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2.2. Related work on consensus functions
mation (φ (NMI) ) with respect to all the partitions in the cluster ensemble. The authors
prove that, by maximizing the number of shared objects in the consolidated clusters, the
EAC consensus function maximizes the aforementioned information theoretical objective
function, although reaching its global optimum is not ensured in all situations. Moreover,
cutting the dendrograms resulting from the application of the single-link clustering on the
co-association matrix at the highest lifetime level leads to a minimization of the variance
of the average φ (NMI) , which guarantees the robustness of the clustering solution to small
variations in the composition of the cluster ensemble —furthermore, this also avoids making
assumptions on the number of clusters, a significant advantage with respect to other
consensus functions. A compendium on the evidence accumulation consensus clustering
approach is presented in (Fred and Jain, 2005), extending the previous consensus functions
through the application of other hierarchical clustering algorithms on the pairwise object
co-association matrix.
The clustering of high dimensional data is the main motivation of the work presented
in (Fern and Brodley, 2003). In this scenario, Random Projection (RP) is an efficient
dimensionality reduction technique, although it often gives rise to highly unstable clustering
results. In order to reduce this variability, the authors propose creating cluster ensembles
by compiling partitions resulting from distinct RP runs, combining them using a consensus
function very similar to EAC, as it applies an agglomerative clustering algorithm on an
object similarity matrix.
One of the two consensus functions presented by Dudoit and Fridlyand (2003), named
BagClust2, resembles evidence accumulation, as it builds a pairwise object dissimilarity
matrix which is subject to a partitioning process for obtaining the consensus clustering.
However, BagClust2 and EAC differ in that the former requires that the desired number
of clusters is passed as a parameter to the consensus function (the same happens with
BagClust1).
In (Greene et al., 2004), consensus clustering was conducted by means of variants of the
EAC consensus functions using distinct hierarchical clustering algorithms (i.e. single-link,
complete-link and average-link) for partitioning the pairwise object co-association matrix,
as proposed in (Fred and Jain, 2005). However, the central matter of study in that work
is the analysis of the diversity of the cluster ensemble as a factor determining the quality
of the consensus clustering. In this sense, the authors focused on random techniques for
introducing diversity in the cluster ensemble, such as random subspacing, random algorithm
initialization, random number of clusters or random feature projection.
A related work is the Majority Rule consensus function of (Goder and Filkov, 2008),
which is also based on clustering the pairwise object co-dissociation matrix, which can be
done by simply setting a dissimilarity threshold like in the the first version of EAC (Fred,
2001), or by applying the average-link hierarchical clustering algorithm —like in the latest
versions of EAC (Fred and Jain, 2005).
Moreover, there exist several consensus functions that make indirect use of pairwise
object co-association (or co-dissociation) matrices, despite the way the consensus clustering
is obtained differs from that of EAC. Examples of this include some graph partition-based
consensus functions, such as CSPA (Strehl and Ghosh, 2002) and BALLS (Gionis, Mannila,
and Tsaparas, 2007), the Iterative Pairwise Consensus (IPC) (Nguyen and Caruana, 2007)
(a consensus function based on cluster centroids in which objects are iteratively reassigned
to the clusters of the consensus partition according to their similarity), or the CC Pivot
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