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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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