Fuzzy Logic and Its Applications

jabgoldfishAI and Robotics

Oct 19, 2013 (3 years and 7 months ago)

57 views

Fuzzy Logic and Its Applications

Textbook:
Fuzzy Logic with Engineering
Applications,
2
nd

Ed. John
-
Wiley,
2004
, T.J.Ross,


References
:


Fuzzy

Logic

with

Engineering

Applications,

1995

T
.
J
.
Ross,

McGraw
-
Hill


Fuzzy

Set

Teory,

1997

G
.
Klir

et

al
.

Prentice

Hall


Fuzzy

Sets

and

Fuzzy

Logic

1995

G

Klir

et

al
.

Prentice

Hall


Introduction

In

1965
,

Prof
.

Lofti

Zadeh

published

the

first

article

“Fuzzy

Sets”
.

It

becomes

billions

of

dollars

business
.


America


Europe


Asia


Thousands

of

patents


Uncertainty
:

Incomplete

Ambiguity
:

Imprecise


Applications
:


Air

Conditioner


Washing

Machine

Subway

System


Camera

Aerospace



Nuclear

Submarine

Pattern

Recognition


Control

Image

Processing


Computer

Vision

They reflect a recent trend to view fuzzy logic (FL),
neurocomputing (NC), genetic computing (GC), Rough
Sets (RS) and probabilistic computing (PC) as an
association of computing methodologies falling under
the rubric of so
-
called soft computing.


Among the basic concepts that underlie human
cognition, three stand out in importance: granulation,
organization, and causation.

Granulation involves a partitioning of a whole into parts;
organization involves an integration of parts into a
whole; and causation relates to an association of causes
with effects

A granule may be viewed as a clump of points (objects)
drawn together by indistinguishability, similarity, or
functionality. Modes of information granulation (IG) in
which granules are crisp play an important role in many
theories, methods and techniques, among them interval
analysis, quantization, rough set theory, qualitative
process theory, and chunking.

In fuzzy logic, fuzzy IG underlies the basic concepts of
linguistic variables, fuzzy if
-
then rules, and fuzzy graphs

This perception is reinforced by viewing it in the context
of generalization. More specifically, any theory, method,
technique, or problem may be fuzzified (or f
-
generalized)
by replacing the concept of a crisp set with that of a
fuzzy set.

Similarly, any theory, method, technique, or problem can
be granulated (g
-
generalized) by partitioning variables,
functions, and relations into granules.

Furthermore, we can combine fuzzification with
granulation, which gives rise to fuzzy granulation (f
-
granulation). Fuzzy granulation, then, provides a basis
for what might be called f.g
-
generalization.

The generalization of two
-
valued logic leads to
multivalued logic and parts of fuzzy logic. But fuzzy logic
in its wide sense

which is the sense in which it is used
today

results from f.g
-
generalization. This crucial
difference between multivalued logic and fuzzy logic
explains why fuzzy logic has so many applications,
whereas multivalued logic does not.

Introduction

Fuzzy

set

theory

provides

a

means

for

representing

uncertainties
.

Probablity


random uncertainty

But

some

uncertainty

is

non
-
random

In

fact,

a

huge

amount!

Natural

Language

is

vague

and

imprecise
.

Fuzzy

set

theory

uses

Linguistic

variables,

rather

than

quantitative

variables

to

represent

imprecise

concepts
.


Fuzzy Logic

Fuzzy

Logic

is

suitable

to

Very

complex

models

Judgemental

Reasoning

Perception

Decision

making

Requiring

precision



high

cost,

long

time

Statistics

and

random

processes


Based

on

Randomness
.

Fuzziness

Example
.

Random

Errors

generally

average

out

over

time

or

space

Non
-
random

errors

will

not

generally

average

out

and

likely

to

grow

with

time
.

Information World

Information World

Crisp

set

has

a

unique

membership

function



A
(x)

=

1

x



A




0

x



A



A
(x)



{
0
,

1
}


Fuzzy

Set

can

have

an

infinite

number

of

membership

functions



A



[
0
,
1
]


Fuzziness

Examples
:

A

number

is

close

to

5

Fuzziness

Examples
:

He/she

is

tall

Fuzziness

Randomness

versus

Fuzziness

Drinking

Water

Problem

Classical

Sets

Fuzzy

Sets

Operations on Classical Sets

Union
:


A



B

=

{x

|

x



A

or

x



B}

Intersection
:


A



B

=

{x

|

x



A

and

x



B}

Complement
:


A’

