Evolutionary Computation - Computational Intelligence in Time Series Forecasting - page 200

For instance, in real-coded GAs the parameters (such as mean and variance
parameters of Gaussian membership functions, and singleton rule consequents in
the training of a neuro-fuzzy network) or the variables appear directly in the
chromosomes (see Figure 5.2) and are modified using special genetic operators.
Various real-coded GAs were recently reviewed by Herrera et al . (1998). The main
aspects of the proposed GA are discussed below, and implementation for compact,
transparent and accurate fuzzy models is also summarized.

5.2.3.1 Real Genetic Operators
Two classical operators, simple arithmetic crossover and uniform mutation, and
four special real-coded operators are used in this GA application. These operators
have been successfully applied by Michalewicz (1998), Setnes and Roubos (1999),
and Roubos and Setnes (2001).
In the following, > @

r is a random number (uniform distribution), g = 0, 1, 2,
..., G is the generation number, l = 1, 2, 3, …, N pop is the chromosome number in a
generation, S a and S b are two chromosomes selected for operation, ^

0,1

`

k " 1, 2, , L

chrom

is the position of an element in the chromosome, and and

max max

min ,
k

a

min ,
k

b

b
are the lower and upper bounds of the parameter encoded by the k th element of
chromosomes S a and S b , respectively.

a

k k

5.2.3.1.1 Selection Function
The purpose of the selection function is to create a steady evolutionary pressure;
this, to some extent, favours the well-performing chromosome to have a higher
chance of survival. The RW selection method is used to select n chromosomes for
various genetic operations (Michalewicz, 1994). The chance of winning on a spin

N

pop

of the RW is given by , implying that the higher the ratio of fitness f l of f ¦ f

l l

l 1

the chromosome S l is with respect to total fitness of all chromosomes in the
population, then the larger is the chance that chromosome S l will be selected
through the RW. The fitness f l of the chromosome S l is defined as

^ `

2

,

f 1 J , l ! 1, 2, , N

l l pop

where J l is the performance of the model encoded in chromosome

S measured in

terms of the mean-squared error (MSE):

1

N

s

2

,

¦

ˆ

J yy

i i

N

i 1

s

where y is the desired output, y is the model output, and N s is the number of
training samples. Notice that because of the reciprocal form and square term in
right-hand side of the fitness function, a small difference in MSE values will be
greatly amplified, i.e . if the MSE difference between two chromosomes is 0.1 then
the corresponding fitness difference will be 100. The inverse of the selection