TESI DOCTORAL - La Salle

Chapter 6. Voting based consensus functions for soft cluster ensembles
⎛
1.853 1.853 0.924 0.055 0.033 0.055 0.071 0.072
⎞
0.072
ΣE = ⎝0.067
0.067 0.043 0.017 0.014 0.017 1.014 1.901 1.901⎠
(6.31)
0.08 0.08 1.033 1.928 1.952 1.928 0.914 0.026 0.026
Notice that the higher the value of the (i,j)th entry of ΣE, the more likely is that the
jth object belongs to the ith cluster. Of course, this is due to the fact that the l =2
fuzzy clusterings contained in the soft cluster ensemble of our toy example express
object to cluster associations by means of membership probabilities, which are directly
proportional to the strength of association between objects and clusters.
Moreover, notice that if each column of ΣE is divided by its L1-norm (i.e. the sum
of its elements), each column entries become cluster membership probabilities, and,
therefore, a classic fuzzy consensus clustering solution Λc can be obtained (see equation
(6.32)).
⎛
0.926 0.926 0.462 0.027 0.016 0.028 0.035 0.036
⎞
0.036
Λc = ⎝0.033
0.033 0.021 0.008 0.007 0.008 0.507 0.951 0.951⎠
(6.32)
0.041 0.041 0.517 0.965 0.977 0.964 0.458 0.013 0.013
Furthermore, notice that Λc can be transformed into a crisp consensus clustering
λc by simply assigning each object to the cluster it is most strongly associated to,
breaking hypothetical ties at random. Referring once more to our toy example, the
crisp consensus clustering obtained by hardening Λc is the one presented in equation
(6.33).
λc = 1 1 3 3 3 3 2 2 2
(6.33)
– ProductConsensus (PC): the only difference between this consensus function and
SumConsensus is that the preference values per candidate are multiplied instead of
added. Quite obviously, the product rule is highly sensitive to low values, which could
ruin the chances of a candidate on winning the election, no matter what its other confidence
values are (van Erp, Vuurpijl, and Schomaker, 2002). Equation (6.34) presents
the voting process that constitutes the core of the ProductConsensus consensus function.
It is important to notice that Λi correspond to the cluster ensemble components
once cluster alignment has been conducted, and matrix products are computed entrywise.
As a result, the k × n product matrix ΠE is obtained.
ΠE =
l
i=1
Λi
(6.34)
Algorithm 6.2 presents the schematic description of the ProductConsensus consensus
function. As in the previous consensus function, we propose converting the fuzzy
clusterings Λi into membership probability matrices, which allows to apply the product
voting rule on them once the cluster correspondence problem has been solved by
means of the Hungarian algorithm. It can be observed that ProductConsensus yields
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