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Fuzzy Sets & Expert Systems in Computer Eng. (1):

Fuzzy SetsPiero P. Bonissone

GE Corporate Research & [email protected]

(adapted from slides by Roger Jang

and Enrique Ruspini)

Fuzzy Sets & Expert Systems in Comp. Eng.: Fuzzy Sets

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Outline

Motivation Fuzzy Sets Basic Concepts

Characteristic Function (Membership Function)

Examples

Notation

Semantics and Interpretations

Related crips sets

Support, Bandwidth, Core, -level cut

Decomposition Theorem Features, Properties, and More Definitions

Convexity, Normality

Cardinality, Measure of Fuzziness, First Moment

MF parametric formulation

Fuzzy Set-theoretic Operations Intersection, Union, Complementation

Numerical Examples

T-norms and T-conormsCopyright 1998, Dr. Piero P. Bonissone, All Rights Reserved

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Energy Strategy Draws Congressional Criticism

... the Administration's national energy policy was criticizedtoday as one sided even by those who vigorously support itsplan for drilling in environmentally fragile areas.

... Mr. Johnston, a strong supporter of more oil and gasdrilling, as well as others forms of energy, said You've justgot to do that for balance; I don't care if you believeit or not.

The white population of Manhattan, including some Hispanicresidents, grew by 20,263 during the decade, to 867,227, a gain

of 3.1 percent. Staten Island added 8509 whites for a total of322,043, a gain of 2.7 percent.

(New York Times, February 22, 1991)

New York City Population Gain Attributed to Immigrant Tide

Copyright 1995, Dr. Enrique H. Ruspini, All Rights Reserved - used with authors permission

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MODELING

p, q

(r

s), t f, 32

. . .

Real WorldPressure

Boundary

Economy

......

Conceptual Model

Conceptualization

Symbolic Model

Interpretation


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Approximate Models

Imprecise, Vague, Uncertain Representations of System Behavior

Classical Models:

If Pressure = 10 ATM, then Volume = 2.5 Cm

Imprecise Models:

If Pressure

5 ATM, then Volume 6 Cm Uncertain Models:

If Pressure

5 ATM, then Prob(Vol = 6 Cm) = 0.9

Vague Models:

If Pressure is HIGH, then Volume is LOW


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Approximate Models and Decision/Control Rules

Vague rules may be used to describe characteristics of thesystem:

If Position(t) is NEAR and Velocity is HIGH, thenPosition(t+1) is MEDIUM,

If Shape is ROUND and Gap is SMALL, thenProbability(Symbol=a) is HIGH,

Usually, the problem-solving goal is the generation ofdecision and control rules:

If Position(t) is LOW and Velocity(t) is HIGH, thenthe Acceleration should be SMALL,

If Probability(Symbol1=a) is LOW andProbability(Symbol2=x) is HIGH, thenProbability(Sequence=ex) is HIGH.


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What is Fuzzy Logic ?

Inferential Approach

Oriented towards System Analysis/DecisionSupport

Utilized to develop Intelligent Automated Systems

Capable of dealing with Vague Information Facilitates development of qualitative models

Extension of Classical Logic using Multiple Truth-Values

Exploits notions of similarity between situations

Based on the Theory of Fuzzy Sets (Zadeh 1965)


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Outline



Examples

Notation Semantics and Interpretations

Related crips sets





MF parametric formulation Fuzzy Logic Operations

Intersection, Union, Complementation

Numerical Examples

T-norms and T-conorms

Copyright 1998, Dr. Piero P. Bonissone, All Rights Reserved

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Fuzzy Sets

Characteristic function of sets A(x) and B(x)

A = { x X| x> 10} Boolean Set A: X {0, 1}B = { x X| x >>10} Fuzzy Set B: X [0, 1]

1

0X10

A(x) B(x)


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Boolean Algebra (Run through)

Assign binary truth value to statements

Combine statements using AND and OR operators

A statement1 true

0 false

A B A B0 0 0

0 1 11 0 0

1 1 1

A B AvB

0 0 0

0 1 11 0 1

1 1 1

A A1 0

0 1

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Fuzzy Sets

Binary Logic vs. Fuzzy Logic:

Sets with crisp and fuzzy boundaries, respectively

A = Set of tall people

Heights510

1.0

Crisp set A

Membershipfunction

Heights510 62

.5

.9

Fuzzy set A

1.0

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Membership Functions (MFs)

Characteristics of MFs: Subjective measures

Not probability functions

MFs

Heights510

.5

.8

.1

ttall in Asia

ttall in the US

ttall in NBA

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Fuzzy Sets

Formal definition:A fuzzy set Ain Xis expressed as a set of ordered pairs:

[ ]A x x x X x XA A= {( , ( ))| ( ) } , : , 0 1

Universe oruniverse of discourse

Fuzzy set

Membership

function(MF)

A fuzzy set is totally characterized by a

membership function (MF).

