Biased Mutation for Evolution Strategies - Self-Adaptive Heuristics for Evolutionary Computation - page 61

1e+010

BMO on rosenbrock
sBMO on rosenbrock
cBMO on rosenbrock
DMO on rosenbrock
ES on rosenbrock

1e+008

1e+006

10000

100

1

0.01

0.0001

1e-006

1e-008

0 500 1000 1500 2000

generations

Fig. 4.6. Fitness development of typical runs of the ES, the BMO, its variants and the
DMO on the rosenbrock function. BMO and cBMO show the fastest approximation
capabilities while the standard ES is comparatively slow.

mentioned variants. The variant sBMO shows rather weak results, but is still
better than the ES as the Wilcoxon rank-sum test will show in the following.
These results lead to the conclusion that the additional degree of freedom ob-
tained by the bias offers an advantage in the multimodal fitness landscape of the
rosenbrock function.

Wilcoxon Rank-Sum Test

To prove the significance of the experimental results, we perform a pairwise
comparison of the operators with a Wilcoxon rank-sum test [68]. The Wilcoxon
rank-sum test is non-parametric and tests the hypothesis that two sample sets
are drawn from a single population, i.e. that their probability distributions are
equal. The Wilcoxon rank-sum test does not require any assumptions like the
student t-test. The latter is only applicable if the source distribution is a normal
distribution, which is usually not the case for EAs. It is applicable if the distri-
bution of the data is not known. The Wilcoxon rank-sum test is more sensitive
for small data sets, as it is based on the combined ranking of both data sets.
The summed rank of the smallest sample is the value for the level of significance
calculation. The exact level of significance makes use of all possible permutations
of ranks and can be approximated with a normal distribution.