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;
/
ð
d
i
Þ
¼exp
d
i
=
b
2
i
i ¼
1
;
2
; ...;
K
ð
10
Þ
where d
i
¼
k
x
c
i
k
and the real constant
ʲ
i
is a scaling or
“
width
”
parameter
(Chen et al.
1991
).
Minimal Resource Allocating Network Algorithm
The learning process of MRAN involves allocation of new hidden units and a
pruning strategy as well as adaptation of network parameters (Kadirkamanathan and
Niranjan
1993
; Platt
1991
; Sundararajan et al.
2002
). The network starts with no
hidden units and as input-output data
ð
ðÞ;
ðÞÞ
are received, some of them are
used to generate new hidden units based on a suitably de
x
y
ned growth criteria. In
particular at each time instant n the following three conditions are evaluated to
decide if the input x
ð
n
Þ
should give rise to a new hidden unit:
kk
¼
e
ð
n
k
y
ð
n
Þ
f
ð
x
ð
n
ÞÞ
k
[
E
1
ð
11
Þ
t
X
2
n
ð
Þ
e
j
e
rms
ð
n
Þ
¼
[
E
2
ð
12
Þ
M
j¼n
ð
M
1
Þ
ð
Þ
¼
ð
Þ
c
r
ð
Þ
ð
13
Þ
d
n
k
x
n
n
k
[
E
3
where c
r
ð
n
Þ
and M represents
the number of past network outputs for calculating the output error e
rms
(n). The
terms E
1
, E
2
and E
3
are thresholds to be suitably selected. As stated in Sundararajan
et al. (
2002
), Yingwei et al. (
1998
) these three conditions evaluate the novelty in the
data. If all the criteria of relations (
11
)
is the centre of the hidden unit that is nearest to x
ð
n
Þ
-
(
13
) are satis
ed, a new hidden unit is added
and the following parameters are associated with it:
ð
Þ
ð
14
Þ
k
K
þ
1
¼
e
n
c
K
þ
1
¼
x
ð
n
Þ
ð
15
Þ
b
K
þ
1
¼ a
k
x
ð
n
Þ
c
r
ð
n
Þ
k
ð
16
Þ
where
ʱ
determines the overlap of the response of a hidden unit in the input space
as speci
ed in Kadirkamanathan and Niranjan (
1993
), Sundararajan et al. (
2002
). If
the observation
ð
ð
Þ;
ð
ÞÞ
-
x
n
y
n
does not satisfy the criteria of relations (
11
)
(
13
), an
EKF is used to update the following parameters of the network:
T
c
1
;
b
1
; ...;
k
N
;
c
N
;
b
N
w
¼ k
0
;
k
1
;
:
ð
17
Þ
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