=

{x

|

x



A,

x



X}





X



Universal

Set

Set

Difference
:


A

|

B

=

{x

|

x



A

and

x



B}


Set

difference

is

also

denoted

by

A

-

B

Properties of Classical Sets

A



B

=

B



A

A



B

=

B



A

A



(B



C)

=

(A



B)



C

A



(B



C)

=

(A



B)



C

A



(B



C)

=

(A



B)



(A



C)

A



(B



C)

=

(A



B)



(A



C)

A



A

=

A

A



A

=

A

A



X

=

X

A



X

=

A

A





=

A

A





=



Properties of Classical Sets

If

A



B



C,

then

A



C

De Morgan

s Law:


(A


B)


= A




B



(A


B)


= A




B



Proof:

LHS=
{x | x


(A and B)}=
{x | x


A or
x


B)}=


A




B

= RHS




Can be extended to
n sets


Generalized De Morgan Law:

A


A






X

X
















Using ( ) to keep original processing order






Generalized Duality Law:





X

X
















Using ( ) to keep original processing order

Law of the excluded middle:


A


A


= X

Law of the Contradiction:


A


A


=


These laws are not true for Fuzzy
Sets!


Fuzzy Sets

Characteristic

function

X,

indicating

the

belongingness

of

x

to

the

set

A


X(x)

=

1

x



A




0

x



A

or

called

membership

Hence,

A



B



X
A



B
(x)


=

X
A
(x)



X
B
(x)


=

max(X
A
(x),X
B
(x))

Note
:

Some

books

use

+

for


,

but

still

it

is

not

ordinary

addition!

Some

more

explanations

follow


Fuzzy Sets

A



B



X
A



B
(x)


=

X
A
(x)



X
B
(x)


=

min(
X
A
(x),
X
B
(x))

A’



X
A’
(x)


=

1



X
A
(x)

A



B



X
A
(x)



X
B
(x)

A’’

=

A


Fuzzy Sets

Note


(x)



[
0
,
1
]



not

{
0
,
1
}

like

Crisp

set

A

=

{

A
(x
1
)

/

x
1

+


A
(x
2
)

/

x
2

+


}


=

{



A
(xi)

/

xi
}

Note
:

‘+’



add


‘/





divide

Only

for

representing

element

and

its

membership
.

Also

some

books

use


(x)

for

Crisp

Sets

too
.

Fuzzy Set Operations


A



B
(x)

=


A
(x)




B
(x)



=

max(

A
(x),


B
(x))



A



B
(x)

=


A
(x)




B
(x)



=

min(

A
(x),


B
(x))


A’
(x)

=

1

-


A
(x)


De

Morgan’s

Law

also

holds
:



(A



B)’

=

A’



B’



(A



B)’

=

A’



B’

But,

in

general


A



A’



A



A’


Properties of Fuzzy Sets

A



B

=

B



A

A



B

=

B



A

A



(B



C)

=

(A



B)



C

A



(B



C)

=

(A



B)



C

A



(B



C)

=

(A



B)



(A



C)

A



(B



C)

=

(A



B)



(A



C)

A



A

=

A




A



A

=

A

A



X

=

X




A



X

=

A

A





=

A




A





=



If

A



B



C,

then

A



C

A’’

=

A

Sets as Points in Hypercubes

Explore

to

n
-
dimension

Classical

Relations

Fuzzy

relations

Logic,

Approximate

reasoning,

Rule
-
based

learning

systems,

Nonlinear

Simulation,

Classification,

Pattern

Recognition,

etc
.

Cartesian Product

A

=

{a,b}

B

=

{
0
,
1
}

A

x

B

=

{

(a,
0
)

(a,
1
)

(b,
0
)

(b,
1
)

}

Ordered

Pairs

Consider

A

x

A

or

A

x

B

x

C

if

C

is

given


Based

on

the

above,

Crisp

Relations

are

discussed

next


Crisp Relations

A

subset

of

a

Cartesian

Product

A
1

x

A
2

x



x

Ar

is

called

an

r
-
ary

relation

over

A
1
,A
2
,

,Ar

If

r

=

2
,

the

relation

is

a

subset

of

A
1

x

A
2

Binary

relation

from

A
1

into

A
2

The

strength

of

a

relation
:

Characteristic

Function


X(x,y)

=

1

(x,y)



X

x

Y




0

(x,y)



X

x

Y

For

Classical

relations,

the

value

is

1

or

0

If

the

universes

or

sets

are

finite,

we

can

use

relational

matrix

to

represent

it
.

