TESI DOCTORAL - La Salle

6.3. Voting based consensus functions
Input: Soft cluster ensemble E containing l fuzzy clusterings Λi (∀i =1...l)
Output: Product voting matrix ΠE
Data: k clusters, n objects
Hungarian (E)
ΠE = 1k×n for i =1...l do
if Λi not membership probabilities then
Probabilize (Λi)
end
ΠE = ΠE ◦ Λi
end
Algorithm 6.2: Symbolic description of the soft consensus function ProductConsensus.
Probabilize and Hungarian are symbolic representations of the conversion of fuzzy clusterings
to membership probability matrices and the cluster disambiguation procedures,
respectively, while 1 k×n represents a k × n unit matrix, and ◦ represents the Hadamard
(or entrywise) matrix product.
the product voting matrix ΠE as its output. However, as in the case of SumConsensus,
it can be transformed into a fuzzy or a crisp consensus clustering, a process that
can be included as the final step of ProductConsensus.
The results of applying the product rule on the toy cluster ensemble of equation (6.29)
yields the product matrix ΠE presented in equation (6.35).
⎛
ΠE = ⎝
0.858 0.858 0.017 7.5·10 −4 2.7·10 −4 7.5·10 −4 7.9·10 −4 9.3·10 −4 9.3·10 −4
0.001 0.001 1.9·10−4 6.6·10−5 4.5·10−5 6.6·10−5 0.037 0.903 0.903
0.001 0.001 0.056 0.929 0.953 0.929 0.008 1.6·10−4 1.6·10−4 ⎞
⎠
(6.35)
Dividing each column of ΠE by its L1-norm gives rise to the fuzzy consensus clustering
solution Λc based on membership probabilities of equation (6.36), and assigning each
object to the cluster it is most strongly associated to (breaking ties randomly) yields
the crisp consensus clustering λc of equation (6.37).
⎛
Λc = ⎝
0.997 0.997 0.235 8.1·10−4 2.8·10−4 8.1·10−4 0.017 0.001 0.001
0.001 0.001 0.003 7.1·10−5 4.7·10−5 7.1·10−5 0.806 0.998 0.998
0.002 0.002 0.762 0.999 0.999 0.999 0.177 1.8·10−4 1.8·10−4 λc = 1 1 3 3 3 3 2 2 2
⎞
⎠ (6.36)
(6.37)
Notice that the differences between the fuzzy consensus clusterings Λc obtained by
SumConsensus and ProductConsensus (equations (6.32) and (6.36) ) –due to the
different way voters’ preferences are combined– are lost when they are transformed
into crisp consensus clusterings.
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