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 GEPESM experiment a threegenic 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 GEPESM 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 (GEPOS) and the enhanced STROGANOFF using a unigenic system
(GEPESU) and a multigenic system (GEPESM).

GEPOS 
GEPESU 
GEPESM 
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 
Onepoint
recombination rate 
0.3 
0.3 
0.3 
Twopoint
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
bestofrun fitness 
86069.183 
86566.298 
86881.997 
Average
bestofrun Rsquare 
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 
Onepoint
recombination rate 
0.3 
Twopoint
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
bestofrun fitness 
73218 
Several conclusions can be drawn from the experiments presented here. First, a STROGANOFFlike system exploring secondorder 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 highorder 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.
