TESI DOCTORAL - La Salle

Chapter 6
Voting based consensus functions
for soft cluster ensembles
As outlined in section 1.2.1, clustering algorithms can be bisected into two large categories,
depending on the number of clusters every object is assigned to. On one hand, hard (or
crisp) clustering algorithms assign each object to a single cluster. For this reason, the
result of applying a hard clustering process on a data set containing n objects is usually
represented as a n-dimensional integer-valued row vector of labels (or labeling) λ, each
component of which identifies to which of the the k clusters each object is assigned to, that
is:
λ =[λ1 λ2 ... λn] (6.1)
where λi ∈ [1,k], ∀i ∈ [1,n].
On the other hand, soft (or fuzzy) clustering algorithms allow the objects to belong to
all clusters to a certain extent. Thus, the results of their application for partitioning a data
set containing n objects into k clusters is usually represented by means of a k×n real-valued
clustering matrix Λ –see equation (6.2)–, the (i,j)th entry of which indicates the degree of
association between the jth object and the ith cluster.
⎛
⎜
Λ = ⎜
⎝
λ11 λ12 ... λ1n
λ21 λ22 ... λ2n
.
. ..
λk1 λk2 ... λkn
.
⎞
⎟
⎠
(6.2)
where λij ∈ R, ∀i ∈ [1,k]and∀j ∈ [1,n].
For illustration purposes, we resort to the toy clustering example presented in chapter
2, in which clustering is conducted on the two-dimensional artificial data set presented in
figure 6.1. This toy data collection contains n = 9 objects, and the desired number of
clusters k is set to 3.
If the classic k-means hard clustering algorithm is applied on this data set, the label
vector presented in equation (6.3) is obtained. Recall that the labels λi contained in λ
are purely symbolic (i.e. the labelings λ =[111222333]orλ =[333222111]
represent exactly the same partition of the data).
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