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L
) to summation part of
m
th
RSP neuron (
m
M
) and
W
RB
=
...
=
...
=
(
l
1
1
w
RB
Lm
is a vector of weights from input layer to radial basis part of
m
th
RSP neuron. All weights, bias, and input-output signals are complex numbers.
By convention
w
lm
is a weight that connects
l
th
neuron to
m
th
neuron. The net
potential of this neuron is defined by following aggregation function:
m
w
RB
2
m
w
RB
1
m
,
...
exp
2
Z
T
W
RB
m
W
m
ʩ
m
(
z
1
,
z
2
...
z
L
)
=
ʳ
m
×
−
Z
−
ʻ
m
×
,
(4.1)
where
Z
W
R
m
=
Z
W
R
m
×
Z
W
R
m
ℵ
. Here superscript
2
represents
the matrix complex conjugate transposition. Output of the neuron may be expressed
as:
Y
m
=
−
−
−
ℵ
f
C
(ʩ
m
(
z
1
,
z
2
...
z
L
))
4.3.3 Learning Rules for Model-1
A multilayer network can be constructed by new neurons similar to network of
conventional neurons. The task of learning is to tune the parameters of the operator
f
and to model the underlying parametric relationship between the inputs and the
output through the weight parameter
W
. We assume that the neuron observes L
input-output pairs
(
z
1
,
y
1
), . . . (
z
n
,
y
n
)
, and generates a function space that maps the
vector space Z
.
Consider a commonly used three-layer network (L-M-N). First layer has L inputs,
second layer has M proposed neurons and the output layer consists of N conventional
neurons. Definitely this network is used in all the applications presented in this topic
based on
SRP
or
C
RSP
neuron model. Let
(
z
∈
Z
)
, into the corollary responding output space Y
(
y
∈
Y
)
be the learning rate and
f
be
ʷ
∈[
0
,
1
]
=
+
the derivative of function
f
.
w
0
is a bias and
z
0
1
j
is the bias input, where
=
√
−
1 is an imaginary unity. The weight update rules for various parameters of
a considered feedforward network of the
C
RSP
neuron are given here:
Let
V
m
j
be the net potential of
m
th
RSP neuron in the hidden layer then from
Eq. (
4.1
)
+
ʳ
m
exp
2
V
m
=
ʻ
m
W
m
Z
T
W
RB
m
−
Z
−
ʳ
m
exp
2
+
ʻ
m
W
m
Z
T
W
RB
m
−
Z
−
+
w
0
m
z
0
(4.2)
This net internal potential of RSP neuron may also be expressed termwise as follows:
V
m
V
ˀ
1
m
V
ˀ
2
m
V
ˀ
1
m
V
ˀ
2
m
=
+
+
+
w
0
m
z
0
(4.3)
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