TESI DOCTORAL - La Salle

Chapter 6. Voting based consensus functions for soft cluster ensembles
their version for soft cluster ensembles becomes pretty straightforward, as it simply requires
substituting the object co-association matrix derived from a hard cluster ensemble OEλ by
its fuzzy counterpart, OEΛ . Notice, that, despite taking object co-association matrices
derived upon a soft cluster ensemble as their input, all these consensus functions output a
hard consensus clustering solution λc.
As mentioned at the introduction of this section, we have established an analogy between
hard and soft consensus functions based on the similarities between object co-association
matrices, assuming that the results of fuzzy clustering are expressed in terms of membership
probabilities. However, we consider that the present study is quite generic, as it could be
extended to the case fuzzy clustering results were expressed in any other form by transforming
them into membership probabilities prior to computing the corresponding object
co-association matrices.
6.3 Voting based consensus functions
In this section, we put forward a set of proposals in the shape of a family of novel consensus
functions especially devised for their application on soft cluster ensembles. These consensus
functions are inspired in voting strategies, which have also been a source of inspiration for
the development of systems for combining supervised classifiers (van Erp, Vuurpijl, and
Schomaker, 2002), search engines (Aslam and Montague, 2001), or word sense disambiguation
systems (Buscaldi and Rosso, 2007). A distinguishing factor of the consensus functions
we propose in this section is that they yield fuzzy consensus clustering solutions, whereas
other soft consensus functions found in the literature ouput crisp consensus clusterings,
despite they are applied on soft cluster ensembles (Punera and Ghosh, 2007).
In fact, voting is a formal way of combining the opinions of several voters into a single
decision (e.g. the election of a president). Therefore, it seems quite logical that voting
strategies can be readily applied for combining the outcomes of multiple decision systems,
using the voting strategy that best fits the way these decisions are expressed.
Given that a clusterer is an unsupervised classifier, the most natural parallelism we can
establish is related to voting based supervised classifier combination (aka classifier ensembles).
In this case, each classifier is a voter, the possible categories objects can be filed under
are the candidates, and an election is the classification of an object (van Erp, Vuurpijl, and
Schomaker, 2002). Quite obviously, the voting strategy employed for combining the votes
–and thus obtain the winner of the election (i.e. the resulting classification of the object by
the ensemble of classifiers)– depends on how votes are expressed, be it an assignment to a
single class (i.e. single-label classification (Sebastiani, 2002)), or an either ranked or scored
list of classes. The former case calls for the application of unweighed voting methods such
as plurality or majority voting, whereas the latter make it possible to apply more sophisticated
voting strategies such as confidence and ranking voting methods (van Erp, Vuurpijl,
and Schomaker, 2002).
Nevertheless, prior to the application of any voting strategy on soft cluster ensembles,
there is a crucial problem to be solved. This has to do with the inherent unsupervised
nature of clustering processes, which makes clusters be ambiguously identified. Therefore,
it is necessary to perform a cluster alignment (or disambiguation) process before voting.
Notice that this is not an issue of concern when applying voting strategies on supervised
171