Crisp Relations

Example
:

If

X

=

{
1
,
2
,
3
}


Y

=

{a,b,c}

R

=

{

(
1

a),(
1

c),(
2

a),(
2

b),(
3

b),(
3

c)

}




a b c


1 1 0 1

R =

2 1 1 0


3 0 1 1

Using

a

diagram

to

represent

the

relation

Crisp Relations

Relations

can

also

be

defined

for

continuous

universes

R

=

{

(x,y)

|

y



2
x,

x



X,

y



Y
}

X

=


1

y



2
x


0

otherwise

Crisp Relations

Cardinality
:

N
:

#

of

elements

in

X

M
:

#

of

elements

in

y

Cardinality

of

R


n
X

x

Y

=

n
X



n
Y

=

M



N

Cardinality

of

the

Power

set

of

this

relation


n
P(X

x

Y)

=

2
M

N

Operations on Crisp Relations

Complete

Relation

Matrix

1 1 1

1
1 1

1 1 1

Null

relation

Matrix

0 0 0

0 0 0

0 0 0

Operations on Crisp Relations

Union


R



S



X
R



S
(x,y)


X
R



S
(x,y)

=

max{

X
R
(x,y),X
S
(x,y)

}

Intersection


R



S



X
R


S
(x,y)


X
R



S
(x,y)

=

min{

X
R
(x,y),X
S
(x,y)

}

Complement


R’



X
R’
(x,y)


X
R’
(x,y)

=

1



X
R
(x,y)

Containment


R



S



X
R
(x,y)



X
S
(x,y)

Identity






0


X



E

Properties of Crisp Relations

Commutativity

Associativity

Distributivity

Idempotency




All

hold

De

Morgan

Law

Excluded

middle

Law

Etc
.

Properties of Crisp Relations

Composition

Let

R

be

a

relation

representing

a

mapping

from

X

to

Y

X

Y

University

sets

Let

S

be

a

relation,

a

mapping

from

Y

to

Z

Can

we

find

T

from

R

to

S?

Properties of Crisp Relations

T
:

mapping

from

X

to

Z

T

=

R



S

Two

ways

to

compute

X
T
(xz)

1.
X
T
(xz)

=



(X
R
(xy)



X
s
(yz))




=

max(min{X
R
(xy),X
S
(yz)})

Max
-
min composition

2.
X
T
(xz)

=



(X
R
(xy)



X
s
(yz))

Max
-
product composition

multiplication

y


Y

y


Y

y


Y

Properties of Crisp Relations

Using

Matrix

representation
:



y
1

y
2

y
3

y
4


x
1

1

0

1

0

R

=


x
2

0

0

0

1


x
3

0

0

0

0



z
1

z
2


y
1

0

1




z
1

z
2


y
2

0

0



x
1

0

0


S

=

y
3

0

1



T

=



x
2

0

0


y
4

0

0




x
3

0

0


T
(x
1
,z
1
)

=

max[min(
1
,
0
)

min(
0
,
0
)

min(
1
,
0
)

min(
0
,
0
)]



=

max[
0
,
0
,
0
,
0
]

=

0

Similar,

but

not

the

same

as

matrix

multiplication!

Fuzzy Relations

Cardinality

of

Fuzzy

Relations

Since

the

cardinality

of

fuzzy

sets

on

any

universe

is

infinity,

the

cardinality

of

a

fuzzy

relation

is

also

infinity
.




Note
:

other

books

have

different

discussions!

Operations on Fuzzy Relations

Union
:



R



S

=

max{


R
(x,y),

S
(x,y)

}

Intersection
:




R



S

=

min{


R
(x,y),

S
(x,y)

}

Complement
:




R

(x,y)

=

1

-


R
(x,y)

Containment
:


R



S




R
(x,y)




S
(x,y)

Properties of Fuzzy Relations

Commutativity

Associativity

Distributivity

Idempotency




All

hold

De

Morgan

Law

Excluded

middle

Law

Etc
.

Note
:

R



R’



E


R



R’



0

In

general
.

Properties of Fuzzy Relations

Fuzzy

Cartesian

Product

and

Composition



R
(x

y)

=


A

x

B
(x

y)

=

min(

A
(x),


B
(y))

Example
:

A

=

0
.
2
/x
1

+

0
.
5
/x
2

+

1
/x
3

B

=

0
.
3
/y
1

+

0
.
9
/y
2







y
1


y
2



0
.
2



x
1

0
.
2

0
.
2



A

x

B

=


0
.
5

0
.
3

0
.
9

=

x
2

0
.
3

0
.
5



1



x
3

0
.
3

0
.
9

Properties of Fuzzy Relations

Vector

Outer

Product

If

R

is

a

fuzzy

relation

on

the

space

X

x

Y


S

is

a

fuzzy

relation

on

the

space

Y

x

Z

Then,

fuzzy

composition

is

T

=

R



S

1.
Fuzzy

max
-
min

composition




T
(xz)

=



(

R
(xy)




s
(yz))

2
.

