FUZZY SETS - Tun Hussein Onn University of...
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FUZZY SETS
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Membership Value Assignment
• There are possible more ways to assignmembership values or function to fuzzyvariables than there are to assign probabilitydensity functions to random variables [Duboisand Prade, 1980]
Fuzzy Logic with Engineering Applications: Timothy J. Ross
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Membership Value Assignment
• Intuition• Inference• Rank ordering• Angular fuzzy sets• Neural networks• Genetic algorithms• Inductive reasoning• Soft partitioning
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Intutition
• Derived from the capacity of humans todevelop membership functions through theirown innate intelligence and understanding.
• Involves contextual and semantic knowledgeabout an issue; it can also involve linguistictruth values about this knowledge.
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Types of Membership Functions
• The most commonly used in practice are– Triangles– Trapezoids– Bell curves– Gaussian, and– Sigmoidal
Fuzzy Logic:Intelligence, Control, and Information, J. Yen and R. Langari, PrenticeHall
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Triangular MF
Specified by three parameters {a,b,c} as follows:
Fuzzy Logic:Intelligence, Control, and Information, J. Yen and R. Langari, PrenticeHall
triangle(x : a,b,c) =
0 x < a
(x − a) (b − a) a ≤ x ≤ b
(c − x) (c − b) b ≤ x ≤ c
0 x > c
a
b
c
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Trapezoidal MF
Specified by four parameters {a,b,c,d} as follows:
Fuzzy Logic:Intelligence, Control, and Information, J. Yen and R. Langari, PrenticeHall
trapezoidal(x : a,b,c,d) =
0 x < a
(x − a) (b − a) a ≤ x < b
1 b ≤ x < c
(d − x) (d − c) c ≤ x ≤ d
0 x ≥ d
a
b c
d
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Gaussian MF
Specified by two parameters {m,σ} as follows:
Where m and σ denote the center and width of the function, respectivelyA small σ will generate a “thin”MF, while a big σ will lead to a “flat”MF.
Fuzzy Logic:Intelligence, Control, and Information, J. Yen and R. Langari, PrenticeHall
gaussian(x : m,σ ) = exp −(x − m)2
σ 2
m
σ
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Bell-shaped MF
Specified by three parameters {a,b,c} as follows:
Where the parameter b is usually positive and we can adjust c and a to vary thecenter and width of the function and then use b to control the slopes.
Fuzzy Logic:Intelligence, Control, and Information, J. Yen and R. Langari, PrenticeHall
bell(x : a,b,c) =1
1 + x − ca
2b
c
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Bell-shaped MF
Fuzzy Logic:Intelligence, Control, and Information, J. Yen and R. Langari, PrenticeHall
bell(x : a,b,c) =1
1 + x − ca
2b
c
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Sigmoidal MF
Specified by two parameters {a, c} as follows:
Where c is the center of the function and a control the slope.
Fuzzy Logic:Intelligence, Control, and Information, J. Yen and R. Langari, PrenticeHall
Sigmoidal(x : a,c) =1
1+ e− a(x − c)
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Sigmoidal MF
Fuzzy Logic:Intelligence, Control, and Information, J. Yen and R. Langari, PrenticeHall
Sigmoidal(x : a,c) =1
1+ e− a(x− c)
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Hedges: a modifier to a fuzzy set
• Hedge modifies the meaning of the originalset to create a compound fuzzy set
– Example:• Very (Concentration)• More or Less(Dilation)
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Hedges: Very & MoreOrLess
Fuzzy Logic:Intelligence, Control, and Information, J. Yen and R. Langari, PrenticeHall
Very :
µveryA x( ) = µA x( )[ ]2
MoreorLess :
µMoreOrLessA x( ) = µA x( )
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Hedges: Very
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Hedges: VeryVeryVery (Extreme)
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Inference
• Use knowledge to perform deductivereasoning, i.e . we wish to deduce or infer aconclusion, given a body of facts andknowledge.
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Inference : Example• In the identification of a triangle
– Let A, B, C be the inner angles of a triangle• Where A ≥ B≥C
– Let U be the universe of triangles, i.e.,• U = {(A,B,C) | A≥B≥C≥0; A+B+C = 180˚}
– Let ‘s define a number of geometric shapes• I Approximate isosceles triangle• R Approximate right triangle• IR Approximate isosceles and right triangle• E Approximate equilateral triangle• T Other triangles
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Inference : Example• We can infer membership values for all of
these triangle types through the method ofinference, because we possess knowledgeabout geometry that helps us to make themembership assignments.
• For Isosceles,µi (A,B,C) = 1- 1/60* min(A-B,B-C)
– If A=B OR B=C THEN µi (A,B,C) = 1;– If A=120˚,B=60˚, and C =0˚ THEN µi (A,B,C) = 0.
Fuzzy Logic with Engineering Applications: Timothy J. Ross
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Inference : Example• For right triangle,
µR (A,B,C) = 1- 1/90* |A-90˚|– If A=90˚ THEN µi (A,B,C) = 1;– If A=180˚ THEN µi (A,B,C) = 0.
• For isosceles and right triangle– IR = min (I, R)
µIR (A,B,C) = min[µI (A,B,C), µR (A,B,C)] = 1 - max[1/60min(A-B, B-C), 1/90|A-90|]
Fuzzy Logic with Engineering Applications: Timothy J. Ross
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Inference : Example• For equilateral triangle
µE (A,B,C) = 1 - 1/180* (A-C)– When A = B = C then µE (A,B,C) = 1,
A = 180 then µE (A,B,C) = 0
• For all other triangles– T = (I.R.E)’ = I’.R’.E’
= min {1 - µI (A,B,C) , 1 - µR (A,B,C) , 1 - µE (A,B,C)
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Inference : Example
– Define a specific triangle:• A = 85˚ ≥ B = 50˚ ≥ C = 45˚
µR = 0.94 µI = 0.916 µIR = 0.916 µE = 0. 7
µT = 0.05
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Rank ordering
• Assessing preferences by a single individual, acommittee, a poll, and other opinion methodscan be used to assign membership values to afuzzy variable.
• Preference is determined by pairwisecomparisons, and these determine theordering of the membership.
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Rank ordering: Example
Fuzzy Logic with Engineering Applications: Timothy J. Ross