Evolutionary Computation - Computational Intelligence in Time Series Forecasting - page 213

is generated. The angle brackets D denote the modulo function with modulus D .
The starting index n in the above equation is a randomly chosen integer from the
interval [0, D -1]. The integer L , which denotes the number of parameters that are
going to be exchanged, is drawn from the interval [1, D ]. The algorithm that
determines L works according to the following lines of pseudo code, where rand ()
is supposed to generate a random number within the interval [0, 1]:

L = 0;

do {

L = L +1;

} while (( rand () < CR ) and ( L < D ));

Hence, the probability t ! CR is taken from the interval [0,

1] and constitutes a control variable in the design process. The random decisions
for both n and L are always made afresh for each newly generated vector

v
Lv CR v

1

Pr , 0.

u X . , 1

X i,G X v,G+1 X u,G+1

j = 0 j = 0 j = 0

j = 1 j = 1 j = 1

n=2

j = 2 j = 2 j = 2

n=3

j = 3 j = 3 j = 3

n=4

j = 4 j = 4 j = 4

j = 5 j = 5 j = 5

j = 6 j = 6 j = 6

vector containing the parameters
x j , j = 0,1,2,…,D-1

Figure 5.7. Crossover process in DE1 for D = 7, n = 2, L = 3 for new vector generation

Note that, in Figure 5.7, since L = 3, three parameters are exchanged; they are
numbered as ( n = 2), ( n +1 = 3), ( n + L -1 = 4), because the modulo function ( n and
D ) = 2, modulo function ( n +1 and D ) = 3 and modulo function ( n + L -1 and D ) = 4,
for D = 7.
To decide whether or not the newly generated vector should become a member
of generation G +1, the new vector

u X is compared with

X . If the newly

, 1 iG

generated vector yields a smaller objective value than X , then X

is set to iG iG , 1

u X , otherwise the old vector

X is retained.

, 1 iG