TESI DOCTORAL - La Salle

Chapter 6. Voting based consensus functions for soft cluster ensembles
probabilities estimated from the reference clustering solution (i.e. it finds the optimal
cluster permutation that yields the largest probability mass over all cluster assignment
probabilities (Fischer and Buhmann, 2003)). Depending on whether the aforementioned
clustering solutions correspond to hard or fuzzy partitions, cluster permutations amount to
label reassignments or to row order rearrangements, respectively.
The Hungarian algorithm poses the cluster correspondence problem as a weighted bipartite
matching problem, solving it in O(k3 ) time. A beautiful analysis of its error probability
can be found in (Topchy et al., 2004). In this work, we have employed the implementation
of (Buehren, 2008), which bases the clusters disambiguation process upon a measure of
the dissimilarity between the clusters of the two clustering solutions under consideration.
Cluster dissimilarity is usually embodied in a k × k matrix, the (i,j)th entry of which is
proportional to the degree of dissimilarity between the ith cluster of one of the clustering
solutions and the jth cluster of the other one.
Cluster dissimilarity can easily be derived upon the considered pair of clustering solutions,
regardless of whether they are hard or fuzzy partitions, as we show next. In the crisp
case, a cluster similarity matrix S λ1 ,λ 2 can be obtained by simple matrix products between
the incidence matrices of both clusterings, denoted as λ1 and λ2 —see equation (6.15).
Sλ1 ,λ = Iλ1 2 Iλ2
T
(6.15)
For illustration purposes, consider the two crisp clustering solutions of equation (6.16):
λ1 =[222111333]
λ2 =[113333322] (6.16)
The incidence matrices corresponding to λ1 and λ2 are presented in equation (6.17).
Iλ1 =
⎛
0 0 0 1 1 1 0 0
⎞
0
⎝1
1 1 0 0 0 0 0 0⎠
0 0 0 0 0 0 1 1 1
Iλ2 =
⎛
1 1 0 0 0 0 0 0
⎞
0
⎝0
0 0 0 0 0 0 1 1⎠
(6.17)
0 0 1 1 1 1 1 0 0
The cluster similarity matrix derived upon these two clustering solutions is the one
presented in equation (6.18).
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