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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
Copyright 1995, Dr. Enrique H. Ruspini, All Rights Reserved - used with authors permission
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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
Copyright 1995, Dr. Enrique H. Ruspini, All Rights Reserved - used with authors permission
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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.
Copyright 1995, Dr. Enrique H. Ruspini, All Rights Reserved - used with authors permission
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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)
Copyright 1995, Dr. Enrique H. Ruspini, All Rights Reserved - used with authors permission
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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 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)
Copyright 1998, Dr. Piero P. Bonissone, All Rights Reserved
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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
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 Logic Operations
Intersection, Union, Complementation
Numerical Examples
T-norms and T-conormsCopyright 1998, Dr. Piero P. Bonissone, All Rights Reserved
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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)
Copyright 1995, Dr. Enrique H. Ruspini, All Rights Reserved - used with authors permission
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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 Logic Operations
Intersection, Union, Complementation
Numerical Examples
T-norms and T-conormsCopyright 1998, Dr. Piero P. Bonissone, All Rights Reserved
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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
Copyright 1995, Dr. Enrique H. Ruspini, All Rights Reserved - used with authors permission
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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
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 Logic Operations
Intersection, Union, Complementation
Numerical Examples
T-norms and T-conormsCopyright 1998, Dr. Piero P. Bonissone, All Rights Reserved
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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
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 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
0
1
X
Core(x)
Core(A)=
x| A(x)=
1
}
A
Copyright 1998, Dr. Piero P. Bonissone, All Rights Reserved
0.5
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Fuzzy Sets
0
1
XSupport
(x)
Support(A)=
x| A(x)>
}
A
Copyright 1998, Dr. Piero P. Bonissone, All Rights Reserved
0.5
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Fuzzy Sets
0
1
X
Bandwidth
(x)
Bandwidth(A)=
x| A(x)
5
}
A
Copyright 1998, Dr. Piero P. Bonissone, All Rights Reserved
0.5
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Fuzzy Sets as Collections of Conventional Sets
0
1
X
(x)
A
=
x| A(x )
}
,
A
Copyright 1995, Dr. Enrique H. Ruspini, All Rights Reserved - used with authors permission
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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
Copyright 1998, Dr. Piero P. Bonissone, All Rights Reserved
=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
Copyright 1995, Dr. Enrique H. Ruspini, All Rights Reserved - used with authors permission
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Fuzzy Partition
Fuzzy partitions formed by the linguistic valuesyoung, middle aged, and old:
lingmf.m
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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, Fuzzy Singletons
Cardinality, Measure of Fuzziness, First Moment
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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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
Copyright 1998, Dr. Piero P. Bonissone, All Rights Reserved
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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 [
Copyright 1998, Dr. Piero P. Bonissone, All Rights Reserved
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Scalar Cardinality
0.4
0.2
0.6
0.81.0
X
Count (A) = Card(A) = 5.4
Copyright 1995, Dr. Enrique H. Ruspini, All Rights Reserved - used with authors permission
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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
Copyright 1995, Dr. Enrique H. Ruspini, All Rights Reserved - used with authors permission
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Measure of Fuzziness
0
1
X
Bandwidth(A) (x)
Measure of Fuzziness = Cardinality {|Bandwidth(A)- A(x)|} = Cardinality { }
A
Copyright 1998, Dr. Piero P. Bonissone, All Rights Reserved
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( )
( ) *
( )=
Copyright 1998, Dr. Piero P. Bonissone, All Rights Reserved