TESI DOCTORAL - La Salle

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Chapter 2. Cluster ensembles and consensus clustering
E
Consensus
function c
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Figure 2.2: Schematic representation of the obtention of a consensus labeling λc by applying
a consensus function F on a hard cluster ensemble E containing l = 3 individual label
vectors, being n =9andk =3.
⎛
0.921 0.932 0.905 0.006 0.005 0.011 0.009 0.016
⎞
0.010
⎜
0.054
⎜
0.025
⎜
⎜0.025
E = ⎜
⎜0.920
⎜
⎜0.054
⎜
⎜0.054
⎝0.920
0.042
0.026
0.026
0.932
0.042
0.042
0.932
0.057
0.038
0.038
0.905
0.057
0.057
0.905
0.025
0.969
0.969
0.006
0.025
0.025
0.006
0.019
0.976
0.976
0.005
0.019
0.019
0.005
0.030
0.959
0.959
0.011
0.030
0.030
0.011
0.976
0.014
0.014
0.009
0.976
0.976
0.009
0.929
0.055
0.055
0.016
0.929
0.929
0.016
0.972 ⎟
0.017 ⎟
0.017 ⎟
0.010 ⎟
0.972 ⎟
0.972 ⎟
0.010⎠
0.025 0.026 0.038 0.969 0.976 0.959 0.014 0.055 0.017
(2.4)
Notice that, just like it was observed in section 1.2.1 regarding crisp and fuzzy clustering
solutions, soft cluster ensembles can be converted to hard cluster ensembles by assigning
each object to the cluster it is more strongly associated to. In fact, by doing so, the soft
ensemble in equation (2.4) would be converted to the hard cluster ensemble in equation
(2.2). Moreover, notice that the l = 3 components that make up both cluster ensembles are
identical, given the symbolic nature of cluster labels.
As for consensus clustering, it is defined as the process of obtaining a consolidated
clustering solution through the application of a consensus function F on a cluster ensemble
E (Strehl and Ghosh, 2002). In other words, consensus clustering can be regarded as the
problem of combining several clustering solutions without accessing the features representing
the clustered objects. Figure 2.2 depicts a schematic representation of a consensus clustering
process conducted on the hard cluster ensemble resulting from our toy example. In this
case, the result of the consensus clustering process is a consensus label vector λc which,
quite obviously, represents the same partition as the individual label vectors that compose
the cluster ensemble. However, in a real context, a higher degree of diversity among the
clustering solutions embedded in the cluster ensemble can be expected (which, in fact, is
desirable), a situation consensus clustering algorithms take advantage of for consolidating
richer consensus clustering solutions (Pinto et al., 2007).
Quite obviously, the design of the consensus function F is a central issue as regards
consensus clustering. Most works in the consensus clustering literature focus on combining
the outcomes of hard clustering processes (as in the example depicted in figure 2.2), although
some consensus functions can be applied to either hard or soft cluster ensembles indistinctly,
possibly after introducing some minor modifications (Strehl and Ghosh, 2002; Fern and
Brodley, 2004; Lange and Buhmann, 2005). However, little effort has been conducted
towards the design of specific consensus functions for soft cluster ensembles that generate
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