GeneXproTools 4.0 implements the Absolute/Hits fitness function
both with and without parsimony pressure. The version with
parsimony pressure puts a little pressure on the size of the
evolving solutions, allowing the discovery of more compact models.
For some Function Finding problems it is important to evolve a model that performs well for all fitness cases within a certain absolute error (the
precision) of the correct value.
For those cases, the fitness f(ij) of an individual program
i for fitness case j is evaluated by the formula:
where p is the precision and E(ij) is the
absolute error of an individual program
i for fitness case j evaluated by the equation:
where P(ij) is the value predicted by the individual program
i for fitness case j and Tj is the target value for fitness case
j.
And overall, the fitness fi of an
individual program
i is expressed by the equation:
where n is the total number of fitness cases.
So, for this fitness function, maximum fitness fmax is given by: fmax =
n Its counterpart with parsimony pressure, uses this fitness
measure fi
as raw fitness rfi and complements
it with a parsimony term.
Thus, in this case, raw maximum fitness rfmax = n.
And the overall fitness fppi (that is, fitness with parsimony pressure) is evaluated by the formula:
where Si is the size of the program, Smax and
Smin represent, respectively, maximum and minimum program sizes and are evaluated by the formulas:
Smax = G (h + t)
Smin = G
where G is the number of genes, and h and t are the head and tail sizes (note that, for simplicity, the linking function was not taken into account). Thus, when
rfi = rfmax and Si =
Smin (highly improbable, though, as this can only happen for very simple functions as this means that all the sub-ETs are composed of just one node),
fppi = fppmax, with fppmax evaluated by the formula:
|