Squared Accuracy

Choosing the Fitness Function

Squared Accuracy

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 A_i 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 f_i of an individual program i is expressed by the equation:

f_i = 1000*A_i*A_i

and therefore ranges from 0 to 1000, with 1000 corresponding to the ideal.

Its counterpart with parsimony pressure, uses this fitness measure f_i as raw fitness rf_i and complements it with a parsimony term.

Thus, in this case, raw maximum fitness rf_max = 1000. And the overall fitness fpp_i (that is, fitness with parsimony pressure) is evaluated by the formula:

where S_i is the size of the program, S_max and S_min represent, respectively, maximum and minimum program sizes and are evaluated by the formulas:

S_max = G (h + t)

S_min = 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 rf_i = rf_max and S_i = S_min (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), fpp_i = fpp_max, with fpp_max evaluated by the formula:

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