TESI DOCTORAL - La Salle

Chapter 2. Cluster ensembles and consensus clustering
problem that is solved by identifying and consolidating groups of clusters (meta-clusters),
which is done by applying graph-based clustering to hyperedges in the hypergraph representation
of the cluster ensemble. In (Strehl and Ghosh, 2002), the authors apply the proposed
consensus functions on hard cluster ensembles, suggesting that they could be extended to a
fuzzy clustering integration scenario. Such extensions (in particular, the soft versions of the
CSPA and MCLA consensus functions, sCSPA and sMCLA, respectively) were introduced
in (Punera and Ghosh, 2007; Sevillano, Alías, and Socoró, 2007b).
The clustering combination problem was also formulated as a graph partitioning problem
in (Fern and Brodley, 2004). In particular, a bipartite graph is built from a hard cluster
ensemble, although the authors suggest that the same method can be applied for combining
soft clustering solutions after introducing minor modifications on the proposed consensus
function, which is called HBGF for Hybrid Bipartite Graph Formulation. As in previous
graph partitioning approaches to clustering combination, the desired number of clusters
must be set aprioriandpassed as a parameter of the consensus function. In contrast, HBGF
considers object and cluster similarity simultaneously when creating the consensus clustering
solution, an issue not considered by other graph partition based consensus functions as
CSPA and MCLA (Strehl and Ghosh, 2002).
More recently, the BALLS consensus function (Gionis, Mannila, and Tsaparas, 2007) operates
on a graph representation of the pairwise object co-dissociation matrix, viewing its
vertices as the objects in the data set, its edges being weighted by the pairwise object
distances. The rationale of the consensus clustering creation process is the iterative construction
of consensus clusters from compact and relatively isolated sets of close vertices,
which are then removed from the graph.
2.2.3 Consensus functions based on object co-association measures
The approach to consensus clustering based on object co-association measures is based on
the assumption that objects belonging to a natural cluster are likely to be co-located in
the same cluster by different clusterings of the cluster ensemble. Therefore, pairwise object
co-occurrences are deemed as votes which are aggregated in a n × n object co-association
matrix (where n is the number of instances contained in the data collection). A great
advantage of this type of methods is that it avoids cluster disambiguation processes, as the
cluster ensemble is inspected object-wise, rather than cluster-wise. However, a downside of
consensus functions based object co-association is that their time and space complexities
scale quadratically with n, thus making their application on large data sets highly costly or
even unfeasible.
One of the pioneering works on the combination of hard clustering solutions based on
object co-association metrics is the evidence accumulation (EAC) approach. In the original
form of the evidence accumulation consensus function, the consensus clustering solution is
obtained by applying a simple majority voting scheme on the co-association matrix (Fred,
2001). In subsequent versions, consensus is derived by applying the single-link hierarchical
clustering algorithm on the object co-association matrix, regarding it as a measure of the
similarity between objects (Fred and Jain, 2002a)—a virtually identical proposal is found
in a contemporary work (Zeng et al., 2002). In (Fred and Jain, 2003), the evidence accumulation
approach is formulated in an information-theoretic framework, defining the optimal
consensus clustering solution as the one maximizing the sum of normalized mutual infor-
37