

evaluation

of

the

system

should

be

based

on

three

performance

measures

pattern

class

specification

acquisition

-LRB-

performed

by

the

learning

component

-RRB-

pattern

class

matching

accuracy

and

pattern

recognition

efficiency

Conclusion

tFPR

is

a

simple

yet

powerful

pattern

recognition

system

Its

main

features

are

the

adoption

of

sigmoidal

functions

to

approximate

arbitrary

fuzzy

set

membership

and

the

use

of

an

efficient

segmentation

algorithm

tFPR

can

represent

and

recognize

arbitrary

pattern

classes

that

can

be

static

or

dynamic

depending

on

time

or

any

other

parameter-space

The

direct

evaluation

of

static

pattern

classes

and

the

linear

time

complexity

of

our

segmen

tation

method

provide

the

groundwork

of

a

system

that

may

be

appropriate

for

real

time

applications

Although

tFPR

is

a

completely

functioning

system

it

is

still

under

development

We

expect

to

solve

or

address

most

of

the

problems

mentioned

here

and

others

we

may

encounter

Each

of

these

solutions

should

add

to

tFPR

's

capabilities

without

significantly

sacrificing

its

performance

The

overall

goal

is

a

system

or

architecture

with

a

wide

set

of

tools

and

methods

for

pattern

class

specification

and

recognition

Acknowledgements

This

research

was

supported

by

NASA

grant

-LRB-

ARPA

Order

-RRB-

and

Teknowledge

Federal

Systems

Inc.

contract

-LRB-

ARPA

contract

-RRB-

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Tong

and

H.T.Nguyen

editors

Fuzzy

Sets

and

Applications

Selected

Papers

by

L.A.

Zadeh

-LRB-

Wiley

New

York

-RRB-

-LSB-

-RSB-

T.

Yamakawa

and

M.

Furukawa

A

design

algorithm

of

membership

functions

for

a

fuzzy

neuron

using

example-based

learning

in

Proceed

ings

of

IEEE

International

Conference

on

Fuzzy

Systems

-LRB-

San

Diego

CA

-RRB-

-LCB-

-LSB-

-RSB-

L.A.

Zadeh

Fuzzy

sets

Inf

Control

-LRB-

-RRB-

-LRB-

-RRB-

fa

c

-LRB-

x

-RRB-

S

-LRB-

ax

c

-RRB-

fAND

-LRB-

zk

-RRB-

min

-LRB-

zk

-RRB-

fOR

-LRB-

zk

-RRB-

max

-LRB-

zk

-RRB-

fNOT

-LRB-

x

-RRB-

x

fNOT

-LRB-

y

-RRB-

y

fONE

-LRB-

x

-RRB-

fZERO

-LRB-

x

-RRB-

fWS

-LRB-

yk

wk

-RRB-

Pkj

ykwk

Pkj

wj

fAV

G

-LRB-

yk

-RRB-

k

kX

j

yj

fGRP

-LRB-

zk

-RRB-

-LSB-

zk

-RSB-

fBND

-LRB-

x

bk

yk

-RRB-

if

x

if

x

yk

if

bk

x

bk

yk

if

bk

x

fV

ALUE

-LRB-

-LRB-

t

-RRB-

zk

-LRB-

t

-RRB-

-RRB-

-LSB-

-LRB-

t

-RRB-

zk

-LRB-

t

-RRB-

-RSB-

fRATE

-LRB-

z

-LRB-

t

-RRB-

-RRB-

z

-LRB-

t

dt

-RRB-

z

-LRB-

t

-RRB-

dt

fRATE

-LRB-

z

-RRB-

frate

-LRB-

z

-RRB-

fRATE

i

-LRB-

z

-RRB-

frate

i

-LRB-

frate

-LRB-

z

-RRB-

-RRB-

Table

Operator

evaluation

functions

in

tFPR

x

is

used

to

denote

x-domain

values

y

is

used

to

denote

y-domain

values

and

z

is

used

to

denote

values

in

either

domain

-LSB-

-RSB-

are

used

to

denote

lists

np

-LRB-

a

c

-RRB-

-LRB-

and

np-list

-RRB-

-LRB-

mand

np-list

-RRB-

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or

np-list

-RRB-

-LRB-

mor

np-list

-RRB-

-LRB-

not

np

-RRB-

-LRB-

one

-RRB-

-LRB-

zero

-RRB-

-LRB-

ws

np-list

wp-list

-RRB-

-LRB-

mws

np-list

wp-list

-RRB-

-LRB-

avg

np-list

-RRB-

-LRB-

mavg

np-list

-RRB-

-LRB-

grp

np-list

-RRB-

-LRB-

mgrp

np-list

-RRB-

-LRB-

bnd

np-list

bp-list

-RRB-

-LRB-

mbnd

np-list

bp-list

-RRB-

-LRB-

value

np

-RRB-

-LRB-

rate

-LSB-

number

-RSB-

np

-RRB-

-LRB-

sgm

np-list

wp-list

cp-list

-RRB-

Table

Syntactic

definition

of

np

functions

in

tFPR

pattern

class

specification

tFPR

pattern

class

HYPOXIA

whenever

mmHg

or

-LRB-

da

-RRB-

or

-LRB-

Mp

-LSB-

-RSB-

Ap

identity

function

vp

-LRB-

p

o

f

-RRB-

-LSB-

p

o

p

f

-RSB-

np

-LSB-

mOR

produce

le

-LRB-

-RRB-

produce

le

-LRB-

-RRB-

produce

le

-LRB-

-RRB-

-RSB-

ap

dp

Cp

True

-RRB-

POST

BYPASS

MYOCARDIAL

DEPRESSION

WITH

ADQ

VOLUME

whenever

CO

l

min

-LRB-

da

-RRB-

and

SV

R

-LRB-

da

-RRB-

and

PAD

-LRB-

da

-RRB-

and

CV

P

-LRB-

da

-RRB-

and

PCWP

-LRB-

da

-RRB-

where

the

relative

importance

of

the

above

measurements

is

-LRB-

Mp

-LSB-

CO

SV

R

PAD

CV

P

PCWP

-RSB-

Ap

identity

function

vp

identity

function

np

-LSB-

mWS

produce

le

-LRB-

-RRB-

produce

gt

-LRB-

-RRB-

produce

gt

-LRB-

-RRB-

produce

gt

-LRB-

-RRB-

produce

gt

-LRB-

-RRB-

-RSB-

ap

dp

Cp

True

-RRB-

SUDDEN

INCREASE

OF

CHEST

TUBE

OUTPUT

whenever

a

short

period

of

constant

chest

tube

output

is

followed

by

a

sudden

increase

in

chest

tube

output

-LRB-

Mp

-LSB-

PULMONARY.CT-output

-RSB-

Ap

identity

function

vp

identity

function

np

-LSB-

SGM

produce

RATE

-LRB-

produce

eq

-LRB-

-RRB-

-RRB-

produce

RATE

-LRB-

produce

gt

-LRB-

-RRB-

-RRB-

duration

of

flat

area

-LRB-

S

LT

-LRB-

S

DT

-RRB-

-RRB-

-RSB-

ap

dp

Cp

True

-RRB-

Table

Three

pattern

class

definitions

and

their

corresponding

pattern

objects

in

tFPR

artificial intelligence
fuzzy sets
medicine
pattern recognition

knowledge acquisition and representation