Choosing the Fitness Function
PPV/NPV
 
GeneXproTools 4.0 implements the Positive Predictive Value / Negative Predictive Value (PPV/NPV) 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 PPV/NPV fitness function of GeneXproTools 4.0 is, as expected, based both on the positive predictive value (PPV) and negative predictive value (NPV).

The PPV/NPV PNi of an individual program i is evaluated by the equation:

where PPVi is the positive predictive value and NPVi is the negative predictive value of the individual program i, and are given by the formulas:

where TPi, TNi, FPi, and FNi represent, respectively, the number of true positives, true negatives, false positives, and false negatives.

TPi, TNi, FPi, and FNi are the four different possible outcomes of a single prediction for a two-class case with classes “1” (“yes”) and “0” (“no”). A false positive is when the outcome is incorrectly classified as “yes” (or “positive”), when it is in fact “no” (or “negative”). A false negative is when the outcome is incorrectly classified as negative when it is in fact positive. True positives and true negatives are obviously correct classifications.

Keeping track of all these possible outcomes is such an error-prone activity, that they are usually shown in what is called a confusion matrix. And for all classification problems, regardless of the fitness function, GeneXproTools 4.0 shows the confusion matrix for all the evolved models and updates it continuously during the run.

 

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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