A New Fuzzy Neural Network with Trapezoidal Fuzzy Weights

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Oct 20, 2013 (3 years and 7 months ago)

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A New Fuzzy Neural
Network
with Trapezoidal Fuzzy Weights


Jee
-
Haeng Lee and Sung
-
Bae Cho


Computer Science Department, Yonsei University

134 Shinchon
-
dong, Sudaemoon
-
ku, Seoul 120
-
749, Korea

Phone: +82
-
2
-
361
-
2720 Fax: +82
-
2
-
365
-
2579

Email: [easygo, sbcho
]@candy.yonsei.ac.kr


Key Word:

fuzzy neural network, trapezoidal membership function, learning algorithm



Abstract

Fuzzy neural networks can be used to solve the
problems that have both fuzzy number and real number
information
. T
his paper

propose
s

a
new

learning
algorithm of fuzzy neural networks with trapezoidal
fuzzy weights. We construct
the
trapezoidal fuzzy
weights by composition of two triangles, and devise a
learning algorithm using the two triangular membership
functions.
This model
can
learn

a v
ariety

of

fuzzy
numbers efficiently.
The results of computer simulations
on numerical data show that the
proposed fuzzy neural
networks produce better results than the conventional
methods.



1.

I
ntroduction


N
eural networks have been widely applied to solve
various problems.
However, conventional
neural
networks can handle only numerical data represented by
real number vectors. Therefore, it is very difficult to
represent linguistic information from human expert
s

and
ambiguous information
with conventional

ne
ural
networks. To express linguistic and ambiguous
information, fuzzy set theory
has been

used in various
ways

[1]
.
Since
fuzzy neural networks can use fuzzy
numbers as well as real numbers, and represent linguistic
information better than standard neural
networks
, they

can be applied
to

many problems.

Various approaches to
construct fuzzy neural networks have been proposed
,

but

there are few architecture
s

and learning algorithms
that
can be used as standard models [2]
.

This paper proposes a learning algor
ithm and
architecture of

fuzzy neural networks which utilize
trapezoidal fuzzy numbers as weights. We construct a
trapezoid as composition of two triangles, and devise a
learning algorithm
with the two

triangular membership
functions

effectively
. The main
reason of using new
weights is to solve the problems that have triangular and
trapezoidal fuzzy numbers
together
for inputs and
outputs. This paper shows new architecture and learning
algorithm of
the
fuzzy neural network.
W
e show
simulation results
that c
ompares different fuzzy neural
networks for
learning fuzzy
numbers.



2.

Fuzzy Neural Networks


2.1 Architecture


The input
-
output relation of a two
-
layer feed
-
forward neural network can be formulated as follows.
This neural network has input, hidden, and o
utput units.
The fuzzy weights and fuzzy biases are trapezoidal fuzzy
numbers, and the inputs and target outputs can be any
shape of fuzzy numbers [
3,4
]. Fig. 1 shows the
architecture of this fuzzified neural network.



















Fig. 1. Architectur
e of fuzzy neural network.


Input units:


,




Hidden units:


,

n
H

1

i

n
I

j

1

n
O

k

1

Input
units

Hidden units

Output
units

Fuzzy weights
W
ji

Fuzzy biases

Fuzzy weights
W
kj

Fuzzy biases



Output units
:


,



where
X
pi

is a fuzzy input,
W
ji

and
W
kj

are trapezoidal
fuzzy weights, and

j

and

k

are trapezoidal fuzzy biases.


2.2 Fuzzy Numbers


The operations of fuzzy numbers can be defined by
the extension principle of Zadeh. Our a
pproach is based
on the level sets of fuzzy numbers where arbitrary
number of
h

can be used. The fuzzy outputs are
numerically calculated by interval arithmetic for level
sets of fuzzy weights and fuzzy inputs [1]. For example,
Fig. 2 represents a fuzzy nu
mber denoted by 6 level sets
where
h

is 0.0, 0.2,

, 1.0.











Fig. 2. Fuzzy numbers in fuzzy neural network.



3. Learning Algorithm


3.1 Cost Function


Let
T
p

be the
n
O

dimensional fuzzy target output and
O
pk

be the actual output. Then, the cost f
unction for the
k
th output of level
h

is as follows.






Then cost function of
h

level sets is



Hence, we can write the cost function of (
X
p
,

T
p
) as




3.2 Derivation of Learning Algorithm


In this section we derive a learning algorithm of the
neural network from the cost function described in the
previous section. The fuzzy weights and biases are
adjusted to minimize the value of cost functi
on. Since the
adjustment should not distort the trapezoidal shape of
weights and biases, we do not change the
h
-
level set of
the fuzzy weights independently [
3,4
]. In this paper, we
construct a trapezoid as composition of two symmetric
triangles (shown in
Fig. 3). We can update each
trapezoidal fuzzy weights without destructing its shape
by the movement of the two triangles and the change of
their width. Then, let the trapezoidal fuzzy weights have
six parameters as follows.





Fig. 3 shows the parameters of trapezoidal fuzzy weights.












Fig. 3. Trapezoidal fuzzy weights.



We derive a rule of adjustment of the weights for
each parameter. According to the standard back
-
propagation algorithm, parameters

are updated by the
following formula, and the other weights are updated in
the same way.







