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C. FERREIRA 7th Online World Conference on Soft Computing in Industrial Applications, 2002

Function Finding and the Creation of Numerical Constants in Gene Expression Programming

Setting the System
 

For the sequence induction problem, the first 10 positive integers an and their corresponding term N were used as fitness cases. The fitness function was based on the relative error with a selection range of 20% and maximum precision (0% error), giving maximum fitness fmax = 200 (Ferreira 2001).

For the “V” shaped function problem, a set of 20 random fitness cases chosen from the interval [-1, 1] was used. The fitness function used was also based on the relative error but in this case a selection range of 100% was used, giving fmax = 2000.

For the time series prediction problem, using an embedding dimension of 10 and a delay time of one, the sunspots series presented in Table 1 result in 90 fitness cases. In this case, a wider selection range of 1000% was chosen, giving fmax = 90,000.

In all the experiments, the selection was made by roulette-wheel sampling coupled with simple elitism and the performance was evaluated over 100 independent runs. The six experiments are summarized in Table 2.

Table 2
General settings used in the sequence induction (SI), the “V” function, and sunspots (SS) problems. The “*” indicates the explicit use of random constants.

  SI* SI V* V SS* SS
Number of runs 100 100 100 100 100 100
Number of generations 100 100 5000 5000 5000 5000
Population size 100 100 100 100 100 100
Number of fitness cases 10 10 20 20 90 90
Function set + - * / + - * / + - * / L E K ~ S C + - * / L E K ~ S C 4 (+ - * /) 4 (+ - * /)
Terminal set a, ? a a, ? a a - j, ? a - j
Random constants array length 10 -- 10 -- 10 --
Random constants range {0, 1, 2, 3} -- [-1,1] -- [-1,1] --
Head length 6 6 6 6 8 8
Number of genes 7 7 5 5 3 3
Linking function + + + + + +
Chromosome length 140 91 100 65 78 51
Mutation rate 0.044 0.044 0.044 0.044 0.044 0.044
One-point recombination rate 0.3 0.3 0.3 0.3 0.3 0.3
Two-point recombination rate 0.3 0.3 0.3 0.3 0.3 0.3
Gene recombination rate 0.1 0.1 0.1 0.1 0.1 0.1
IS transposition rate 0.1 0.1 0.1 0.1 0.1 0.1
IS elements length 1,2,3 1,2,3 1,2,3 1,2,3 1,2,3 1,2,3
RIS transposition rate 0.1 0.1 0.1 0.1 0.1 0.1
RIS elements length 1,2,3 1,2,3 1,2,3 1,2,3 1,2,3 1,2,3
Gene transposition rate 0.1 0.1 0.1 0.1 0.1 0.1
Random constants mutation rate 0.01 -- 0.01 -- 0.01 --
Dc specific transposition rate 0.1 -- 0.1 -- 0.1 --
Dc specific IS elements length 1,2,3 -- 1,2,3 -- 1,2,3 --
Selection range 20% 20% 100% 100% 1000% 1000%
Precision 0% 0% 0% 0% 0% 0%
Average best-of-run fitness 179.827 197.232 1914.8 1931.84 86215.27 89033.29
Average best-of-run R-square 0.977612 0.999345 0.957255 0.995340 0.713365 0.811863
Success rate 16% 81% -- -- -- --


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