fuzzy sets
DESCRIPTION
A set is a collection of objects. A special kind of set. Fuzzy Sets. Jan Jantzen www.inference.dk 2013. Summary: A set . This object is not a member of the set. This object is a member of the set. A classical set has a sharp boundary. ... and a fuzzy set. - PowerPoint PPT PresentationTRANSCRIPT
Fuzzy Sets
Jan Jantzenwww.inference.dk
2013
A set is a collection of objects
A special kind of set
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Summary: A set ...This object is not a member of the set
This object is a member of the set
A classical set has a sharp boundary
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... and a fuzzy set
A fuzzy set has a graded boundary
This object is not a member of the set
This object is a member of the set to a degree, for instance 0.8. The membership is between 0 and 1.
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Example: High and low pressures
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Example: "Find books from around 1980"
This could include 1978, 1979, 1980, 1981, and 1982
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Maybe even 41 or 42 could be all right?
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Example: A fuzzy washing machine
• If you fill it with only a few clothes, it will use shorter time and thus save electricity and water.
Samsung J1045AV capacity 7 kg
There is a computer inside that makes decisions depending on how full the machine is and other information from sensors.
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A rule (implication)• IF the machine is full THEN wash long time
Condition Action.
The internal computer is able to execute an if—then rule even when the condition is only partially fulfilled.
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IF the machine is full …• Example: Load = 3.5 kg clothes
true
false
Three examples of functions that define ‘full’. The horizontal axis is the weight of the clothes, and the vertical axis is the degree of truth of the statement ‘the machine is full’.
Classical set
Linear fuzzy set
Nonlinear fuzzy set
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… THEN wash long time
• long time could be t = 120 minutes
The duration depends on the washing program that the user selects.
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Decision Making (inference)• Rule. IF the machine is full THEN wash long time• Measurement. Load = 3.5 kg • Conclusion. Full(Load) × t = 0.5 × 120 = 60 mins
)(
)(
00
0
xfyxxxfy
Analogy
The dots mean 'therefore'
The machine is only half full, so it washes half the time.
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A rule base with four rules1. IF machine is full AND clothes are dirty THEN wash long time2. IF machine is full AND clothes are not dirty THEN wash medium time3. IF machine is not full AND clothes are dirty THEN wash medium time4. IF machine is not full AND clothes are not dirty THEN wash short time
There are two inputs that are combined with a logical 'and'.
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THEORY OF FUZZY SETS
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Lotfi Zadeh’s Challenge
Clearly, the “class of all real numbers which are much greater than 1,” or “the class of beautiful women,” or “the class of tall men,” do not constitute classes or sets in the usual mathematical sense of these terms (Zadeh 1965).
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Sets
}3,2,1{}3|{ zZz
{Live dinosaurs in British Museum} =
}2,1,0{}2,1,1,0{
The set of
The set of positive integers
belonging tofor which
The empty set
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Fuzzy Sets
}1|)(,{ xxx
{nice days}
{adults}
Membership function
Much greater than
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Tall Persons
150 160 170 180 190 2000
0.2
0.4
0.6
0.8
1
Height [cm]
Mem
bers
hip
fuzzy
crisp
Universe
Degree of membership
Membership function
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Fuzzy (http://www.m-w.com)
adjectiveSynonyms: faint, bleary, dim, ill-defined,
indistinct, obscure, shadowy, unclear, undefined, vague
Unfortunately, they all carry a negative connotation.
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'Around noon'Trapezoidal
Triangular
Smooth versions of the same sets.
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The 4 Seasons
0
0.5
1
Time of the year
Mem
bers
hip
Spring Summer Autumn Winter
we are here
Seasons have overlap; the transition is fuzzy.
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Summary
A set A fuzzy set
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OPERATIONS ON FUZZY SETS
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Set Operations
BA BA BA
Classical
Fuzzy
Union Intersection Negation
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Fuzzy Set Operations
)(),(max xx BA
)(),(min xx BA
)(),(max1 xx BA
A B
x
)(xB
)(xA
Union
Intersection
Negation
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Example: Age
Primary term Primary term
The square of Young
The square root of Old
The negation of 'very young'
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Operations
)(),(max)(
)(),(min)(
)(1)(
)()(
)()(
)()(
)()(
3/1
3
2/1
2
xxx
xxx
xx
xx
xx
xx
xx
BABorA
BABandA
AAnot
AAslightly
AAextremely
AAmorl
AAvery
Here is a whole vocabulary of seven words.
Each operates on a membership function and returns a membership function. They can be combined serially, one after the other, and the result will be a membership function.
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Cartesian ProductThe AND composition of all possible combinations of memberships from A and B
The curves correspond to a cut by a horizontal plane at different levels
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Example: Donald Duck's family
• Suppose,– nephew Huey resembles nephew Dewey– nephew Huey resembles nephew Louie– nephew Dewey resembles uncle Donald– nephew Louie resembles uncle Donald
• Question: How much does Huey resemble Donald?
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Solution: Fuzzy Composition of Relations
Dewey Louie
Huey
Dew
ey Louie
=
Donald
Huey
Donald
0.8 0.9
0.5
0.6
Huey Dewey DonaldHuey Louie Donald
CompositionRelation
Relation
Relation
0.8 0.5
0.9 0.6
?
?
?
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If—Then Rules
1. If x is Neg then y is Neg2. If x is Pos then y is Pos
xy
Rule 2
Rule 1
approximately equal
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Key Concepts
• Universe• Membership function• Fuzzy variables• Set operations• Fuzzy relations • All of the above are parallels to classical
set theory
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Application examples
• Database and WWW searches• Matching of buyers and sellers• Rule bases in expert systems