1

0

Membership

Membe
r
ship

1

x

0

h

where
t

is the index of number of iterations through
learning,


is
positive real constant, and


is positive real
constant less than 1.0. Then the derivatives are written as
follows.






and





where



.


In this way, the amount of adjustment is calculated.
Finally, the weights can be updated as follows.






Since t
he two triangles in a trapezoidal fuzzy weight are
symmetric,
can be calculated easily.


and

are modified in the same way. An
example for the adjustment of weights is illustrated in
Fig.

4. All the weights and biases of the fuzzy neural
network are modified in this way.









Fig. 4. Modification of weight.


When fuzzy neural network is updated, any weights
should hold following constraints.

,

for all
,
.

If the inequalities are not satisfied after updated, the
modification the two weights are swaped.



4. Simulation Results


Fuzzy neural networks can learn fuzzy numbers. We
construct various fu
zzy numbers that have shapes of
triangle only, trapezoid only, and the mixture of both
triangle and trapezoid. Every experiment has done on the
same condition that learning rate is 0.5, momentum
constant is 0.5, 10 hidden nodes and 6
-
level sets, 0.0, 0.2,

, 1.0. The neural networks have been learnedfor 3000
times to make a comparison with other models.

First experiment is to learn simple triangular fuzzy
numbers. Let us consider next fuzzy if
-
then rules :


If
x

is small then
y

is small.

If
x

is large then
y

is large.


Fig. 5(a) shows the membership functions of
the
fuzzy
numbers, and Fig. 5(b) shows actual outputs.


(a) Membership functions.


(b) Actual outputs.

Fig. 5. Learning triangular fuzzy numbers.


Next experiment is to learn simple triangular fuzzy
numbers. Let us consider next fuzzy if
-
then rules :


If
x

is medium small then
y

is medium small.

If
x

is medium large then
y

is medium large.


Fig. 6(a) shows membership functions of the fuzzy
numbers, a
nd Fig. 6(b) shows actual outputs.


Final

experiment is to learn
the mixed

fuzzy
1

h

0

1

h

0

numbers

of triangle and trapzoid
. Let us consider next
fuzzy if
-
then rules :


If
x

is medium small then
y

is medium small.

If
x

is medium large then
y

is medium large.

If
x

is

middle

then
y

is
m
i
ddle
.


Fig. 7(a) shows the membership functions of
the
fuzzy
numbers, and Fig. 7(b) shows actual outputs.


(a) Membership functions.


(b) Actual outputs.

Fig. 6. Learning trapezoidal f
uzzy numbers.


(a)

Membership functions.


(b) Actual outputs.

Fig.
7
.
Learning

mixed fuzzy numbers.


Regarding the modeling power, we cannot conclude that
the proposed fuzzy neural network is better than othe
r
neural network with pure triangular and trapezoidal
fuzzy weights. How
e
ver,
with re
s
pect to the mean
-
squared error,
our model is better than other models.
Table 1 shows the mean
-
squared errors after 3000
iteration
s
.

For further information, refer to our
recent
publication

[
5
]
.


Types of
w
eight

Shapes of
f
uzzy numbers

for inputs and outputs

Triangle

Trapezoid

Mixture

Triangle

0.00083

0.00762

0.00860

Trapezoid

0.00103

0.00866

0.00969

Proposed

0.00148

0.00656

0.00796

Table 1. Mean
-
squared errors for le
arning

various shapes of fuzzy numbers.



5. Conclusions


In this paper, we propose a new fuzzy neural
network with trapezoidal fuzzy weights which are made
by two triangular fuzzy weights
,

and derive a learning
algorithm for the neural network.
With

compu
ter
simulations, our model has high fitting ability for more
complex fuzzy numbers.

Fuzzy neural network can learn both numerical
and
linguistic values represented by fuzzy numbers, so it can
be applied
to
various problems
where
human knowledge

can be prov
ided
. However, there are
models of fuzzy
neural networks regarded as standard models. A general
architecture and algorithms should be reported
with

various experiments in many environments.



References


[1]

A. Kaufmann, M.M. Gupta,
Introduction to Fuzzy
Arith
metic: Theory and Applications
, Van Nostrand
Reinhold, New York, 1985.

[2]

J.J. Buckley, and Y. Hayashi,

Fuzzy neural
networks: A survey,


Fuzzy Sets and Systems
, vol.
66, pp. 1
-
13, 1994.

[3]

H. I
shibuchi,

M.

Morioka,
and I.B. Turksen,
“Learning by
f
uzzified
n
eur
al
n
etworks,”
Internat. J.
Approximate Reasoning
, vol.
13,
p
p. 327
-
358, 1995.

[4]

H. Ishibuchi, K. Kwon, and H. Tanaka,

A learning
algorithm of fuzzy neural networks with triangular
fuzzy weights,


Fuzzy Sets and Systems
, vol. 71, pp.
277
-
293, 1995.

[5]

K. H.

Lee
,
and
S. B.

Cho
,

A
learning algorithm of
f
uzzy
n
eural
n
etwork
s

with
t
rapezoidal
f
uzzy
w
eights
,


Proc. of A
sian Fuzzy Systems Symposium
,
Masan, Korea,
June
1998.