Evolving Neural and Fuzzy Systems - Computational Intelligence in Time Series Forecasting - page 307

made up of a linear combination of the input values, Lee and Takagi (1993b) found
it more advantageous to encode both the membership and the fitness functions in
chromosomes. To each rule with N input variables and n membership functions in
the genotype they assigned a gene to encode the N+ 1 weights in a linear
combination of the input variables for the rule concerned. The drawback of the
encoding approach is that n combinations have to be encoded.
Tettamanzi (1995) implemented his fuzzy control evolving system on a WARP
fuzzy processor capable of supporting up to 256 rules with up to four antecedent
clauses and one consequent clause, as well as antecedent membership functions of
arbitrary shapes. To define the appropriate fitness function he used the concept of
competition , defined later (Tettamanzi, 1994). The concept registers the number of
competitions c undergone by an individual, the number of its wins w , and the
number of successes s . Using this statistical data the membership function of
fitness for a given individual is defined as

n b

P

f ()

xNabx (,) (1 x )

with

( ab )

( ab )

Nab (,)

ab

ab

as a normalization factor, in which a = w + s and b = c - s .
Recently, a new evolutionary road to fuzzy systems design was paved by Shi et
al . (1999), who, along with the membership function shapes and the fuzzy rule set,
also encoded the membership function type and the number of rules inside the set.
Two types of membership function have been considered: linear and nonlinear
(Gaussian, triangle and their combination). Each membership function was
completely defined by its start point, its end point, and the function type.
In order to make the evolving process easier, the fitness function, which
measures the performance of the system, was carefully defined. Depending on the
application, the fitness functions taken are

1 N
i

2

E ot

¦

i i

f

N

1

and

2

1 N § ot ·

i i

E ¦ ¨

¸ f

N t

© ¹

i 1

i

o and t being the i th obtained and target outputs respectively.
For control of crossover and mutation as the most critical parameters, an
adaptive tuning approach, made up of eight fuzzy rules, was integrated into the