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x Gaussian mutation, in which all elements of a chromosome are mutated
such that
c
and
k
f
is a random
number drawn from a Gaussian distribution with zero mean and an
""
where
f
s
g
a
c
1
aaa
1
,
,
c
,
,
c
,
aa
k
L
k
k
k
chrom
Gg
a
§ ·
max
min
§
·
a
adaptive variance
k
k
¹
. It can be seen that
V
V
¨
¸
¨
¸
k
G
3
©
¹
©
decreases as the generation counter
g
increases. Therefore, parameter
tuning performed by a Gaussian mutation operator becomes finer as the
generation counter
g
increases.
5.2.4 Forecasting Example
In this section we briefly describe a binary-coded GA that can be used to train a
neuro-fuzzy system that will be considered once again in Chapter 6. For
convenience we restrict our discussion to a Takagi-Sugeno-type neuro-fuzzy
network, but with singleton rules consequent only, which has been used
extensively by Wang (1994) for a variety of identification and modeling
applications. Furthermore, the fuzzy logic system selected is based on GMFs, the
product inference rule and a weighted-average defuzzifier. Mathematically, the
Takagi-Sugeno-type fuzzy logic system selected can be written as
^
`
M
M
n
2
2
¦¦
, where
l
l
l
l
y
y
E
E
E
exp
x
l
l
c
V
i
i
i
l
1
l
1
i
1
with
i
= 1, 2, ...,
n
; and
l
= 1, 2, ...,
M
.
Here, we assume that
c
!
, where
U
i
and
V
are the input and
output universes of discourse respectively.
The corresponding
l
th rule of the fuzzy logic system can be written as follows:
R
l
: If
x
1
is
l
l
U
,
l
0 and
y
V
i
i
i
Then
y
is
y
l
and
x
2
is
and
...
and
x
n
is
l
l
l
n
G
G
G
1
2
where
x
i
with
i
= 1, 2, ...,
n
represent the
n
number of inputs to the system,
l
= 1, 2,
...,
M
are the
M
number of fuzzy rules that construct the fuzzy system,
G
with
i
=
1, 2, ...,
n
and
l
= 1, 2, ...,
M
are the GMFs with corresponding mean and variance
parameters
c
V
respectively that partition the
i
th input domain, and
y
l
represents the (singleton) output from the
l
th rule. It will be shown in Chapter 6
that a similar fuzzy system can be represented as a three-layer multi-input single-
output feedforward network form. Because of neuro implementation of fuzzy logic
systems, the same feedforward network actually represents a Takagi-Sugeno-type
neuro-fuzzy network.
Given a set of
N
input-output training samples of the form
l
l
and
i
i
p
p
X
,
d
, where the
p
n
input pattern
X
ª
pp
,
,...,
p
n
º
U
\
and the corresponding desired output
xx
x
¬
¼
12
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