234 S. Silva et al.
0400Frequency inGeneration 15IM−100400Frequency inGeneration 15VOW0400Frequency inGeneration 25
0400Frequency inGeneration 250400Frequency inGeneration 50
0400Frequency inGneration 50(^0051015202530)
400
Dimensions
Frequency inGeneration 100
(^0051015202530)
400
Dimensions
Frequency inGeneration 100
Fig. 4 Distribution of the number of dimensions in the population in generations 15, 25, 50 and
100 (toptobottom) for M3GP. On theleft, a typical run of problem IM-10. On theright, a typical
run of problem VOW
different values, depending on the run. The second one is that M3GP tends to use a
larger number of dimensions than M2GP. What these numbers do not show is that
different problems result in very different behaviors with respect to the evolution of
the number of dimensions. Figure 4 illustrates two main types of behavior, described
next. In most problems the distribution of the number of dimensions moves rapidly
to higher values in the beginning of the run, and then remains stable and more or
less in the same range until the end of the run (exemplified on the left in Fig. 4 ).
However, in some problems, like WAV and VOW, the distribution of the number
of dimensions does not settle during the 100 generations of the run, and instead
keeps moving towards higher values (exemplified on the right in Fig. 4 ). The WAV
problem goes as high as 37 dimensions, and curiously this is one of the problems
where M3GP produces substantially smaller trees than M2GP.
The comparison between M3GP and the state-of-the-art classifiers is based only
on training and test fitness, once again considering fitness to be accuracy. Based
on the comparison previously done between M2GP and several state-of-the-art
methods (see Sect.7.1), we have decided to compare M3GP with a tree based