TESI DOCTORAL - La Salle

Chapter 6. Voting based consensus functions for soft cluster ensembles
the ith cluster according to λ, and 0 otherwise —see equation (6.7), which presents the
incidence matrix corresponding to the label vector λ of equation (6.2).
⎛
0 0 0 1 1 1 0 0
⎞
0
Iλ = ⎝1
1 1 0 0 0 0 0 0⎠
(6.7)
0 0 0 0 0 0 1 1 1
Notice that the information contained in Iλ is somehow comparable to the contents of
a soft clustering matrix Λ, in the sense that they both express the degree of association
between objects and clusters. For illustration purposes, equation (6.8) presents the fuzzy
clustering matrix Λ output by the FCM clustering algorithm on the artificial data set of
figure 6.1. In fact, rounding each element of this clustering matrix Λ to the nearest integer
would indeed yield the incidence matrix Iλ of equation (6.7).
⎛
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⎠
(6.8)
0.025 0.042 0.038 0.006 0.005 0.011 0.976 0.929 0.972
The construction of object co-association matrices given the incidence matrix Iλ built
upon a hard clustering solution λ is pretty straightforward, and it only requires computing
some matrix products.
In particular, the object co-association matrix Oλ is computed as the product between
the transpose of Iλ and Iλ. The object co-association matrix corresponding to the hard
clustering solution obtained on our toy clustering example is presented in equation (6.9).
In fact, Oλ is a n × n adjacency matrix, the (i,j)th entry of which equals 1 if the ith
and the jth objects are placed in the same cluster, or 0 otherwise (Kuncheva, Hadjitodorov,
and Todorova, 2006). The name object co-association matrix stems from the fact that the
contents of Oλ indicate, from a clustering viewpoint, the degree of similarity between the
n objects in the data set.
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