TESI DOCTORAL - La Salle

6.3. Voting based consensus functions
Input: Soft cluster ensemble E containing l fuzzy clusterings Λi (∀i =1...l)
Output: Sum voting matrix ΣE
Data: k clusters, n objects
Hungarian (E)
ΣE = 0k×n for i =1...l do
if Λi not membership probabilities then
Probabilize (Λi)
end
ΣE = ΣE + Λi
end
Algorithm 6.1: Symbolic description of the soft consensus function SumConsensus.
Probabilize and Hungarian are symbolic representations of the conversion of fuzzy clusterings
to membership probability matrices and the cluster disambiguation procedures,
respectively, while 0 k×n represents a k × n zero matrix.
that all the voters cast for each candidate. As a result, a k × n sum matrix ΣE is
obtained, the (i,j)th entry of which equals the sum of the preference scores of assigning
the jth object to the ith cluster across the l cluster ensemble components, as presented
in equation (6.30).
ΣE =
l
i=1
Λi
(6.30)
where Λi refers to the ith clustering contained in the soft cluster ensemble E, once
the cluster disambiguation process has been conducted.
A schematic and generic description of the SumConsensus consensus function is presented
in algorithm 6.1. As it can be observed, we propose transforming all the fuzzy
clusterings compiled in the cluster ensemble E into membership probability matrices
–which is symbolically represented by the procedure called Probabilize–, thus making
the sum voting method directly applicable on them once the cluster alignment
problem is solved by means of the Hungarian procedure. According to algorithm
6.1, SumConsensus outputs the sum voting matrix ΣE, which can be interpreted
readily as a fuzzy consensus clustering. However, it can easily be converted into a
classic membership probability based fuzzy consensus clustering Λc, oracrispconsensus
clustering λc, as we show in the following paragraphs —as it will be seen, such
postprocessing could also be included as the final step of SumConsensus.
The application of SumConsensus on the toy cluster ensemble of equation (6.29) gives
rise to the sum matrix ΣE presented in equation (6.31). Notice that the execution of
the Probabilize and the Hungarian procedures is not required in this case, as the
l = 2 fuzzy clusterings considered express object to cluster associations by means of
membership probabilities and their clusters are aligned.
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