TESI DOCTORAL - La Salle

Chapter 6. Voting based consensus functions for soft cluster ensembles
in the ensemble) will depend on the voting procedure applied, which, at the same time, is
conditioned by the way the voters’ preferences are expressed.
Given that in fuzzy cluster ensembles voters express their preference for each and every
one of the k candidates, the soft consensus functions proposed in this chapter make use of
confidence and positional voting methods (van Erp, Vuurpijl, and Schomaker, 2002), which
are applicable in voting scenarios in which voters grade candidates according to their degree
of confidence. The former makes direct use of the specific values of the preference scores
the voters emit –thus, they are sensitive to their scaling–, whereas the latter are based on
ranking the candidates according to the degree of confidence expressed by the voters.
As mentioned earlier, the way fuzzy clusterers express their preference for the clusters
can be either directly or inversely proportional to the strength of association between objects
and clusters (e.g. membership probabilities or distances to centroids, respectively). In fact,
it is possible that both types of clusterings are intermingled in E, and, for this reason, the
voting method must somehow be informed of this fact, so that appropriate scale or ranking
transformations are applied —depending on whether a confidence or a positional voting
strategy is employed.
In the following sections, we present four consensus functions for soft cluster ensembles,
each of which is based on a specific voting mechanism. Besides their generic description,
we illustrate them by means of a toy example using the soft cluster ensemble E containing
the l = 2 cluster aligned fuzzy clustering solutions presented in equation (6.28).
E =
Λ1
Λ2
⎛
0.921 0.932 0.905 0.025 0.019 0.030 0.014 0.055
⎞
0.017
⎜
⎜
0.025
⎜
= ⎜0.054
⎜
⎜0.932
⎝0.042
0.042
0.026
0.921
0.025
0.038
0.057
0.019
0.005
0.006
0.969
0.030
0.011
0.005
0.976
0.014
0.009
0.011
0.959
0.025
0.006
0.976
0.009
0.057
0.038
0.929
0.016
0.017
0.972
0.972 ⎟
0.010 ⎟
0.055 ⎟
0.929⎠
0.026 0.054 0.976 0.959 0.976 0.969 0.905 0.010 0.016
(6.29)
Notice that, in this toy example, the l = 2 clustering solutions (voters) compiled in
the ensemble (Λ1 and Λ2) express object to cluster associations (their preferences for candidates)
by means of membership probabilities, which makes the scalar elements of both
clusterings directly comparable, thus avoiding the need for applying any scale transformations.
However, this need not be the general case, and we will address how to deal with
cluster ensembles containing unequal clusterings as the proposed consensus functions are
presented throughout the following paragraphs.
Confidence voting
Consensus functions based on confidence voting methods derive the consolidated clustering
solution upon the values of the confidence scores each clusterer assigns to each cluster. For
this reason, a prerequisite for using these voting methods is that these confidence values are
comparable in magnitude (van Erp, Vuurpijl, and Schomaker, 2002). Assuming this is true,
we propose the application of the sum and product confidence voting rules, which gives rise
to the following two consensus functions:
– SumConsensus (SC): this consensus function is based on the application of the
confidence voting sum rule, which simply consists of adding the confidence values
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