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C. FERREIRA, 2002 (Terms of Use) ISBN: 9729589054

Gene Expression Programming: Mathematical Modeling by an Artificial Intelligence

Simulating STROGANOFF and enhanced STROGANOFF with GEP
 
The performance of both the original and enhanced STROGANOFF was evaluated on the sunspots data of Table 4.5 and is shown in Table 4.17. As expected, the enhanced implementation is considerably better than the original STROGANOFF. Note, however, that in the GEP-ESM experiment a three-genic system was used and, therefore, the system we are simulating does not correspond to the enhanced STROGANOFF as described by Nikolaev and Iba (2001) but, rather, is a much more efficient algorithm, as it benefits from the multigenic nature of gene expression programming. Indeed, the multigenic system works considerably better than the unigenic one. It is worth emphasizing that the implementation of multiple parse trees in GP is unfeasible and so is a system similar to the one used in the GEP-ESM experiment. And, of course, the facility for the manipulation of random constants in GP is much less versatile than the one used in GEP and, in fact, the coefficients in GP can only be discovered a posteriori, usually by a neural network. This obviously raises the question of what is exactly the GP doing in this case for, without the coefficients, the polynomials are useless (see Table 4.18 below).


Table 4.17
Settings used in the GEP simulation of the original STROGANOFF (GEP-OS) and the enhanced STROGANOFF using a unigenic system (GEP-ESU) and a multigenic system (GEP-ESM).

  GEP-OS GEP-ESU GEP-ESM
Number of runs 100 100 100
Number of generations 5000 5000 5000
Population size 100 100 100
Number of fitness cases 90 90 90
Function set (F9)16 F1 - F16 F1 - F16
Terminal set d0 - d9 d0 - d9 d0 - d9
Random constants array length 120 120 40
Random constants range [-1,1] [-1,1] [-1,1]
Head length 21 21 7
Number of genes 1 1 3
Linking function -- -- +
Chromosome length 169 169 171
Head/tail mutation rate 0.044 0.044 0.044
Dc mutation rate 0.06 0.06 0.06
One-point recombination rate 0.3 0.3 0.3
Two-point recombination rate 0.3 0.3 0.3
Gene recombination rate -- -- 0.1
IS transposition rate 0.1 0.1 0.1
IS elements length 1,2,3 1,2,3 1,2,3
RIS transposition rate 0.1 0.1 0.1
RIS elements length 1,2,3 1,2,3 1,2,3
Gene transposition rate -- -- 0.1
Random constants mutation rate 0.25 0.25 0.25
Dc specific transposition rate 0.1 0.1 0.1
Dc specific IS elements length 5,7,9 5,7,9 5,7,9
Selection range 1000% 1000% 1000%
Precision 0% 0% 0%
Average best-of-run fitness 86069.183 86566.298 86881.997
Average best-of-run R-square 0.4949506 0.6712016 0.7631128


Indeed, continuing our discussion about the importance of random constants in evolutionary symbolic regression, there is a simple experiment we could do. We could implement the bivariate polynomials presented in Table 4.16 and try to evolve complex polynomial models with them (Table 4.18). On the one hand, the comparison of this experiment with the experiments summarized in Table 4.17 clearly show that coefficients are indeed fundamental to the evolution of polynomials. But genuine polynomials with coefficients are much less efficient than a simple GEA (compare Table 4.10 with Table 4.17) and, therefore, the role of numerical constants in evolutionary symbolic regression is a marginal one and has only theoretical interest.


Table 4.18
The role of coefficients in polynomial evolution.

Number of runs 100
Number of generations 5000
Population size 100
Number of fitness cases 90
Function set F1 - F16
Terminal set d0 - d9
Head length 7
Number of genes 3
Linking function +
Chromosome length 45
Mutation rate 0.044
One-point recombination rate 0.3
Two-point recombination rate 0.3
Gene recombination rate 0.1
IS transposition rate 0.1
IS elements length 1,2,3
RIS transposition rate 0.1
RIS elements length 1,2,3
Gene transposition rate 0.1
Selection range 1000%
Precision 0%
Average best-of-run fitness 73218


Several conclusions can be drawn from the experiments presented here. First, a STROGANOFF-like system exploring second-order bivariate basis polynomials, although mathematically appealing, is extremely inefficient in evolutionary terms. Not only is its performance significantly worse but also its structural complexity is considerably more complicated. Second, the finding of the coefficients is, as expected, fundamental to the construction of models based on high-order multivariate polynomials. And, finally, a simple GEP system with the usual set of mathematical functions is much more efficient than the complicated and computationally expensive STROGANOFF systems.

In the next section we are going to use the settings of the winner algorithm (the simple GEA with the basic arithmetic functions) to evolve a model to make predictions about sunspots.

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