Transcript Chapter 02 for Neuro-Fuzzy and Soft Computing

Neuro-Fuzzy and Soft Computing: Fuzzy Sets
Slides for Fuzzy Sets, Ch. 2 of
Neuro-Fuzzy and Soft Computing
J.-S. Roger Jang (張智星)
CS Dept., Tsing Hua Univ., Taiwan
http://www.cs.nthu.edu.tw/~jang
[email protected]
Neuro-Fuzzy and Soft Computing: Fuzzy Sets
Fuzzy Sets: Outline
Introduction
Basic definitions and terminology
Set-theoretic operations
MF formulation and parameterization
• MFs of one and two dimensions
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Neuro-Fuzzy and Soft Computing: Fuzzy Sets
Fuzzy Sets
Sets with fuzzy boundaries
A = Set of tall people
Crisp set A
1.0
Fuzzy set A
1.0
.9
Membership
.5
function
5’10’’
3
Heights
5’10’’ 6’2’’
Heights
Neuro-Fuzzy and Soft Computing: Fuzzy Sets
Membership Functions (MFs)
Characteristics of MFs:
• Subjective measures
• Not probability functions
“tall” in Asia
MFs
.8
“tall” in the US
.5
“tall” in NBA
.1
5’10’’
4
Heights
Neuro-Fuzzy and Soft Computing: Fuzzy Sets
Fuzzy Sets
Formal definition:
A fuzzy set A in X is expressed as a set of ordered
pairs:
A  {( x,  A ( x ))| x  X }
Fuzzy set
Membership
function
(MF)
Universe or
universe of discourse
A fuzzy set is totally characterized by a
membership function (MF).
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Neuro-Fuzzy and Soft Computing: Fuzzy Sets
Fuzzy Sets with Discrete Universes
Fuzzy set C = “desirable city to live in”
X = {SF, Boston, LA} (discrete and nonordered)
C = {(SF, 0.9), (Boston, 0.8), (LA, 0.6)}
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)}
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Neuro-Fuzzy and Soft Computing: Fuzzy Sets
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(x) 
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1
 x  50 
1 

 10 
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Neuro-Fuzzy and Soft Computing: Fuzzy Sets
Alternative Notation
A fuzzy set A can be alternatively denoted as
follows:
X is discrete
X is continuous
A

A
( xi ) / xi
xi X
A    A( x) / x
X
Note that S and integral signs stand for the union of
membership grades; “/” stands for a marker and does
not imply division.
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Neuro-Fuzzy and Soft Computing: Fuzzy Sets
Fuzzy Partition
Fuzzy partitions formed by the linguistic values
“young”, “middle aged”, and “old”:
lingmf.m
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Neuro-Fuzzy and Soft Computing: Fuzzy Sets
More Definitions
Support
Core
Normality
Crossover points
Fuzzy singleton
a-cut, strong a-cut
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Convexity
Fuzzy numbers
Bandwidth
Symmetricity
Open left or right, closed
Neuro-Fuzzy and Soft Computing: Fuzzy Sets
MF Terminology
MF
1
.5
a
0
Core
Crossover points
a - cut
Support
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X
Neuro-Fuzzy and Soft Computing: Fuzzy Sets
Convexity of Fuzzy Sets
A fuzzy set A is convex if for any l in [0, 1],
A ( lx1  ( 1  l ) x2 )  min( A ( x1 ), A ( x2 ))
Alternatively, A is convex is all its a-cuts are
convex.
convexmf.m
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Neuro-Fuzzy and Soft Computing: Fuzzy Sets
Set-Theoretic Operations
Subset:
A  B  A  B
Complement:
A  X  A  A ( x )  1  A ( x )
Union:
C  A  B  c ( x )  max( A ( x ), B ( x ))  A ( x ) B ( x )
Intersection:
C  A  B  c ( x )  min( A ( x ), B ( x ))  A ( x ) B ( x )
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Neuro-Fuzzy and Soft Computing: Fuzzy Sets
Set-Theoretic Operations
subset.m
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fuzsetop.m
Neuro-Fuzzy and Soft Computing: Fuzzy Sets
MF Formulation

Triangular MF:
Trapezoidal MF:

d  x 
x a
trapmf ( x ; a , b , c , d )  max min
, 1,
 , 0
b a
d c  

Gaussian MF:

gaussmf ( x ; a , b , c )  e
Generalized bell MF:
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x a c  x 
,
 , 0
b a c b 
trimf ( x ; a , b , c )  max min 
gbellmf ( x ; a , b , c ) 
1  x c 
 

2  
2
1
x c
1
b
2b
Neuro-Fuzzy and Soft Computing: Fuzzy Sets
MF Formulation
disp_mf.m
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Neuro-Fuzzy and Soft Computing: Fuzzy Sets
MF Formulation
Sigmoidal MF:
sigmf ( 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
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Neuro-Fuzzy and Soft Computing: Fuzzy Sets
MF Formulation
L-R MF:
Example:
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 c  x
 F L  a  , x  c

LR ( x ; c , a ,  )  
 F  x  c  , x  c
R

   
F L ( x )  max( 0 , 1  x 2 )
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FR ( x )  exp( x )
c=65
c=25
a=60
a=10
b=10
b=40
difflr.m
Neuro-Fuzzy and Soft Computing: Fuzzy Sets
Cylindrical Extension
Base set A
Cylindrical Ext. of A
cyl_ext.m
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Neuro-Fuzzy and Soft Computing: Fuzzy Sets
2D MF Projection
Two-dimensional
Projection
Projection
MF
onto X
onto Y
 R ( x, y )
 A( x) 
max  R ( x , y )
B( y) 
max  R ( x , y )
project.m
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y
x
Neuro-Fuzzy and Soft Computing: Fuzzy Sets
2D MFs
2dmf.m
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