Choosing the Fitness Function
Mean Squared Error
 
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.

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 MSE fitness function of GeneXproTools 4.0 is, as expected, based on the standard mean squared error.

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); 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 measure fi as raw fitness rfi and complements it with a parsimony term.

Thus, in this case, raw maximum fitness rfmax = 1000. 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:



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