TESI DOCTORAL - La Salle

Chapter 2
Cluster ensembles and consensus
clustering
In our quest for overcoming clustering indeterminacies in a multimodal context, the notions
of cluster ensembles and consensus clustering play a central role. As mentioned at the
end of chapter 1, our strategy for clustering multimodal data in a robust manner is based
on the massive creation of multiple partitions of the target data set and the subsequent
combination of these into a single consensus clustering solution. Therefore, an appropriate
way to start this chapter is by formally defining the two closely related concepts of cluster
ensembles and consensus clustering1 .
For starters, a cluster ensemble E is defined as the compilation of the outcomes of l
clustering processes. For simplicity, we assume in this work that the l clustering processes
group the data into the same number of clusters, namely k, although this is not a strictly
necessary constraint2 . Depending on whether the clustering processes are crisp or fuzzy, E
will be a hard or a soft cluster ensemble.
In the former case, E is mathematically defined as a l×n integer-valued matrix compiling
l row label vectors λi (∀i ∈ [1,...,l]) resulting from the respective hard clustering processes
(see equation (2.1)).
⎛
λ1
λl
⎞
⎛
⎜
⎜λ2
⎟
E = ⎜ ⎟
⎝ . ⎠ =
⎜
⎝ .
λ11 λ12 ... λ1m
λ21 λ22 ... λ2m
. ..
λl1 λl2 ... λlm
⎞
⎟
⎠
(2.1)
where λij ∈{1,...,k} (∀i ∈ [1,...,l], and ∀j ∈ [1,...,n]), i.e. each component of each
1 In some works, the term ‘cluster ensemble’ is used to designate the framework for combining multiple
partitionings obtained from separate clustering runs into a final consensus clustering (Strehl and Ghosh,
2002; Punera and Ghosh, 2007). In this work, however, we stick to the literal meaning of this expression,
and use it to design the result of gathering several clustering solutions.
2 Since our goal is to combine partitions differing only in the way data are represented and clustered, we
set the number of clusters k to be equal across the l clustering processes. However, combining clustering
solutions with a variable number of clusters is a common practice in the cluster ensembles literature. This
can be useful for clustering complex data sets upon simple individual partitions (Fred and Jain, 2005), or
for discovering the natural number of clusters in the data set (Strehl and Ghosh, 2002), although these
potentialities are not exploited in this work.
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