chapter 2: fuzzy sets introduction (2.1) basic definitions &terminology (2.2) set-theoretic...
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Chapter 2: FUZZY SETS Introduction (2.1)
Basic Definitions &Terminology (2.2)
Set-theoretic Operations (2.3)
Membership Function (MF) Formulation & Parameterization (2.4)
More on Fuzzy Union, Intersection & Complement (2.5)
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BİL 561 Fuzzy Neural Networks Ch.2: Fuzzy setsDr. Yusuf OYSAL
Introduction (2.1)
Sets with fuzzy boundaries
A = Set of tall people
Heights5’10’’
1.0
Crisp set A
Membershipfunction
Heights5’10’’ 6’2’’
.5
.9
Fuzzy set A1.0
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BİL 561 Fuzzy Neural Networks Ch.2: Fuzzy setsDr. Yusuf OYSAL
Membership Functions (MFs)
Characteristics of MFs:Subjective measuresNot probability functions
MFs
Heights
.8
“tall” in Asia
.5 “tall” in the US
5’10’’
.1 “tall” in NBA
Introduction (2.1) (cont.)
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BİL 561 Fuzzy Neural Networks Ch.2: Fuzzy setsDr. Yusuf OYSAL
Basic definitions & Terminology (2.2)
Formal definition:A fuzzy set A in X is expressed as a set of ordered
pairs:
A x x x XA {( , ( ))| }
Universe oruniverse of discourse
Fuzzy setMembership
function(MF)
A fuzzy set is totally characterized by aA fuzzy set is totally characterized by amembership function (MF).membership function (MF).
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Dr. Yusuf OYSAL
Fuzzy Sets with Discrete Universes
Fuzzy set C = “desirable city to live in”X = {SF, Boston, LA} (discrete and non-ordered)
C = {(SF, 0.9), (Boston, 0.8), (LA, 0.6)}
(subjective membership values!)
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)}
(subjective membership values!)
Basic definitions & Terminology (2.2) (cont.)
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BİL 561 Fuzzy Neural Networks Ch.2: Fuzzy setsDr. Yusuf OYSAL
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
15010
2
Basic definitions & Terminology (2.2) (cont.)
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BİL 561 Fuzzy Neural Networks Ch.2: Fuzzy setsDr. Yusuf OYSAL
Alternative Notation
A fuzzy set A can be alternatively denoted as follows:
A x xA
X
( ) /X is continuous
A x xAx X
i i
i
( ) /
X is discrete
Note that and integral signs stand for the union of membership grades; “/” stands for a marker and does not imply division.
Basic definitions & Terminology (2.2) (cont.)
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BİL 561 Fuzzy Neural Networks Ch.2: Fuzzy setsDr. Yusuf OYSAL
Fuzzy Partition
Fuzzy partitions formed by the linguistic values “young”, “middle aged”, and “old”:
Basic definitions & Terminology (2.2) (cont.)
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BİL 561 Fuzzy Neural Networks Ch.2: Fuzzy setsDr. Yusuf OYSAL
Support(A) = {x X | A(x) > 0}
Core(A) = {x X | A(x) = 1}
Normality: core(A) A is a normal fuzzy set
Crossover(A) = {x X | A(x) = 0.5}
- cut: A = {x X | A(x) }
Strong - cut: A’ = {x X | A(x) > }
Basic definitions & Terminology (2.2) (cont.)
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BİL 561 Fuzzy Neural Networks Ch.2: Fuzzy setsDr. Yusuf OYSAL
Crossover points
Support
- cut
Core
MF
X0
.5
1
MF Terminology
Basic definitions & Terminology (2.2) (cont.)
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BİL 561 Fuzzy Neural Networks Ch.2: Fuzzy setsDr. Yusuf OYSAL
Convexity of Fuzzy Sets
A fuzzy set A is convex if for any in [0, 1],
A A Ax x x x( ( ) ) min( ( ), ( ))1 2 1 21
Alternatively, A is convex if all its -cuts are convex.
Basic definitions & Terminology (2.2) (cont.)
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BİL 561 Fuzzy Neural Networks Ch.2: Fuzzy setsDr. Yusuf OYSAL
Fuzzy numbers: a fuzzy number A is a fuzzy set in IR that satisfies normality & convexity
Bandwidths: for a normal & convex set, the bandwidth is the distance between two unique crossover points
Width(A) = |x2 – x1|
With A(x1) = A(x2) = 0.5
Symmetry: a fuzzy set A is symmetric if its MF is symmetric around a certain point x = c, namely
A(x + c) = A(c – x) x X
Basic definitions & Terminology (2.2) (cont.)
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BİL 561 Fuzzy Neural Networks Ch.2: Fuzzy setsDr. Yusuf OYSAL
Open left, open right, closed:
0)x(lim)x(lim set A fuzzy closed
1)x(lim and 0)x(lim set A fuzzyright open
0)x(lim and 1)x(lim set A fuzzyleft open
Ax
A-x
Ax
A-x
Ax
A-x
Basic definitions & Terminology (2.2) (cont.)
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BİL 561 Fuzzy Neural Networks Ch.2: Fuzzy setsDr. Yusuf OYSAL
Set-Theoretic Operations (2.3)
Subset:A B A B
C A B x x x x xc A B A B ( ) max( ( ), ( )) ( ) ( )
C A B x x x x xc A B A B ( ) min( ( ), ( )) ( ) ( )
A X A x xA A ( ) ( )1
Complement:
Union:
Intersection:
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BİL 561 Fuzzy Neural Networks Ch.2: Fuzzy setsDr. Yusuf OYSAL
Set-Theoretic Operations (2.3) (cont.)
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BİL 561 Fuzzy Neural Networks Ch.2: Fuzzy setsDr. Yusuf OYSAL
MF Formulation & Parameterization (2.4)
Triangular MF: trimf x a b cx ab a
c xc b
( ; , , ) max min , ,
0
trapmf x a b c dx ab a
d xd c
( ; , , , ) max min , , ,
1 0
gbellmf x a b cx cb
b( ; , , )
1
12
2cx
2
1
e),c;x(gaussmf
Trapezoidal MF:
Gaussian MF:
Generalized bell MF:
MFs of One Dimension
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BİL 561 Fuzzy Neural Networks Ch.2: Fuzzy setsDr. Yusuf OYSAL
MF Formulation & Parameterization (2.4) (cont.)
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BİL 561 Fuzzy Neural Networks Ch.2: Fuzzy setsDr. Yusuf OYSAL
Change of parameters in the generalized bell MF
MF Formulation & Parameterization (2.4) (cont.)
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BİL 561 Fuzzy Neural Networks Ch.2: Fuzzy setsDr. Yusuf OYSAL
Physical meaning of parameters in a generalized bell MF
MF Formulation & Parameterization (2.4) (cont.)
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BİL 561 Fuzzy Neural Networks Ch.2: Fuzzy setsDr. Yusuf OYSAL
Gaussian MFs and bell MFs achieve smoothness, they are unable to specify asymmetric Mfs which are important in many applications
Asymmetric & close MFs can be synthesized using either the absolute difference or the product of two sigmoidal functions
MF Formulation & Parameterization (2.4) (cont.)
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BİL 561 Fuzzy Neural Networks Ch.2: Fuzzy setsDr. Yusuf OYSAL
Sigmoidal MF:)cx(ae1
1)c,a;x(sigmf
Extensions:
Product
of two sig. MF
Abs. difference
of two sig. MF
MF Formulation & Parameterization (2.4) (cont.)
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BİL 561 Fuzzy Neural Networks Ch.2: Fuzzy setsDr. Yusuf OYSAL
A sigmoidal MF is inherently open right or left & thus, it is appropriate for representing concepts such as “very large” or “very negative”
Sigmoidal MF mostly used as activation function of artificial neural networks (NN)
A NN should synthesize a close MF in order to simulate the behavior of a fuzzy inference system
MF Formulation & Parameterization (2.4) (cont.)
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BİL 561 Fuzzy Neural Networks Ch.2: Fuzzy setsDr. Yusuf OYSAL
Left –Right (LR) MF:
LR x cF
c xx c
Fx c
x c
L
R
( ; , , ),
,
Example: F x xL ( ) max( , ) 0 1 2 F x xR ( ) exp( ) 3
c=65
a=60
b=10
c=25
a=10
b=40
MF Formulation & Parameterization (2.4) (cont.)
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BİL 561 Fuzzy Neural Networks Ch.2: Fuzzy setsDr. Yusuf OYSAL
The list of MFs introduced in this section is by no means exhaustive
Other specialized MFs can be created for specific applications if necessary
Any type of continuous probability distribution functions can be used as an MF
MF Formulation & Parameterization (2.4) (cont.)
