|GeneXproTools 4.0 implements the Squared Accuracy
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 all classification problems, in order to be able to apply a particular fitness function,
the learning algorithms of GeneXproTools 4.0 must convert the value returned by the evolved model into “1” or “0” using the
0/1 Rounding Threshold. If the value returned by the evolved model is equal to or greater than the rounding threshold, then the record is classified as “1”, “0” otherwise.
Thus, the 0/1 Rounding Threshold is an integral part of all fitness functions used for classification and must be appropriately set in the Settings Panel -> Fitness Function Tab.
The Squared Accuracy
fitness function of
GeneXproTools is, as expected, based on the classification accuracy.
The classification accuracy Ai of an individual program
i depends on the number of fitness cases correctly classified
(true positives plus true
negatives) and is evaluated by the formula:
where t is the number of sample cases correctly classified, and
n is the total number of sample cases.
The fitness fi of an individual program
i is expressed by the equation:
fi = 1000*Ai*Ai
and therefore 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: