TESI DOCTORAL - La Salle

Chapter 2. Cluster ensembles and consensus clustering
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Figure 2.1: Scatterplot of an artificially generated two-dimensional toy data set containing
n = 9 objects grouped into k = 3 natural clusters. Each object is identified by a numerical
label.
labeling is an integer label identifying to which of the k clusters each of the n objects in
the data set is assigned to.
For illustration purposes, and resorting to the toy clustering example presented in section
1.2.1, equation (2.2) presents a hard cluster ensemble created by compiling the outcomes of
l = 3 independent runs of the k-means clustering algorithm on the two-dimensional data set
presented in figure 2.1, which contains n = 9 objects, setting the desired number of clusters
k equal to 3.
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⎛
1 1 1 3 3 3 2 2
⎞
2
E = ⎝2
2 2 1 1 1 3 3 3⎠
(2.2)
2 2 2 3 3 3 1 1 1
On its part, a soft cluster ensemble E is defined as the compilation of the outcomes of l
fuzzy clustering processes, and as such, it is mathematically expressed as a kl×n matrix, as
presented in equation (2.3) (Punera and Ghosh, 2007; Sevillano, Alías, and Socoró, 2007b).
⎛ ⎞
Λ1
⎜
⎜Λ2
⎟
E = ⎜ ⎟
⎝ . ⎠
Λl
(2.3)
where Λi is the k × n real-valued clustering matrix resulting from the ith soft clustering
process (∀i ∈ [1,...,l]).
Continuing with the same toy example, equation (2.4) presents a soft cluster ensemble
created by collecting the outcomes of l = 3 independent executions of the fuzzy c-means
clustering algorithm on the same data set as before. As k = 3, the first three rows of
E correspond to the clustering probability membership matrix output by the first soft
clustering process, the next three are the outcome of the second fuzzy clusterer, and so on.
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