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BİL 561 Fuzzy Neural Networks Ch.2: Fuzzy setsDr. Yusuf OYSAL
MFs of two dimensions
In this case, there are two inputs assigned to an MF: this MF is a twp dimensional MF. A one input MF is called ordinary MF
Extension of a one-dimensional MF to a two-dimensional MF via cylindrical extensions
If A is a fuzzy set in X, then its cylindrical extension in X*Y is a fuzzy set C(A) defined by:
C(A) can be viewed as a two-dimensional fuzzy setY*X
A )y,x(|)x()A(C
MF Formulation & Parameterization (2.4) (cont.)
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BİL 561 Fuzzy Neural Networks Ch.2: Fuzzy setsDr. Yusuf OYSAL
Cylindrical extension
Base set A Cylindrical Ext. of A
MF Formulation & Parameterization (2.4) (cont.)
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BİL 561 Fuzzy Neural Networks Ch.2: Fuzzy setsDr. Yusuf OYSAL
Projection of fuzzy sets (decrease dimension)
Let R be a two-dimensional fuzzy set on X*Y. Then the projections of R onto X and Y are defined as:
and
respectively.
x|)y,x(maxRX
Ry
X
YR
xY y|)y,x(maxR
MF Formulation & Parameterization (2.4) (cont.)
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BİL 561 Fuzzy Neural Networks Ch.2: Fuzzy setsDr. Yusuf OYSAL
Two-dimensional
MF
Projection
onto X
Projection
onto Y
MF Formulation & Parameterization (2.4) (cont.)
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BİL 561 Fuzzy Neural Networks Ch.2: Fuzzy setsDr. Yusuf OYSAL
Composite & non-composite MFs
Suppose that the fuzzy A = “(x,y) is near (3,4)” is defined by:
This two-dimensional MF is composite The fuzzy set A is composed of two statements:
“x is near 3” & “y is near 4”
)1,4;y(G*)2,3;x(G
14y
exp2
3xexp
4y2
3xexp)y,x(
22
22
A
MF Formulation & Parameterization (2.4) (cont.)
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BİL 561 Fuzzy Neural Networks Ch.2: Fuzzy setsDr. Yusuf OYSAL
These two statements are respectively defined as:
near 3 (x) = G(x;3,2) & near 4 (x) = G(y;4,1)
If a fuzzy set is defined by:
it is non-composite.
A composite two-dimensional MF is usually the result of two statements joined by the AND or OR connectives.
5.2A|4y||3x|1
1)y,x(
MF Formulation & Parameterization (2.4) (cont.)
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BİL 561 Fuzzy Neural Networks Ch.2: Fuzzy setsDr. Yusuf OYSAL
Composite two-dimensional MFs based on min & max operations
Let trap(x) = trapezoid (x;-6,-2,2,6) trap(y) = trapezoid (y;-6,-2,2,6)
be two trapezoidal MFs on X and Y respectively
By applying the min and max operators, we obtain two-dimensional MFs on X*Y.
MF Formulation & Parameterization (2.4) (cont.)
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BİL 561 Fuzzy Neural Networks Ch.2: Fuzzy setsDr. Yusuf OYSAL
Two dimensional MFs defined by the min and max operators
MF Formulation & Parameterization (2.4) (cont.)
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BİL 561 Fuzzy Neural Networks Ch.2: Fuzzy setsDr. Yusuf OYSAL
More on Fuzzy Union, Intersection & Complement (2.5)
Fuzzy complement
Another way to define reasonable & consistent operations on fuzzy sets
General requirements:
Boundary: N(0)=1 and N(1) = 0Monotonicity: N(a) > N(b) if a < bInvolution: N(N(a) = a
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BİL 561 Fuzzy Neural Networks Ch.2: Fuzzy setsDr. Yusuf OYSAL
More on Fuzzy Union, Intersection & Complement (2.5) (cont.)
Two types of fuzzy complements:
Sugeno’s complement:
(Family of fuzzy complement operators)
Yager’s complement:
N a aww w( ) ( ) / 1 1
N aa
sas( )1
1(s > -1)
(w > 0)
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BİL 561 Fuzzy Neural Networks Ch.2: Fuzzy setsDr. Yusuf OYSAL
N aa
sas( )1
1
Sugeno’s complement:
N a aww w( ) ( ) / 1 1
Yager’s complement:
More on Fuzzy Union, Intersection & Complement (2.5) (cont.)
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BİL 561 Fuzzy Neural Networks Ch.2: Fuzzy setsDr. Yusuf OYSAL
Fuzzy Intersection and Union:
The intersection of two fuzzy sets A and B is specified in general by a function
T: [0,1] * [0,1] [0,1] with
This class of fuzzy intersection operators are called T-norm (triangular) operators.
