Databases Reference
In-Depth Information
where
x
i
is the
i
th input feature,
sm
i
is the low value of the
i
th input
feature,
bg
i
is the high value of the
i
th input feature,
me
i
is the medium
value of the
i
th input feature and
th
i
is taken as (
bg
i
−
sm
i
)
/
3. Each input
data is the input of these membership functions (low, medium, high) and
the outputs of the units
O
i,j
(
i
=1
n, j
=1
,
2
,
3) are the grades of
the membership functions. The inputs of the unit in the output layer are
w
i,j
O
i,j
. The output unit has a sigmoid function
f
given by:
−
1
1+
e
−
s
o
=
F
(
s
)=
(9.4)
where
s
is the sum of the inputs of the output unit i.e.:
N
3
s
=
O
i,j
W
i,j
(9.5)
i
=1
j
=1
o
is the output of this network. The connection weights
w
i,j
are modified
by the
δ
rule.
9.3. Particle Swarm Optimization
The particle swarm algorithm is an optimization technique inspired by the
metaphor of social interaction observed among insects or animals. The
kind of social interaction modeled within a PSO is used to guide a
population of individuals (called particles) moving towards the most
promising area of the search space.
42-47
PSO was developed and first
introduced as a stochastic optimization algorithm by Eberhart and
Kennedy.
48
During this period, PSO gained increasing popularity due to
its effectiveness in performing dicult optimization tasks. Among other
applications, it has been applied to tackle multi-objective problems,
49
minimax problems,
50,51
integer programming problems,
52
noisy and
continuously changing environments,
53-55
errors-in-variables problems,
56
existence of function zeros,
57
power systems
58-64
parameter learning of
neural networks (NNs),
65,66
control,
67-70
prediction,
71-73
modeling
74-76
and
numerous engineering applications.
77-88
In a PSO algorithm, each particle is a candidate solution equivalent to
apointina
d
-dimensional space, so the
i
th particle can be represented as
x
i
=(
x
i,
1
,x
i,
2
,...,x
i,d
). Each particle “flies” through the search space,
depending on two important factors,
p
i
=(
p
i,
1
,p
i,
2
,...,p
i,d
), the best
position found so far by the current particle and
p
g
=(
p
g
1
,p
g
2
,...,p
gd
),
