Choosing the Fitness Function

Correlation Coefficient

 GeneXproTools 4.0 implements the Correlation Coefficient 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 Correlation Coefficient fitness function of GeneXproTools 4.0 is, as expected, based on the standard correlation coefficient, which is a dimensionless index that ranges from -1 to 1 and reflects the extent of a linear relationship between the predicted values and the target values. The correlation coefficient Ci of an individual program i is evaluated by the equation: where Cov(T,P) is the covariance of the target and model outputs; and st and sp are the corresponding standard deviations, which are given by: where P(ij) is the value predicted by the individual program i for sample case j (out of n fitness cases or sample cases); Tj is the target value for fitness case j; andandare given by the formulas: The correlation coefficient is confined to the range [-1, 1]. When Ci = 1, there is a perfect positive linear correlation between T and P, that is, they vary by the same amount. When Ci = -1, there is a perfect negative linear correlation between T and P, that is, they vary in opposite ways (when T increases, P decreases by the same amount). When Ci = 0, there is no correlation between T and P. Intermediate values describe partial correlations and the closer to 1 or -1 the better the model. The fitness fi of an individual program i is expressed by the equation: fi = 1000*Ci*Ci and therefore 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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