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Fuzzy Sets with Discrete Universes

Fuzzy set C = desirable city to live in

X = {SF, Boston, Troy} (discrete and nonordered)

C = {(SF, 0.9), (Boston, 0.8), (Troy, 0.6)}

Fuzzy set A = sensible number of children

X = {0, 1, 2, 3, 4, 5, 6} (discrete universe)

A = {(0, .1), (1, .3), (2, .7), (3, 1), (4, .6), (5, .2), (6, .1)}

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Fuzzy Sets with Cont. Universes

Fuzzy set B = about 50 years old

X = Set of positive real numbers (continuous)B = {(x,

B(x)) | x in X}

B

xx

( )

= +

1

150

10

2

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Outline



Examples

Notation


Related crips sets


Decomposition Theorem

Features, Properties, and More Definitions





Numerical Examples


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Examples of Fuzzy Sets

Tall Persons (Height)

Dangerous Maneuvers (Action Sequences) Blonde Individuals (Hair color)

Loud Noises (Sound Intensity)

Large Investments (Money) High Speeds (Speed)

Close Objects (Distance)

Large Numbers (Numbers) Desirable Actions (Decision or Control Space)


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Outline



Examples

Notation


Related crips sets







Numerical Examples


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Denoting Fuzzy Sets

f(x) | x

X

f : X

[0,1] : x

f(x)

f(x1)|x1 + f(x2)|x2 + ... + f(xn )|xn

Memberships Objects/Points


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Alternative Notation

A fuzzy set A can be alternatively denoted asfollows:A x xA

x X

i i

i

= ( ) /

A x xAX

= ( ) /

X is discrete

X is continuous

Note that

and integral signs stand for the union ofmembership grades; / stands for a marker and doesnot imply division.

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Outline



Examples

Notation


Related crips sets







Numerical Examples


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Probability vs. Fuzziness

Randomness:Uncertainty described by tendency

(frequency) of a random variable to take on avalue in a specified regionInterpretations: frequency -> willingness to accept bet(subjective probability)

Fuzziness:

Degree to which the element satisfiesproperties characterized by a fuzzy set.

Interpretations: Possibility -> similarity -> desirability

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Outline



Examples


Related crips sets







Numerical Examples



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Fuzzy Sets

0

1

X

Core(x)

Core(A)=

x| A(x)=

1

}

A


0.5

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Fuzzy Sets

0

1

XSupport

(x)

Support(A)=

x| A(x)>

}

A


0.5

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Fuzzy Sets

0

1

X

Bandwidth

(x)

Bandwidth(A)=

x| A(x)

5

}

A


0.5

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Fuzzy Sets as Collections of Conventional Sets

0

1

X

(x)

A

=

x| A(x )

}

,

A


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Identity Principle (Decomposition Theorem)

A fuzzy set A can be represented by the union ofall its

-cut sets, weighted by their value:

A x A X( ) ( )[ , ]= 0 1


=0.5A

5(16-7(A)]TJ/F81Tf012-12313428.70224.24Tm2Tr0.12w0Tc0Tw0))Tj0122TD.)

C

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Identity Principle (Decomposition Theorem)

X

1

(x)

X

=1 (x) 1 * A1

X

1(x)

0

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0

1

Age10 20 30 40

Young Persons


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Fuzzy Partition

Fuzzy partitions formed by the linguistic valuesyoung, middle aged, and old:

lingmf.m

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Outline



Examples


Related crips sets



Convexity, Normality, Fuzzy Singletons


MF parametric formulation

Fuzzy Logic Operations


Numerical Examples



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Convexity of Fuzzy Sets

A fuzzy set Ais convex if for any in [0, 1],

A A Ax x x x( ( ) ) min( ( ), ( ))1 2 1 21+

Alternatively, Ais convex is all its -cuts areconvex.

convexmf.m

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Normality of Fuzzy Sets

A fuzzy set Ais normal if

H e ig h t A M a x A xx( ) ( )= = 1

0

1

X

Height

(x)A

0.5


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Fuzzy Set Representation of

Crisp Numbers and Crisp Intervals

A crisp number ais represented by a fuzzy singleton

A x

x

x( ) =

1

0

i f = a

i f a

A crisp interval [b,c]is represented by a fuzzy set

0

1

X

(x)A

0.5

a

(x)B

b c

B xx

x b c( )

, ]=

1

0

i f [ b , c ]

i f [


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Scalar Cardinality

0.4

0.2

0.6

0.81.0

X

Count (A) = Card(A) = 5.4


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Fuzzy Cardinality

[ The answer is a fuzzy set in the set of integer numbers ]

|A|(n) = , if exists an - cut with |A|= n,0, otherwise.

0.4

0.2

0.60.8

1.0

Card

1 2 3 4 5 6 7 8 9N


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Measure of Fuzziness

0

1

X

Bandwidth(A) (x)

Measure of Fuzziness = Cardinality {|Bandwidth(A)- A(x)|} = Cardinality { }

A


0.5

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First Moment of a Fuzzy Sets

The First Moment of a discrete Fuzzy Set A(xi) is:

F ir s tM o m e n t A

A x x

A x

i ii

n

ii

n( )

( ) *

( )

= =

=

1

1

The First Moment of a continuous Fuzzy Set A(x) is:

F i r s tM o m e n t AA x x d x

A x d x( )

( ) *

( )=


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