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.
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