Fuzzy

max
-
production

composition



T
(xz)

=



(

R
(xy)




s
(yz))

Note
:

R



S



S



R

y


Y

y


Y

Properties of Fuzzy Relations

Example
:



y
1

y
2




z
1

z
2

z
3

R

=


x
1

0
.
7

0
.
5



S

=


y
1

0
.
9

0
.
6

0
.
2


x
2

0
.
8

0
.
4



y
2

0
.
1

0
.
7

0
.
5







z
1

z
2

z
3

Using

max
-
min,

T

=


x
1

0
.
7

0
.
6

0
.
5





x
2

0
.
8

0
.
6

0
.
4






z
1

z
2

z
3

Using

max
-
product,

T

=


x
1

0
.
63

0
.
42

0
.
25





x
2

0
.
72

0
.
48

0
.
20

Note
:

Set,

Relation,

Composition


How

to

find

new

membership

from

the

given

ones!

Tolerance and Equivalence Relation

Crisp

Equivalence

Relation


R



X

x

X

Relation

has

the

following

properties
:

Reflexivity


(xi

xi)



R

or

X
R
(xi

xi)

=

1

Symmetry


(xi

xj)



R



(xj

xi)



R


or

X
R
(xi

xj)

=

X
R
(xj

xi)

Transitivity


(xi

xj)



R

and

(xj

xk)



R



(xi

xk)



R


or

X
R
(xi

xj)

=

1

and

X
R
(xj

xk)

=

1



X
R
(xi

xk)

=

1

Tolerance and Equivalence Relation

Graph

representation
:

Crisp Tolerance Relation

(or proximity relation)

Only

has

reflexivity

and

symmetry

A

tolerance

relation,

R
1

can

become

an

Equivalence

Relation

by

at

most

(n
-
1
)

compositions

(


n
-
1
),

n

is

the

cardinal

member

of

X
.

R
1
n
-
1

=

R
1



R
1







R
1

=

R


Crisp Tolerance Relation

(or proximity relation)

Example
:


1

1

0

0

0


1

1

0

0

1

R
1

=

0

0

1

0

0


0

0

0

1

0


0

1

0

0

1




1

1

0

0

1




1

1

0

0

1

Try

R
1
2

=

0

0

1

0

0




0

0

0

1

0




1

1

0

0

1


Note
:

symmetric,

reflexive,

but

not

transitive,

why?

X(x
1

x
2
)

=

1

X(x
2

x
5
)

=

1


but

X(x
1

x
5
)



1

(=
0
)

Now,

it

is

transitive!

Fuzzy Tolerance and Equivalence Relation

A

fuzzy

relation

R

has
:

1
.

Reflexivity



R
(xi

xi)

=

1

2
.

Symmetry



R
(xi

xj)

=


R
(xj

xi)

3
.

Transitivity



R
(xi

xj)

=


1





R
(xj

xk)

=


2




R
(xi

xk)

=






where





min{

1
,


2
}

Fuzzy

tolerance

relation

R
1

has

reflexivity,

symmetry
.

It

can

be

transformed

into

a

fuzzy

equivalence

relation

by

at

most

(n
-
1
)

(


n
-
1
)

compositions
.


R
1
n
-
1

=

R
1



R
1







R
1

=

R

Fuzzy Tolerance and Equivalence Relation

Example
:



1

0
.
8

0

0
.
1

0
.
2



0
.
8

1

0
.
4

0

0
.
9


R
1

=


0

0
.
4

1

0

0



0
.
1

0

0

1

0
.
5



0
.
2

0
.
9

0

0
.
5

1


R
1
(x
1

x
2
)

=

0
.
8


R
1
(x
2

x
5
)

=

0
.
9

But


R
1
(x
1

x
5
)

=

0
.
2



min(
0
.
8
,
0
.
9
)

not

transitive

Fuzzy Tolerance and Equivalence Relation

Value

Assignment

How

to

find

the

membership

values

for

the

relation?

1.
Cartesian

Production

Note
:

you

have

to

know

the

membership

value

for

the


sets!

Will

discuss

in

chapter

4
.

