Outline • Motivation • Fuzzy Sets Basic Concepts Characteristic Function (Membership Function) Examples Notation Semantics and Interpretations Related crips sets » Support, Bandwidth, Core, α-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
Membership Function (MF) Formulation Triangular MF:
x −a c−x trim f ( x ; a , b , c ) = max min , , 0 b −a c −b
Trapezoidal MF:
d − x x −a trapm f ( x ; a , b , c , d ) = m ax m in ,1, , 0 b −a d −c
Gaussian MF:
gaussm f ( x ; a , b , c ) = e
Generalized bell MF:
9/1/99
gbellm f ( x ; a , b , c ) =
1 x −c − 2 σ
2
1 x −c 1+ b
2b
2
MF Formulation
disp_mf.m 9/1/99
3
MF Formulation
Sigmoidal MF:
sigm f ( x ; a , b , c ) =
1 1 + e −a( x −c )
Extensions: Abs. difference of two sig. MF
Product of two sig. MF disp_sig.m 9/1/99
4
MF Formulation L-R MF:
Example:
c − x F ,x < c L α L R ( x ; c ,α , β ) = F x − c , x ≥ c R β
FL ( x ) =
max( 0 , 1 − x 2 )
FR ( x ) = exp( − x ) 3
c=25 a=10 b=40
c=65 a=60 b=10 difflr.m 9/1/99
5
Cylindrical Extension
Base set A
Cylindrical Ext. of A
cyl_ext.m 9/1/99
6
2D MF Projection Two-dimensional MF
Projection onto X
Projection onto Y
µ R ( x, y )
µ A( x) = max µ R ( x, y )
µB( y) = max µ R ( x , y )
project.m 9/1/99
y
x
7
2D MFs
2dmf.m 9/1/99
8
Outline • Motivation • Fuzzy Sets Basic Concepts Characteristic Function (Membership Function) Examples Notation Semantics and Interpretations Related crips sets » Support, Bandwidth, Core, α-level cut » Decomposition Theorem – Features, Properties, and More Definitions » Convexity, Normality, Fuzzy Singletons » Cardinality, Measure of Fuzziness, First Moment – Fuzzy Logic Operations » Intersection, Union, Complementation » Numerical Examples » T-norms and T-conorms
– – – – –
Copyright 1998, Dr. Piero P. Bonissone, All Rights Reserved
Intersection of Fuzzy Sets
A
B
1
0 (A ∩ B) (x) = min ( A(x), B(x) ) Copyright 1995, Dr. Enrique H. Ruspini, All Rights Reserved - used with author’s permission
Intersection of Fuzzy Sets
A
1
0 (A ∩ B) (x) = min ( A(x), B(x) ) Copyright 1995, Dr. Enrique H. Ruspini, All Rights Reserved - used with author’s permission
Union of Fuzzy Sets
A
B
1
0 (A ∪ B) (x) = max ( A(x), B(x) ) Copyright 1995, Dr. Enrique H. Ruspini, All Rights Reserved - used with author’s permission
Complement of a Fuzzy Set
bb
¬ A(x)
A(x)
¬ A(x)
1
0 A (x) ≡ ¬ A(x) = 1 - A(x) Copyright 1995, Dr. Enrique H. Ruspini, All Rights Reserved - used with author’s permission
Fuzzy Set Operations 1
A(x)
0 Intersection
B(x)
X
1
(A ∩ B)(x) 0 X Union
1
(A ∪ B)(x) 0
X
Complementation 1 A(x) 0 Copyright 1998, Dr. Piero P. Bonissone, All Rights Reserved
X
Fuzzy Set Operations • Zadeh’s Original Definitions Intersection:
(A ∩ B)(x) = min [A(x), B(x)]
Union:
(A ∪ B)(x) = max [A(x), B(x)]
Complementation
A(x) = 1 - A(x)
• Other Definitions Intersection:
(A ∩ B)(x) = T-norm [A(x), B(x)]
Union:
(A ∪ B)(x) = T-Conorm [A(x), B(x)]
• Isomorphisms between fuzzy sets, algebra, and logics. • Original definitions form a Brouwerian lattice: {Boolean ring properties} but not {excluded middle, law of non-contradiction} • Beside (min, max ) no other pair (T-norms, T-conorm) satisfies distributivity Copyright 1998, Dr. Piero P. Bonissone, All Rights Reserved
Fuzzy Set Inclusion
bb
1 B
A
0
A(x) ≤ B(x) Copyright 1995, Dr. Enrique H. Ruspini, All Rights Reserved - used with author’s permission
Degree of Inclusion
1
1 A B
0
0
(A→B)(x) = max [ 1-A(x), B(x) ] Copyright 1995, Dr. Enrique H. Ruspini, All Rights Reserved - used with author’s permission