Himpunan fuzzy i
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Transcript of Himpunan fuzzy i
Fuzzy SetsFuzzy Sets
Referensi : Neuro Fuzzy and Soft Computing J.-S. Roger Jang
Dr. Djamel Bouchaffra CSE 513 Soft Computing, Ch.2: Fuzzy sets
Intelligent System
OutlineOutlineIntroductionBasic definitions and terminologySet-theoretic operationsMF formulation and parameterization
MFs of one and two dimensionsDerivatives of parameterized MFs
More on fuzzy union, intersection, and complement
Fuzzy complement Fuzzy intersection and unionParameterized T-norm and T-conorm
Dr. Djamel Bouchaffra CSE 513 Soft Computing, Ch.2: Fuzzy sets 2
Dr. Djamel Bouchaffra CSE 513 Soft Computing, Ch.2: Fuzzy sets 3
Conventional (Boolean) Set Theory:Conventional (Boolean) Set Theory:
Fuzzy Set TheoryFuzzy Set Theory
© INFORM 1990-1998 Slide 3
“Strong Fever”
40.1°C40.1°C
42°C42°C
41.4°C41.4°C
39.3°C39.3°C
38.7°C
37.2°C
38°C
Fuzzy Set Theory:Fuzzy Set Theory:
40.1°C40.1°C
42°C42°C
41.4°C41.4°C
39.3°C39.3°C
38.7°C
37.2°C
38°C
“More-or-Less” Rather Than “Either-Or” !
“Strong Fever”
Introduction (2.1) Sets with fuzzy boundaries
Dr. Djamel Bouchaffra CSE 513 Soft Computing, Ch.2: Fuzzy sets 4
A = Set of tall people
Heights5’10’’
1.0
Crisp set A
Membershipfunction
Heights5’10’’ 6’2’’
.5
.9
Fuzzy set A1.0
Membership Functions (MFs)
Characteristics of MFs:○ Subjective measures○ Not probability functions
Dr. Djamel Bouchaffra CSE 513 Soft Computing, Ch.2: Fuzzy sets 5
MFs
Heights
.8
“tall” in Asia
.5 “tall” in the US
5’10’’.1 “tall” in NBA
Basic definitions & Terminology (2.2)
Formal definition:A fuzzy set A in X is expressed as a set of ordered
pairs:
Dr. Djamel Bouchaffra CSE 513 Soft Computing, Ch.2: Fuzzy sets 6
A x x x XA {( , ( ))| }
Universe oruniverse of discourse
Fuzzy set Membershipfunction
(MF)
A fuzzy set is totally characterized by amembership function (MF).
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!)
Dr. Djamel Bouchaffra 7
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}
Dr. Djamel Bouchaffra CSE 513 Soft Computing, Ch.2: Fuzzy sets 8
B xx
( )
1
1 5010
2
Basic definitions & Terminology
Alternative Notation
A fuzzy set A can be alternatively denoted as follows:
Dr. Djamel Bouchaffra CSE 513 Soft Computing, Ch.2: Fuzzy sets 9
A x xAX
( ) /X is continuous
A x xAx X
i ii
( ) /
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
Fuzzy Partition
Fuzzy partitions formed by the linguistic values “young”, “middle aged”, and “old”:
Dr. Djamel Bouchaffra CSE 513 Soft Computing, Ch.2: Fuzzy sets 10
Basic definitions & Terminology
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) > }
Dr. Djamel Bouchaffra CSE 513 Soft Computing, Ch.2: Fuzzy sets 11
Dr. Djamel Bouchaffra CSE 513 Soft Computing, Ch.2: Fuzzy sets 12
Crossover points
Support
- cut
Core
MF
X0
.5
1
MF Terminology
Convexity of Fuzzy Sets
A fuzzy set A is convex if for any in [0, 1],
Dr. Djamel Bouchaffra CSE 513 Soft Computing, Ch.2: Fuzzy sets 13
A A Ax x x x( ( ) ) min( ( ), ( ))1 2 1 21
Alternatively, A is convex if all its -cuts are convex.
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
Dr. Djamel Bouchaffra CSE 513 Soft Computing, Ch.2: Fuzzy sets 14
Basic definitions & Terminology
Open left, open right, closed:
Dr. Djamel Bouchaffra CSE 513 Soft Computing, Ch.2: Fuzzy sets 15
0)x(lim)x(lim set A fuzzy closed
1)x(lim and 0)x(lim set A fuzzyright open
0)x(limand1)x(limset A fuzzyleft open
AxA-x
AxA-x
AxA-x
Set-Theoretic Operations (2.3) Subset:
Dr. Djamel Bouchaffra CSE 513 Soft Computing, Ch.2: Fuzzy sets 16
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:
MF Formulation & Parameterization
Triangular MF:
Dr. Djamel Bouchaffra CSE 513 Soft Computing, Ch.2: Fuzzy sets
trim f x a b cx a
b a
c x
c b( ; , , ) max m in , ,
0
trapm f x a b c dx a
b a
d x
d c( ; , , , ) m ax m in , , ,
1 0
gbellm f x a b cx c
b
b( ; , , )
1
1
2
2cx21
e),c;x(gaussmf
Trapezoidal MF:
Gaussian MF:
Generalized bell MF:
MFs of One Dimension
Change of parameters in the generalized bell MF
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MF Formulation & Parameterization
Dr. Djamel Bouchaffra CSE 513 Soft Computing, Ch.2: Fuzzy sets 21
Physical meaning of parameters in a generalized bell MF
MF Formulation & Parameterization
○ 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
Dr. Djamel Bouchaffra CSE 513 Soft Computing, Ch.2: Fuzzy sets 22
MF Formulation & Parameterization
○ Sigmoidal MF:
Dr. Djamel Bouchaffra CSE 513 Soft Computing, Ch.2: Fuzzy sets 23
)cx(ae11)c,a;x(sigmf
Extensions:
Productof two sig. MF
Abs. differenceof two sig. MF
MF Formulation & Parameterization
○ 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
Dr. Djamel Bouchaffra CSE 513 Soft Computing, Ch.2: Fuzzy sets 24
MF Formulation & Parameterization
○ Left –Right (LR) MF:
Dr. Djamel Bouchaffra CSE 513 Soft Computing, Ch.2: Fuzzy sets 25
LR x c
Fc x
x c
Fx c
x c
L
R
( ; , , )
,
,
Example: F x xL ( ) max( , ) 0 1 2 F x xR ( ) exp( ) 3
c=65a=60b=10
c=25a=10b=40
MF Formulation & Parameterization
○ 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
Dr. Djamel Bouchaffra CSE 513 Soft Computing, Ch.2: Fuzzy sets 26
MF Formulation & Parameterization