2
.

Y

=

f(x)


X



input

vector





Y



output

vector


3
.

Look

up

table



y
1

y
2

y
3






x
1






x
2






x
3

Fuzzy Tolerance and Equivalence Relation

Value

Assignment

4
.

Linguistic

rule

of

knowledge



chapters

7



9

5
.

Classification



chapter

11

6
.

Similarity

methods

in

data

manipulation


The

more

robust

a

data

set,

the

more

accurate

the

relation

entries!

Cosine Amplitude

X

=

{x
1
,x
2
,

,x
n
}

each

element

is

also

a

vector

X
i

=

{x
i
1
,x
i
2
,

,x
im
}



ij

=


R
(x
i
,x
j
)

It

will

be

n

x

n

symmetric,reflexive


i
.
e
.

a

tolerance

relation!

Note
:

this

relates

to

the

vector

dot

product

for

cosine

function

Cosine Amplitude

Example:



x
1
x
2
x
3
x
4
x
5


xi
1

0.3 0.2 0.1 0.7 0.4


xi
2

0.6 0.4 0.6 0.2 0.6


xi
3

0.1 0.4 0.3 0.1 0

Using the above formula:



1



symm



0.836 1


R
1
=

0.914 0.934 1



0.682 0.6 0.441 1



0.982 0.74 0.818 0.7741 1

Cosine Amplitude



1




symm



0.538 1


R1 =

0.667 0.667 1



0.429 0.333 0.25 1



0.818 0.429 0.538 0.429 1

Computationally simple!

Max
-
min Method:

Other Similarity Methods

Absolute Exponential:

Exponential Similarity Coefficient:

Where, S
k

= any general measure for all the data i.e.
(S
k
)
2


0

Other Similarity Methods

Other methods produce scalar quantities which are similar
to the cosine amplitude, such as the following:

Geometric average minimum:

Scalar Product:


Where:

Other Similarity Methods

Some methods are analogous to popular statistical
quantities, such as:

Correlation Coefficient:

Where:

and

Arithmetic Average Minimum:

Other Similarity Methods

Some methods are based on the inverse relationships,
for example:

Absolute Reciprocal:

Where M is selected to make 0 ≤ r
ij

≤ 1

Absolute subtrahend:

Where c is selected to make 0 ≤ r
ij

≤ 1

Other Similarity Methods

Other

methods

are

nonparametric,

such

as
:

Nonparametric
:


where

x

ik

=

x
ik



x
i

and

x

jk



x
j

n
+

= number of elements > 0 in

{x

i1
x

j1
,x

i2
x

j2
,

,x

im
x

jm
}

n
-

= number of elements < 0 in


{x

i1
x

j1
,x

i2
,x

j2
,

,x

im
,x

jm
}


In the above equations, terms such as x

i1

x

j1

are
products of data elements.

Membership Function

Membership Functions characterize the fuzziness of
fuzzy sets. There are an infinite # of ways to
characterize fuzzy


infinite ways to define fuzzy
membership functions.


Membership function essentially embodies all fuzziness
for a particular fuzzy set, its description is essential to
fuzzy property or operation.

Features of Membership Function

Core
:

comprises

of

elements

x

of

the

universe,

such

that



A
(x)

=

1

Support
:

comprises

of

elements

x

of

universe,

such

that




A
(x)

>

0

Boundaries
:

comprise

the

elements

x

of

the

universe



0

<


A
(x)

<

1

A

normal

fuzzy

set

has

at

least

one

element

with

membership

1

For

fuzzy

set,

if

one

and

only

one

element

has

a

membership

=

1
,

this

element

is

called

as

the

prototype

of

set
.

A

subnormal

fuzzy

set

has

no

element

with

membership=
1
.

Features of Membership Function

Graphically,

Features of Membership Function

A convex fuzzy set has a membership whose value is:


1. strictly monotonically increasing, or


2. strictly monotonically decreasing, or


3. strictly monotonically increasing, then strictly


monotonically decreasing

Or another way to describe:



(y) ≥ min[

(x),

(z)], if x < y < z

If A and B are convex sets, then A


B is also a convex set

Crossover points have membership 0.5

Height of a Fuzzy set is the maximum value of the
membership: max{

A
(x)}

Features of Membership Function

If height <
1
, the fuzzy set is subnormal.

Fuzzy number: like a number is close to
5
. It has to have
the properties:


1
. A must be a normal fuzzy set.


2
.

A must be closed for all

(
0
,
1
].


3
. The support,
0
A must be bounded.