Mf-param-oper2

  • June 2020
  • PDF

This document was uploaded by user and they confirmed that they have the permission to share it. If you are author or own the copyright of this book, please report to us by using this DMCA report form. Report DMCA


Overview

Download & View Mf-param-oper2 as PDF for free.

More details

  • Words: 783
  • Pages: 17
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