TESI DOCTORAL - La Salle

6.3. Voting based consensus functions
The interpretation of the contents of S Λ1 ,Λ 2 is the same as in the crisp scenario, i.e. its
(i,j)th element is proportional to the similarity between the ith cluster of Λ1 and the jth
cluster of Λ2. Again, transforming S Λ1 ,Λ 2 into a cluster dissimilarity matrix is the final
step before solving the weight bipartite matching problem using the Hungarian method
implementation of (Buehren, 2008), thus obtaining the cluster correspondence vector π Λ1 ,Λ 2
of equation (6.27) (notice that this is exactly the same permutation vector of equation (6.19),
as the present toy example is the fuzzy version of the former).
π Λ1 ,Λ 2 = [3 1 2] (6.27)
Although the interpretation of the cluster correspondence vector is equivalent in both
the hard and the soft clustering scenarios (i.e. the cluster that is given the number ‘1’
identifier in Λ1 corresponds to the cluster number ‘3’ of Λ2, and so on), recall that, in the
fuzzy case, cluster permutations are equivalent to row order rearrangements.
Consequently, in order to obtain the cluster permuted version of the fuzzy partition
Λ1, it is necessary to multiply the transpose of the cluster permutuation matrix PΛ1 ,Λ2 associated to the cluster correspondence vector πΛ1 ,Λ by the fuzzy partition Λ1 itself. As
2
a result, the cluster permuted soft clustering Λ πΛ1 ,Λ2 1 is obtained —see equation (6.28) for
the pair of cluster aligned fuzzy clustering solutions of our toy example.
Λ πΛ 1 ,Λ 2
1
⎛
0.921 0.932 0.905 0.025 0.019 0.030 0.014 0.055
⎞
0.017
= ⎝0.025
0.042 0.038 0.006 0.005 0.011 0.976 0.929 0.972⎠
0.054
⎛
0.932
0.026
0.921
0.057
0.019
0.969
0.030
0.976
0.014
0.959
0.025
0.009
0.057
0.016
0.017
0.010
⎞
0.055
Λ2 = ⎝0.042
0.025 0.005 0.011 0.009 0.006 0.038 0.972 0.929⎠
(6.28)
0.026 0.054 0.976 0.959 0.976 0.969 0.905 0.010 0.016
Given a cluster ensemble E containing a set of l soft clustering solutions, the cluster
disambiguation process consists in, taking one of them as a reference, apply the Hungarian
method sequentially on the remaining l − 1 clustering solutions (Topchy et al., 2004). As a
result, a cluster aligned version of the cluster ensemble is obtained, and voting procedures
can be readily applied on it.
6.3.2 Voting strategies
Once the correspondence between the k clusters of each one of the l soft clustering solutions
compiled in the cluster ensemble E has been resolved and the corresponding cluster
permutations have been applied, it is time to derive the consensus clustering solution upon
E, a task we tackle by means of voting procedures. In this section, we describe four voting
methods, which give rise to as many consensus functions.
Before proceeding to their description, recall that the scalar elements of a soft cluster
ensemble E are considered, from a voting standpoint, as the expression of the degree of
preference of each voter (i.e. clusterer) for each candidate (cluster) in the present election
(clusterization of an object). The result of the election (i.e. the consolidated clusterization
of the object under consideration based upon the decisions of the l clusterers comprised
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