TESI DOCTORAL - La Salle

Chapter 2. Cluster ensembles and consensus clustering
ensemble components, which are generated by random feature extraction and selection, weak
clusterers, or random number of clusters, among others. Unfortunately, the fitness function
driving the genetic algorithm used to select the best cluster ensemble is the classification
accuracy of the respective ensemble with respect to the correct object labels (ground truth),
which makes this strategy of limited use in practice.
Another interesting work that incides in the heuristics of the cluster ensemble construction
process is (Kuncheva, Hadjitodorov, and Todorova, 2006). In this sense, the authors
recommend creating individual partitions with a variable number of clusters as a means for
obtaining good quality consensus labelings.
As mentioned earlier, an alternative way of obtaining a good quality consensus clustering
solution is based not on designing the cluster ensemble components according to certain
heuristics, but on refining its contents. The rationale of such strategies is based on the fact
that the quality of the consensus clustering solution is penalized by the inclusion of poor
individual clustering solutions in the ensemble. For this reason, it seems logical to develop
techniques capable of discarding such cluster ensemble components prior to conducting the
consensus clustering process.
In this sense, the application of quality and/or diversity criteria for selecting a small
subset of a large cluster ensemble was evaluated in (Fern and Lin, 2008) as a means for obtaining
a consensus clustering solution that equals or betters the one that would be obtained
using the whole ensemble. Pursuing the same goal, the authors of (Goder and Filkov, 2008)
propose creating smaller subsets of the cluster ensemble that will yield better consensus.
These mini-cluster ensembles are generated by clustering the individual partitions of the
cluster ensemble using a hierarchical agglomerative average-link clustering algorithm.
2.2 Related work on consensus functions
The goal of this subsection is to review the state of the art in the area of consensus clustering.
Although a considerable corpus of theoretical work on combining classifications was
developed in the 80s and earlier (e.g. (Mirkin, 1975; Barthelemy, Laclerc, and Monjardet,
1986; Neumann and Norton, 1986)), it was not until the start of the present decade when
this field experienced a significant flourish of activity.
Despite this relatively recent awakening, multiple approaches to the combination of
several clusterings can be found in the literature. In general terms, consensus clustering
can be posed as an optimization problem the goal of which is to minimize a cost function
measuring the dissimilarity between the consensus clustering solution and the partitions in
the cluster ensemble –often, the cost function is expressed in terms of the number of pairwise
co-clustering disagreements between the individual partitions in the cluster ensemble and
the consensus clustering solution. Unfortunately, finding the partition that minimizes the
proposed symmetric difference distance metric (i.e. the so-called median partition) isa
NP-hard problem (Goder and Filkov, 2008), and this is the reason why it is necessary to
resort to distinct heuristics so as to conduct clustering combination.
Aiming to provide the reader with a global perspective on the distinct existing approaches
in this field, table 2.1 presents a taxonomy of some of the most relevant consensus
functions according to the theoretical foundations that guides the consensus process. Notice
that some consensus functions appear under more than one theoretical approach, as some-
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