TESI DOCTORAL - La Salle

6.3. Voting based consensus functions
classifier ensembles, as categories (i.e. candidates) are univocally defined in that case. We
elaborate on the cluster disambiguation technique employed in this work in section 6.3.1. It
is important to highlight that consensus functions based on object co-association matrices
circumvent this inconvenience (Lange and Buhmann, 2005), although their main drawback
is that the complexity of the object co-association matrix computation is quadratic with
the number of objects in the data set (Long, Zhang, and Yu, 2005).
The problem of combining the outcomes of multiple soft clustering processes by means
of voting strategies implies interpreting the contents of the soft cluster ensemble as the
preference of each voter (clusterer) for each candidate (cluster), as soft clustering algorithms
output the degree of association of each object to all the clusters. For this reason, voting
methods capable of dealing with voters’ preferences (in particular, confidence and ranking
voting strategies) are the basis of our consensus functions, as they lend themselves naturally
to be applied in this context. However, care must be taken as regards how these preferences
are expressed, that is, if they are directly or inversely proportional to the strength of the
association between objects and clusters (e.g. membership probabilities or distances to
centroids, respectively). In section 6.3.2, we describe four voting strategies that give rise to
the proposed consensus functions.
6.3.1 Cluster disambiguation
In this section, we elaborate on the problem of cluster disambiguation, also known as the
cluster correspondence problem.
As pointed out earlier, a single hard clustering solution can be expressed by multiple
equivalent labeling vectors λ, due to the symbolic nature of the labels clusters are identified
with. This also occurs in soft clustering, as the permutation of the rows of a clustering
matrix Λ also gives rise to equivalent fuzzy partitions. Quite obviously, this cluster identification
ambiguity also rises between the multiple clustering solutions compiled in a cluster
ensemble E, and thus, it becomes an issue of concern when it comes to applying voting
strategies for conducting consensus clustering, given the equivalence between clusters and
candidates defined by the previously described analogy with voting procedures. For this
reason, our voting-based consensus functions for soft cluster ensembles make use of a cluster
disambiguation technique prior to proper voting.
In particular, we require from such method the ability to solve the cluster re-labeling
problem —an instance of the cluster correspondence problem in which a one to one correspondence
between clusters is considered (recall that, in this work, all the clusterings in the
ensemble and the consensus clustering are assumed to have the same number of clusters,
namely k).
To solve the cluster re-labeling problem we make use of the Hungarian method (also
known as Kuhn-Munkres algorithm or Munkres assignment algorithm) (Kuhn, 1955), a
technique that allows to obtain the most consistent alignment among the different clusterings
(Ayad and Kamel, 2008).
Given a pair of clustering solutions with k clusters each, the Hungarian method is capable
of finding, among the k! possible cluster permutations, the one that maximizes the overlap
between them. In particular, such cluster permutation is applied on one of the two clustering
solutions, while the other is taken as a reference. Put in probabilistic terms, the Hungarian
algorithm selects the cluster permutation that best fits the empirical cluster assignment
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