TESI DOCTORAL - La Salle

Chapter 6. Voting based consensus functions for soft cluster ensembles
clustering process, the interpretation of the scalar values λij contained in Λ differs (i.e. if λij
represent membership probabilities, the larger their value the stronger the object-cluster
association, whereas the opposite interpretation should be made in case they represent
object to centroid distances).
Either way, fuzzy clustering can be regarded as a version of hard clustering with relaxed
object membership restrictions. Such relaxation is particularly useful when the clusters are
not well separated, or when their boundaries are ambiguous. Moreover, object to cluster
association information may be of help in discovering more complex relations between a
given object and the clusters (Xu and Wunsch II, 2005). Furthermore, notice that soft
clustering can also be regarded as a generalization of its hard counterpart, as a crisp partition
can always be derived from a fuzzy one, whereas the opposite cannot be held.
However, these apparent strengths of soft clustering algorithms have barely been reflected
in the development of consensus functions especially devised for combining the outcomes
of multiple fuzzy clustering processes, as most proposals in this area are oriented
towards their application on hard clustering scenarios. Nevertheless, as described in section
2.2, there exist a few works in the consensus clustering literature devoted to the derivation
of consensus functions for soft cluster ensembles, such as the VMA consensus function of
(Dimitriadou, Weingessel, and Hornik, 2002) or the ITK consensus function of (Punera and
Ghosh, 2007). Moreover, several other consensus functions can be indistinctly applied on
both hard and soft cluster ensembles, such as PLA (Lange and Buhmann, 2005) or HGBF
(Fern and Brodley, 2004), while others, originally devised for hard cluster ensembles, can be
adapted for their use as soft partition combiners with relative ease (e.g. (Strehl and Ghosh,
2002)).
In this chapter, we make several proposals regarding the application of consensus processes
on soft cluster ensembles. For starters, the notion of soft cluster ensembles is reviewed
in section 6.1. Next, in section 6.2, we describe a procedure for adapting the hard consensus
functions employed in this work to soft cluster ensembles. In section 6.3, a family of
novel soft consensus functions based on the application of cluster disambiguation and voting
strategies is proposed. Finally, the results of several experiments regarding the performance
of the proposed soft consensus functions are presented in section 6.4, and the discussion of
section 6.5 puts an end to this chapter.
6.1 Soft cluster ensembles
As described in chapter 2, cluster ensembles are nothing but the compilation of the outputs
of multiple (namely l) clustering processes. Focusing on a fuzzy clustering scenario, and
making the simplyfing assumption that the l clustering processes partition the data into k
clusters, a soft cluster ensemble E is represented by means of a kl × n real-valued matrix:
⎛ ⎞
Λ1
⎜
⎜Λ2
⎟
E = ⎜ ⎟
⎝ . ⎠
Λl
(6.6)
where Λi is the k × n real-valued clustering matrix resulting from the ith soft clustering
process (∀i ∈ [1,l]).
165