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An objective of this study is to quantify the importance and
ideal levels of diversity, recorded by different measures on common problems.
The primary test of the relationship between diversity and fitness will be
the Spearman correlation measure [Siegel, 1956]. The Spearman
measure ranks two sets of variables and tests for a linear relationship
between the variables' ranks.
Correlation is first examined to determine if two runs can be distinguished by
their diversity in terms of which run is better.
As interesting relationships
could easily exist but may not necessarily be linear, a range
of scatter plots of diversity measures and fitness are evaluated.
Figure 4.3:
Examples of ranked correlation scatter plots between fitness and diversity, where low fitness (ideal) is ranked from 1 to 50 (with 50 runs total) and diversity is ranked from high (1) to low (50). The middle graph shows the case of no correlation where the points are aligned vertically or horizontally.

The Spearman correlation coefficient is computed
as follows:
where is the number
of runs and is the distance between each population's
fitness rank and diversity rank.
A value of 1.0 represents negative correlation, 0.0 denotes no correlation
and 1.0 demonstrates
positive correlation. For the measures used here, when low bestofgeneration fitness values, which
will be ranked in ascending order (1=best,,50=worst), occur with high diversity,
ranked in ascending order (1=lowest diversity and 50=highest diversity), the correlation
coefficient should be strongly negative. Alternatively, a positive correlation
indicates that either bad fitness accompanies high diversity or good fitness
accompanies low diversity.
Figure 4.3 shows the relationship between fitness
on the XAxis and diversity on the YAxis and the type of correlation
that a scatter plot in these circumstances would indicate.
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S Gustafson
20040520