T. function the for operator binary a is * where
)x(*)x()x(),x(T)x(~
B
~
ABABA
More on Fuzzy Union, Intersection & Complement (2.5) (cont.)
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BİL 561 Fuzzy Neural Networks Ch.2: Fuzzy setsDr. Yusuf OYSAL
T-norm operators satisfy:
Boundary: T(0, 0) = 0, T(a, 1) = T(1, a) = aCorrect generalization to crisp sets
Monotonicity: T(a, b) < T(c, d) if a < c and b < dA decrease of membership in A & B cannot increase a membership in A B
Commutativity: T(a, b) = T(b, a)T is indifferent to the order of fuzzy sets to be combined
Associativity: T(a, T(b, c)) = T(T(a, b), c)Intersection is independent of the order of pairwise groupings
More on Fuzzy Union, Intersection & Complement (2.5) (cont.)
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BİL 561 Fuzzy Neural Networks Ch.2: Fuzzy setsDr. Yusuf OYSAL
T-norm (cont.)
Four examples (page 37):
Minimum: Tm(a, b) = min (a,b) = a b
Algebraic product: Ta(a, b) = ab
Bounded product: Tb(a, b) = 0 V (a + b – 1)
Drastic product: Td(a, b) =
1ba, if 0,
1 a if b,
1b if ,a
More on Fuzzy Union, Intersection & Complement (2.5) (cont.)
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BİL 561 Fuzzy Neural Networks Ch.2: Fuzzy setsDr. Yusuf OYSAL
T-norm Operator
Minimum:Tm(a, b)
Algebraicproduct:Ta(a, b)
Boundedproduct:Tb(a, b)
Drasticproduct:Td(a, b)
More on Fuzzy Union, Intersection & Complement (2.5) (cont.)
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BİL 561 Fuzzy Neural Networks Ch.2: Fuzzy setsDr. Yusuf OYSAL
T-conorm or S-norm
The fuzzy union operator is defined by a function
S: [0,1] * [0,1] [0,1]
wich aggregates two membership function as:
where s is called an s-norm satisfying:
Boundary: S(1, 1) = 1, S(a, 0) = S(0, a) = aMonotonicity: S(a, b) < S(c, d) if a < c and b < dCommutativity: S(a, b) = S(b, a)Associativity: S(a, S(b, c)) = S(S(a, b), c)
)x()x()x(),x(S B
~
ABABA
More on Fuzzy Union, Intersection & Complement (2.5) (cont.)
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BİL 561 Fuzzy Neural Networks Ch.2: Fuzzy setsDr. Yusuf OYSAL
T-conorm or S-norm (cont.)
Four examples (page 38):
Maximum: Sm(a, b) = max(a,b) = a V b
Algebraic sum: Sa(a, b) = a + b - ab
Bounded sum: Sb(a, b) = 1 (a + b)
Drastic sum: Sd(a, b) =
0ba, if 1,
0a if b,
0b if ,a
More on Fuzzy Union, Intersection & Complement (2.5) (cont.)
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BİL 561 Fuzzy Neural Networks Ch.2: Fuzzy setsDr. Yusuf OYSAL
T-conorm or S-norm
Maximum:Sm(a, b)
Algebraicsum:
Sa(a, b)
Boundedsum:
Sb(a, b)
Drasticsum:
Sd(a, b)
More on Fuzzy Union, Intersection & Complement (2.5) (cont.)
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BİL 561 Fuzzy Neural Networks Ch.2: Fuzzy setsDr. Yusuf OYSAL
Generalized DeMorgan’s Law
T-norms and T-conorms are duals which support the generalization of DeMorgan’s law:T(a, b) = N(S(N(a), N(b)))S(a, b) = N(T(N(a), N(b)))
Tm(a, b)Ta(a, b)Tb(a, b)Td(a, b)
Sm(a, b)Sa(a, b)Sb(a, b)Sd(a, b)
More on Fuzzy Union, Intersection & Complement (2.5) (cont.)
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BİL 561 Fuzzy Neural Networks Ch.2: Fuzzy setsDr. Yusuf OYSAL
Parameterized T-norm and T-conorm
Parameterized T-norms and dual T-conorms have been proposed by several researchers:
YagerSchweizer and SklarDubois and PradeHamacherFrankSugenoDombi
More on Fuzzy Union, Intersection & Complement (2.5) (cont.)