Neuro-fuzzy Approach - Computational Intelligence in Time Series Forecasting - page 232

where,

,

' w k 1 w k w k 1 (6.17e)

and w represents the networks free parameter vector in general. The momentum
constant is usually less than one. Therefore, we can write mo < 1.

6.4.2.2 Improved Backpropagation Training Algorithm
To improve the training performance of the proposed neuro-fuzzy network, we
have modified the momentum version of the backpropagation algorithm by adding
to it the modified error index term/modified performance index term (6.18a), as
proposed by Xiaosong et al. (1995).

2

N

w 0 . 5 J w , (6.18a)

¦

S e e

m r avg

r 1

where,

1

N

¦ w . (6.18b)

e e avg r

N

r 1

and e avg is the average error. Thus, the new error index (new performance index) is
finally defined as

w S w w , (6.19)

S S

new m

where, S ( w ) is the unmodified performance index as defined in (6.11b ). From this,
the corresponding gradient can be defined as

N

S w ¦ e w w e w w w

(6.20a)

r r

r 1

N

w J e w w e w w w

S ¦ e

(6.20b)

m r avg r

r 1

N

w e w J e w w e w w w

¦

S e

, (6.20c)

new r r avg r

r 1

where the constant term (gama) J < 1 has to be chosen appropriately.
With the modified error index extension as per Equation (6.20c) we need only
to add a new vector term with the original error vector

e .
Theoretical justification of the improved training performance of the network by
the use of a modified error index term has been described in Xiaosong et al.
(1995).

J w e

e avg