TESI DOCTORAL - La Salle

6.4. Experiments
λc = 1 1 1 3 3 3 1 2 2
(6.44)
Notice that the fuzzy consensus clusterings Λc output by BordaConsensus and CondorcetConsensus
(equations (6.40) and (6.43)) differ notably from those obtained by
SumConsensus and ProductConsensus (equations (6.32) and (6.36)) —see the double
and triple ties obtained at the clusterization of the third and seventh objects, which
are due to the intrinsic differences between the distinct voting strategies applied.
Moreover, notice that the two positional voting based consensus functions (BC and
CC) yield structuraly similar fuzzy consensus clusterings Λc, although their contents
differ slightly. However, their hardened versions λc (equations (6.41) and (6.44)) differ
in a larger extent, due to the random way ties are broken.
6.4 Experiments
This section presents several consensus clustering experiments evaluating the consensus
functions for soft cluster ensembles proposed in the previous section. These experiments
are conducted according to the following design.
– What do we want to measure? We are interested in comparing both in the quality
of the consensus clustering solutions obtained and the time complexity of the proposed
consensus functions.
– How do we measure it? As regards the time complexity aspect, all consensus
processes follow a flat architecture (i.e. one step consensus), and we measure the
CPU time required for their execution, using the computational resources described
in appendix A.6. As far as the evaluation of the quality of the consensus clustering
results is concerned, despite the proposed consensus functions output fuzzy consensus
clustering solutions, we have compared their hardened version with respect to the
ground truth of each data set in terms of normalized mutual information φ (NMI) .The
reason for this is twofold: firstly, a soft ground truth is not available for these data sets,
so fuzzy consensus clusterings cannot be directly evaluated. And secondly, provided
that the CSPA, HGPA, MCLA and EAC consensus functions output hard consensus
clustering solutions, fair inter-consensus functions comparison requires converting the
soft consensus clustering matrices Λc output by VMA, BC, CC, PC and SC to crisp
consensus labelings λc —recall that this simply boils down to assigning each object
to the cluster it is more strongly associated to.
– How are the experiments designed? In each consensus clustering experiment we
have applied our four voting-based consensus functions –SumConsensus (SC), ProductConsensus
(PC), BordaConsensus (BC) and CondorcetConsensus (CC)–, besides
the fuzzy versions of CSPA, EAC, HGPA and MCLA (see section 6.2) plus one of
the pioneering soft consensus functions, namely VMA (Voting Merging Algorithm)
(Dimitriadou, Weingessel, and Hornik, 2002) —see appendix A.5 for a description.
Experiments have been conducted on the twelve unimodal data collections employed
in this work, which are described in appendix A.2.1. As regards the creation of the
soft cluster ensemble components, we have employed the fuzzy c-means and the kmeans
clustering algorithms. Whereas the former is fuzzy by nature, the latter is not.
184