Document

An Extension to Kolmogorov Theorem and
Functional Link Network.
Hirota Laboratory
Dr. Lopez Gomez Angel
(現在メキシコ自治大学ＵＮＡＭ助手)
Kolmogorov Theorem
 n

, xn       pq  x p  
q 0
 p 1

2n
y  f  x1 ,
f represents
an arbitrary
continuous
given
function of n
number of
variables,
and each one xi
belongs to the
unit interval
[0,1].
Algorithmic Implementation of the Kolmogorov Theorem.

is a real
continuous
function; its
construction is
pointwise and
dependent of
T h e f u n c t i o n s q ' s
are monotonic,
s t r i c t l y
increasing, and
independent
of f
.
f
The constants  0   p  1, p  1, , n 
are linearly independent
over the rational numbers
Algorithmic Implementation of the Kolmogorov Theorem.
Kolmogorov Theorem
y  f  x1 ,
 n

, xn       pq  x p  
q 0
 p 1

2n
1
0.35
0.9
0.3
1
0.8
0.9
0.25
0.8
0.7
0.7
0.6
0.6
0.4
0.3
 q(x)
(x)
y
0.2
0.5
0.5
0.15
0.4
0.2
0.3
0.1
0.1
0
1
0.2
0.8
1
0.6
0.8
0.1
0.6
0.4
0.4
0.2
x2
0.05
0
0.2
0
0
0
x1
0
0.1
0.2
0.3
0.4
0.5
x
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
x
0.6
0.7
0.8
0.9
1
Extension of the Kolmogorov Theorem.
- Kolmogorov Theorem 1
x  I  [0,1]
y  f  x1,
 n

, xn       pq  x p  
q 0
 p1


2n
 x 

0
Fuzzification
- Extended Kolmogorov Theorem -
A  F (I )
  n
Y  f A1, , An       p q Ap
q 0 
  p1


2n
  

 
 
x
1
Extension of the Kolmogorov Theorem.
[Lopez Gomez (2001)]
Kolmogorov Theorem
Definition of concepts in fuzzy topological spaces
Prove the continuity of  and
 with respect Hausdorff metric
Prove the extended Kolmogorov theorem by mathematical induction
Extended Kolmogorov Theorem
Extension of the Kolmogorov Theorem.
Mappings among
singletons
Kolmogorov Theorem
Extended Kolmogorov Theorem
Mappings among
fuzzy sets
Representation Space of
Arbitrary Continuous Functions
Performance Improvements of
System Modeling and Prediction Systems
Extension of the Kolmogorov Theorem.
Simulation of the projection over the functions

Y  f A1,
q
  n
, An       p q Ap
q 0 
  p1

  
2n

 
 
  A 1
0.8
0.6
0.4
0.2
0.2
0.4
0.6
A
0.8
1
Extension of the Kolmogorov Theorem.
Simulation of the projection over the function

Y  f A1,

  n
, An       p q Ap
q 0 
  p1

  
2n
 Z 
1
0.8
0.6
0.4
0.2
Z

 
 
Fuzzification of the Functional Link Network.
Functional Link Networks are single-layer neural networks that are
able to handle linearly nonseparable task using the appropriately
enhanced input representation.
The principal restriction in the set of functions to transform the input in
the networks is that these must be a subset of orthonormal basis
functions.
Examples of functional link successfully applied in diverse problems are:
x, x2 , x3 , ; x,sin  x,cos  x,sin 2 x,cos 2 x,
The idea of the functional link is very close to that of series expansion. It
is clear that the functional link can implement truncated series.
Fuzzy Functional Link Network.
[Lopez Gomez (2002)]
- Architecture of the Fuzzy Functional Link Network -
Y%
 
% %
fX
11
 
% %
L fX
1
n
 L
% %
fX
21
 
% %
fX
2
n
L
 
% %
fX
k
1
 
% %
L fX
kn
Additional
Processes Induced
Functional Link
X% 1
X% n
Fuzzy Functional Link Network.
[Lopez Gomez (2002)]
Functional Link Network
Description of the fuzzy vector space to consider
Prove that a orthogonal basis is also a fuzzy basis
Prove that this fuzzy basis is fuzzy linearly independent
Fuzzy Functional Link Network
Representation of the Extended Kolmogorov Theorem using FFLN
Extended Kolmogorov Theorem
Kolmogorov
Theorem
Functional Link
Network
Fuzzy Functional Link Network
Representation of the Extended Kolmogorov Theorem with a Neural Network
  
 zq
2n
q=0

 z 0

A1
0


1
2
0
1

A2

 z 1


 z2

1
2
1
 z 3

 1
2 3
2

A1

 z 4

 1
2 4
3

A2

A2
4
Application: The function to be approximated is unknown and is constructed using
expert knowledge.
Prediction System to estimate the best moment to crop some fruits
Input
Process Unit
Color
Moments
Zernike
Moments
Output
Estimated Ripeness
of the Fruit

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