A Metaheuristic Approach to Two Dimensional Recursive Digital Filter Design - Advances in Heuristic Signal Processing and Applications - page 170

component of the i th vector as

x j,i, 0 = x j, min + rand i,j ( 0 , 1 )(x j, max − x j, min ) (8.21)

where rand i,j ( 0 , 1 ) is a uniformly distributed random number lying between 0 and 1
and is instantiated independently for each component of the i th vector. The follow-
ing steps are taken next: Reproduction, Seed Dispersal, and Competitive Exclusion
(in that order), which are explained in the following subsections.

8.3.2 Reproduction

The plants will produce seeds depending on their relative fitness which will be
spread out over the problem space. Each seed, in turn, will grow into a flowering
plant. Thus, if S max and S min denote the number of seeds produced by plants with
the best and the worst fitness, respectively, then seed count of plants will increase
linearly from S min to S max depending on their corresponding fitness values. The
number of seeds produced by the i th weed X i,G is therefore given by

F max ,G −

S min )

f( X i,G )
F max ,G −

s i,G = (S max − (8.22)

F min ,G

where F max ,G and F min ,G are the maximum and minimum fitness values at the G th
generation of the weed colony.

8.3.3 Dispersal of Seeds Through the Search Space

The produced seeds are randomly distributed over the D -dimensional search space
by random numbers drawn from a normal distribution with zero mean but with a
varying variance. However, the standard deviation (SD), σ , of the normal distribu-
tion decreases over the generations from an initial value, σ max , to a value, σ min , and
is determined by the following equation,

G max − G
G max

n

σ = (σ max − σ min ) + σ min (8.23)

where σ is the SD at the current generation and G max is the maximum number of
iterations while n is the nonlinear modulation index. This is the adaptation property
of the algorithm.

8.3.4 Competitive Exclusion

If a plant does not reproduce, it will become extinct. Hence this leads to the require-
ment of a competitive exclusion in order to eliminate plants with low fitness values.