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. 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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