TESI DOCTORAL - La Salle

Chapter 6. Voting based consensus functions for soft cluster ensembles
0.6
0.4
0.2
0
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1
−0.2
−0.2 0 0.2 0.4 0.6
Figure 6.1: Scatterplot of an artificially generated two-dimensional data set containing n =9
objects, which are represented by coloured symbols and identified by a number. The black
star symbols represent the position of the cluster centroids found by the k-means algorithm.
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9
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λ =[222111333] (6.3)
As regards the results of applying a fuzzy clustering algorithm on this data collection,
these clearly differ depending on the way the degree of association between objects and
clusters is codified. Usually, the scalar values λij contained in the clustering matrix Λ
represent cluster membership probabilities (i.e. the higher the value of λij, the more strongly
the jth object is associated to the ith cluster). For instance, this is the way the well-known
fuzzy c-means (FCM) clustering algorithm codifies its clustering results (Höppner, Klawonn,
and Kruse, 1999). In fact, if this algorithm is applied on the previously described artificial
data set, the clustering matrix presented in equation (6.4) is obtained.
⎛
0.054 0.026 0.057 0.969 0.976 0.959 0.009 0.016
⎞
0.010
Λ = ⎝0.921
0.932 0.905 0.025 0.019 0.030 0.014 0.055 0.017⎠
(6.4)
0.025 0.042 0.038 0.006 0.005 0.011 0.976 0.929 0.972
It can be observed that any row permutation in Λ would yield an equivalent fuzzy partition.
Moreover, notice that Λ can be transformed into a hard clustering solution by simply
assigning each object to the cluster with maximum membership probability.
However, the degree of association between objects and clusters can be described in
terms of other parameters, such as the distance of each object to the cluster centroids (such
as k-means, that despite being a hard clustering algorithm, can output this information). In
fact, the object to centroid1 distance matrix obtained after applying k-means on the same
toy data set as before is presented in equation (6.5).
⎛
0.362 0.325 0.672 0.002 0.002 0.005 0.160 0.092
⎞
0.125
Λ = ⎝0.010
0.009 0.027 0.397 0.490 0.436 0.251 0.320 0.209⎠
(6.5)
0.170 0.202 0.445 0.090 0.125 0.162 0.002 0.005 0.002
In this case, the conversion of Λ into a crisp partition requires assigning every object
to the closest (i.e. minimum distance) cluster. Thus, depending on the nature of the soft
1 The cluster centroids are represented by means of a black star symbol in figure 6.1
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