TESI DOCTORAL - La Salle

Chapter 6. Voting based consensus functions for soft cluster ensembles
CSPA EAC HGPA MCLA VMA BC CC PC SC
CSPA ——— × × 0.0001 0.0001 0.0001 0.0001 0.0013 0.0012
EAC 0.0001 ——— × 0.0001 0.0002 0.0001 0.0001 0.002 0.0019
HGPA 0.0001 0.0001 ——— 0.0001 0.0001 0.0001 0.0001 0.0017 0.0016
MCLA × 0.0001 0.0001 ——— 0.0001 0.0009 × 0.0003 0.0003
VMA 0.0001 0.0001 0.0001 0.0001 ——— 0.0001 0.0001 0.0001 0.0001
BC 0.0001 0.0001 0.0001 0.0001 × ——— 0.0001 × ×
CC 0.0001 0.0001 0.0001 0.0001 × × ——— 0.0001 0.0001
PC 0.0001 0.0001 0.0001 0.0001 × × × ——— ×
SC 0.0001 0.0001 0.0001 0.0001 × 0.0337 0.0419 × ———
Table 6.2: Significance levels p corresponding to the pairwise comparison of soft consensus
functions using a t-paired test on the Zoo data set. The upper and lower triangular sections
of the table correspond to the comparison in terms of CPU time and φ (NMI) , respectively.
Statistically non-significant differences (p >0.05) are denoted by the symbol ×.
differences is denoted by means of the × symbol.
For instance, let us see how does BordaConsensus (BC) compare to the eight remaining
consensus function in terms of execution CPU time —for an easier identification, the
contents of the corresponding boxes of table 6.2 are italicized. In fact, they tell us that the
differences observed in figure 6.2 (according to which BC is apparently faster than MCLA
and CC, and slower than CSPA, EAC, HGPA, VMA, PC and SC) are statistically significant
with respect to all but the PC and SC consensus functions.
If this comparison is based on the φ (NMI) of the consensus clustering solutions, figure 6.2
suggests that BC performs better than CSPA, EAC, HGPA and MCLA, which is true from a
statistical significance standpoint, as the corresponding entries of table 6.2 (which are typed
in boldface for ease of identification) confirm. In contrast, the small differences between
the φ (NMI) values of BC, VMA, CC and PC appreciated in figure 6.2 are statistically non
significant, whereas it is with respect to SC despite its apparent closeness.
In order to provide the reader with a global perspective that illustrates the performance
of the proposed consensus functions compared to their state-of-the-art counterparts across
the twelve unimodal collections employed in this work, we have computed the total percentage
of experiments in which the latter yield better, equivalent or worse results than the
voting-based consensus functions —considering the statistical significance of the differences
between the compared magnitudes (CPU time and φ (NMI) ).
Firstly, table 6.3 presents the results of such comparative analysis when it is referred
to the quality of the consensus clusterings output by the consensus functions in all the
experiments conducted. It can be observed that the four proposed consensus functions
outperform EAC, HGPA and MCLA in a pretty overwhelming percentage of the experiments
(an average 94.4% of the total). When compared to CSPA and VMA, we can appreciate
certain differences between the performance of the consensus functions based on confidence
voting (PC and SC) and the ones based on positional voting (BC and CC). In general terms,
SC and PC perform slightly better than BC and CC. Moreover, notice that BordaConsensus
and CondorcetConsensus attain exactly the same results, whereas the similarity between
the results of SC and PC is also very noticeable. We conjecture that these high degrees
of resemblance is due to the fact that evaluation is conducted upon a hardened version
of the soft consensus clustering output by these consensus functions. Thus, the intrinsic
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