TESI DOCTORAL - La Salle

Chapter 2. Cluster ensembles and consensus clustering
desambiguation is conducted by matching those clusters sharing the highest percentage
of objects, iterating this cluster matching process across all the partitions in the cluster
ensemble. As a result of the voting step, a fuzzy partition of the data set is obtained.
Subsequently, a merging procedure is conducted on this soft partition, fusing those clusters
which are closest to each other. By imposing a stopping criterion based on the sureness
of the obtained clusters, this merging process is capable of finding the natural number
of clusters in the data set. In (Dimitriadou, Weingessel, and Hornik, 2002), the authors
define the consensus clustering solution as the one minimizing the average square distance
with respect to all the partitions in the cluster ensemble. They demostrate that obtaining
such consensus clustering boils down to finding the optimal re-labeling of the clusters of
all the individual clusterings, which becomes an unfeasible problem if approached directly,
since it requires an enumeration of all possible permutations. Therefore, they resort to
the VMA consensus function of (Dimitriadou, Weingessel, and Hornik, 2001) for finding an
approximate solution to the problem, extending its application to soft cluster ensembles.
One of the two consensus functions proposed in (Dudoit and Fridlyand, 2003), called
BagClust1, is based on applying plurality voting on the labelings in the cluster ensemble
after a label disambiguation process based on measuring the overlap between clusters. The
generation of the cluster ensemble components follows a resampling strategy similar to
bagging, aiming to reduce the variability in the partitioning results via consensus clustering.
A related proposal was the one presented in (Fischer and Buhmann, 2003). In that work,
consensus clustering is viewed as a means for improving the quality and reliability of the
results of path-based clustering, applying bagging for creating the hard cluster ensemble.
The consensus clustering solution is obtained through a maximum likelihood mapping in
which the label permutation problem is solved by means of the Hungarian method (Kuhn,
1955), which somehow resembles the application of plurality voting on the disambiguated
individual partitions in the cluster ensemble (Ayad and Kamel, 2008). Moreover, a related
reliability measure chooses the number of clusters with the highest stability as the preferable
consensus solution.
In (Greene and Cunningham, 2006), a majoritary voting strategy was applied for generating
the consensus clustering solution, after disambiguating the clusters using the Hungarian
algorithm. An additional interest of that work is that it was one of the first research
efforts that considered the problem of creating and combining a large number of clustering
solutions in the context of high dimensional data sets (such as text document collections).
Indeed, the authors point out that using large ensembles boosts computational cost, while
small ensembles tend to produce unstable consensus clustering solutions. In this context,
the authors propose basing the cluster ensemble construction and the consensus clustering
tasks on a prototype reduction technique that allows representing the whole data set by
means of a minimal set of objects, ensuring that the clustering results will approximate
those that would be obtained on the original data set. By doing so, the final clustering
solution can be extended to those objects that have been left out of the reduced data set
representation while alleviating the overall computational cost of the whole process. In particular,
the reduced version of the cluster ensemble is obtained by projecting the pairwise
object similarity matrix by means of a kernel matrix.
The recent work of (Ayad and Kamel, 2008) presented a set of consensus functions based
on cumulative voting –named URCV, RCV and ACV– whose time complexity scales linearly
with the size of the data set. Another interesting feature is their capability of combining
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