Sensitivity/Specificity/PPV/NPV

Choosing the Fitness Function

GeneXproTools 4.0 implements the Sensitivity/Specificity/PPV/NPV (SSPN) 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 SSPN fitness function of GeneXproTools 4.0 is, as expected, based on all the four statistical indexes: sensitivity, specificity, positive predictive value (PPV), and negative predictive value (NPV).

The sensitivity/specificity/PPV/NPV SSPN_i of an individual program i is evaluated by the equation:

where SE_i is the sensitivity, SP_i the specificity, PPV_i the positive predictive value, and NPV_i the negative predictive value of the individual program i, and are given by the formulas:

where TP_i, TN_i, FP_i, and FN_i represent, respectively, the number of true positives, true negatives, false positives, and false negatives.

TP_i, TN_i, FP_i, and FN_i 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 f_i 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 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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