TESI DOCTORAL - La Salle

6.3. Voting based consensus functions
Sλ1 ,λ = Iλ1 2 Iλ2
T
=
⎛
1 0
⎞
0
⎛
0
⎝1
0
0
1
0
0
1
0
1
0
0
1
0
0
1
0
0
0
0
1
0
0
1
⎜
1
⎜
⎞ ⎜
0
0 ⎜
⎜0
0⎠
⎜
⎜0
1 ⎜
⎜0
⎜
⎜0
⎝0
0
0
0
0
0
0
1
0 ⎟
1 ⎟ ⎛
1 ⎟ 0
1 ⎟ = ⎝2
1 ⎟ 0
1 ⎟
0⎠
0
0
2
⎞
3
1⎠
1
(6.18)
0 1 0
Firstly, notice that S λ1 ,λ 2 is not a symmetric matrix (as object co-association matrices
are), due to the fact that its rows and columns correspond to different entities. In fact,
the (i,j)th element of S λ1 ,λ 2 is equal to the number of objects that are assigned to the ith
cluster of λ1 and to the jth cluster of λ2, thus clearly indicating the degree of resemblance of
these clusters. Deriving a cluster dissimilarity matrix from S λ1 ,λ 2 is pretty straightforward,
provided that the implementation of the Hungarian method employed in this work does not
require that the cluster dissimilarity measures verify any special property as regards their
scale. The result of the cluster disambiguation method applied on this toy example is the
cluster correspondence vector π λ1 ,λ 2 presented in equation (6.19).
π λ1 ,λ 2 = [3 1 2] (6.19)
The interpretation of the cluster correspondence vector πλ1 ,λ is that the cluster iden-
2
tified with the ‘1’ label in λ1 corresponds to the cluster with label ‘3’ of λ2, the cluster
labeled as ‘2’ in λ1 must be aligned with the cluster with label ‘1’ of λ2, and the cluster ‘3’
of λ1 is equivalent to cluster with label ‘2’ of λ2.
The most usual way of representing the information contained in the cluster correspondence
vector πλ1 ,λ is by means of a cluster permutation matrix P 2 λ1 ,λ . In general, P 2 λ1 ,λ is 2
a k × k matrix whose entries are all zero except that the πλ1 ,λ (i)-th entry of the ith row is
2
equal to 1. The cluster permutation matrix corresponding to our toy example is presented
in equation (6.20). Notice how all of its entries are zero except for the third entry of the
first row (as πλ1 ,λ (1) = 3), the first entry of the second row (as π 2 λ1 ,λ (2) = 1) and the
2
second entry of the third row (as πλ1 ,λ (3) = 2).
2
P λ1 ,λ 2 =
⎛
0 0
⎞
1
⎝1
0 0⎠
(6.20)
0 1 0
In order to obtain the cluster permuted version of the clustering λ1, it is necessary to
multiply the transpose of the cluster permutation matrix Pλ1 ,λ by the incidence matrix
2
associated to this clustering, Iλ1 , which yields the cluster permuted incidence matrix I πλ1 ,λ2 λ , 1
as indicated in equation (6.21).
I πλ 1 ,λ 2
λ 1
T
= Pλ1 ,λ Iλ1
2 ,λ2 In our example, the cluster permuted incidence matrix is:
174
(6.21)