|GeneXproTools 4.0 implements the Mean Squared Error
(MSE) 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.
The MSE fitness function of
GeneXproTools is, as expected, based on the standard
mean squared error, which, on its turn, is based on the
The mean squared error Ei of an individual program
i is evaluated by the equation:
where P(ij) is the value predicted by the individual program
i for fitness case j (out of n fitness cases
or sample cases); and
Tj is the target value for fitness case j.
For a perfect fit, P(ij) = Tj
and Ei = 0. So, the MSE index ranges from 0 to infinity, with 0 corresponding to the ideal.
As it stands, Ei can not be used directly as fitness since, for fitness proportionate selection, the value of fitness must increase with efficiency.
Thus, for evaluating the fitness fi of an individual program
i, the following equation is used:
which obviously ranges from 0 to 1000, with 1000 corresponding to the ideal.
Its counterpart with parsimony pressure, uses this fitness
as raw fitness rfi and complements
it with a parsimony term.
Thus, in this case, raw maximum fitness rfmax =
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: