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Capturing Relationships in Multi-Objective
Optimization: Extended results
1

I. E XTENDED R ESULTS

0.95

Specificity

In Table I we present the sensitivity and specificity for
each studied correlation metric. We analyse four different
contexts: the relationship between variables, between variables
and objectives and between objectives separately; and all
relationships together. For relationships between objectives
we do not analyse specificity since all objectives are related.
Analysing Kendall’s correlation, it obtains the best specificity
value in all cases, but the worst sensitivity too. Pearson’s
correlation gets the best sensitivity in most cases, the same
cases it gets the worst specificity. Spearman’s correlation gets
the best balance, with better specificity than Pearson’s in most
cases and better sensitivity than Kendall’s.

0.85

0.8

TABLE I
S ENSITIVITY AND S PECIFICITY

var/var
var/obj
obj/obj

Kendall
0.17/1.00
0.08/0.83
0.00/ *
0.08/0.99

Pearson
0.83/0.66
0.10/0.66
0.40/ *
0.19/0.66

0.75
0

Spearman
0.50/0.80
0.12/0.50
0.10/ *
0.14/0.79

Kendall
Spearman
Pearson
200

400
Iteration

600

800

Fig. 1. Specificity along the search process
0.25

* all five objectives are related, specificity are not applicable.

Kendall
Spearman
Pearson

0.2

Sensitivity

We also analyze the correlation metrics specificity and
sensitivity along the search process. Our objective is verify
if it is possible capture the relationships along the search
process, and if the metrics ability is the same in all stages
of the search. In Figure 1 we present the specificity analysis,
it these analysis Kendall’s correlation gets a value a little under
the maximum along all search, mainly the initial stage. The
other two metrics starts with maximum value but decrease a lot
until getting increasing again, and finish with a value close to
0.90 and 0.95. In Figure 2 we present the sensitivity analysis,
that shows that the metrics ability of getting true relationships
increase along the search process. It probably occurs because
the exploration used along initial stages of the search can
generate many erratics solutions, that do not represent the
problem characteristics, but as the solutions evolves it became
easier to get the true relationships. Our analysis shows that
the metrics has difficulty to get poor relationships reflecting a
sensitivity under 0.25 for all metrics.
In Figures 3, 4 and 5 we show the mean of the relationships
along six stages of the search process over 30 runs, captured
by Kendall’s, Pearson’s and Spearman’s respectively. For a
cleaner visualization we shows only the relationships stronger
than a 0.25 threshold.

0.9

0.15

0.1

0.05

0

0

200

400
Iteration

600

Fig. 2. Sensitivity along the search process

800

(a) iteration 0

(b) iteration 156

(c) iteration 312

(d) iteration 468

(e) iteration 598

(f) iteration 754

Fig. 3. Relations found along the search process using Kendalls

(a) iteration 0

(b) iteration 156

(c) iteration 312

(d) iteration 468

(e) iteration 598

(f) iteration 754

Fig. 4. Relations found along the search process using Pearsons

(a) iteration 0

(b) iteration 156

(c) iteration 312

(d) iteration 468

(e) iteration 598

(f) iteration 754

Fig. 5. Relations found along the search process using Spearmans


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