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.

The MSE fitness function of GeneXproTools is, as expected, based on the standard mean squared error, which, on its turn, is based on the absolute 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 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 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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