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Daida et al. (2001) established the tunable nature of the
Binomial3 problem by demonstrating increased difficulty through the reduced
frequency which good solutions are found,
the longer it took for good solutions to be found and the increased
size of good solutions.
In Figure 6.2, the
best fitness of 100 runs for each experiment are
placed into bins and plotted in a histogram. The leftmost bins in each
experiment  corresponding to the number of best solutions found  are
expectedly larger for
easier instances. The Binomial3 results show
a clear trend of less good solutions with increasing ERC ranges.
In the case of the polynomial results, this trend is clear for degree 3 and
degree 11. Note the large number of poor fit runs for the degree 11 polynomial (the rightmost bins). These `failed' runs tend to be present in regression
problems where the lack of initial fitness improvements causes the run to
become stuck in poor local optima. The nature of the degree 11 polynomial
amplifies this effect, as we would expect.
As shown later, the degree 7 polynomial
produces a mixture
of results but generally shows the expected behaviour.
Figure 6.2:
Each problem instance's final population performance values have been discretised and placed into 60 (Binomial3) or 21 (polynomials) bins to show their distribution. Bins to the left in each instance represent better fitness.

Figure 6.3:
Results of the Binomial3 experiments. The average size of the population containing the bestofrun individual is plotted against bestofrun fitness (left column), the generation in which the bestofrun individual occurred (middle column) and the average depth of the population (right column). Each circle corresponds to one run.

Figure 6.4:
Results of the random polynomial experiments. The average size of the population containing the bestofrun individual is plotted against the bestofrun fitness (left column), the generation in which the bestofrun individual occurred (middle column) and the average depth of the population (right column). Each circle corresponds to one run.

Figure 6.5:
The Binomial3 problem adjusted fitness, depth, nodes, edit distance and entropy versus generation for the control, unity, ten, and hundred experiments. Points represent the average over 100 runs.

Figure 6.6:
The random polynomial experiments adjusted fitness, depth, nodes, edit distance and entropy versus generation for degree 3, 7 and 11 polynomials. Points represent the average over 100 runs.

In Figure 6.3 the results from Daida et al. (2001) are
reproduced based on the experiments carried out here. Figure
6.4 shows the similar results for the generated polynomials.
The same trends apply for both the Binomial3
instances and the random polynomials.
Increasing the ERC range worsens the best fitness found and
increases the size and depth of the solutions when best fitness is found.
When the difficulty is increased, the bestofrun solutions are found
in later generations. Note that while the fitness function remains
the same in the Binomial3 problems, the polynomial fitness function
does change.
For the polynomial experiments, the distinguishing feature of run performance
is the amount of
improvement that genetic programming can find. The difference
in this behaviour can be seen in Figure 6.6.
The higher degree polynomials contain smaller variations and the fitness
functions across the instances are not the same. As the higher degree
polynomials
start with lower initial fitness,
the improvement rate from initial generations to later ones is observed
as a sign of
performance. The higher the degree, the less improvement is made during
the evolutionary process, the bigger and deeper solutions are and
the later they are found.
A more indepth analysis of the behaviour of runs
with respect to size, entropy and edit distance follows.
Next: 2 Binomial3 Results
Up: 4 Binomial3 and Random
Previous: 4 Binomial3 and Random
Contents
S Gustafson
20040520