TESI DOCTORAL - La Salle

6.3. Voting based consensus functions
Input: Soft cluster ensemble E containing l fuzzy clusterings Λi (∀i =1...l)
Output: Borda voting matrix BE
Data: k clusters, n objects
Hungarian (E)
BE = 0 k×n
for a =1...l do
for b =1...n do
r = Rank (λab);
for c =1...k do
BE (r (c) ,b)=BE (r (c) ,b)+(k − c +1);
end
end
end
Algorithm 6.3: Symbolic description of the soft consensus function BordaConsensus.
Hungarian and Rank are symbolic representations of the cluster disambiguation and cluster
ordering procedures, respectively, while the vector λab represents the bth column of
the ath cluster ensemble component Λa, r is a clusters ranking vector and 0 k×n represents
a k × n zero matrix.
i.e. a soft consensus clustering solution Λc (see equation (6.40)), and assigning each
object to the cluster it is most strongly associated to –breaking ties randomly– yields
the crisp consensus clustering of equation (6.41).
⎛
0.5 0.5 0.417 0.333 0.333 0.333 0.333 0.333
⎞
0.333
Λc = ⎝0.25
0.25 0.166 0.167 0.167 0.167 0.333 0.5 0.5⎠
(6.40)
0.25 0.25 0.417 0.5 0.5 0.5 0.333 0.167 0.167
λc = 1 1 3 3 3 3 3 2 2
(6.41)
– CondorcetConsensus (CC): just like Borda voting, the Condorcet voting method’s
origin dates from the French revolution period, as an effort to address the shortcomings
of simple majority voting when there are more than two candidates (Condorcet,
1785). Although often considered to be a multi-step unweighed voting algorithm,
the Condorcet voting method can also be regarded as a positional voting strategy, as
it employs the voters’ preference choices between any given pair of candidates (van
Erp, Vuurpijl, and Schomaker, 2002). In particular, this voting method performs an
exhaustive pairwise candidate ranking comparison across voters, and the winner of
each one of these one-to-one confrontations scores a point. The result of this process
is the Condorcet score matrix CE, the(i,j)th element of which indicates how many
candidates does the ith candidate beat in one-to-one comparisons in the jth election
(where candidates are clusters and an election corresponds to the clusterization of an
object).
Algorithm 6.4 presents a description of the CondorcetConsensus consensus function.
As in BordaConsensus, the Rank procedure must take into account whether the scalar
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