TESI DOCTORAL - La Salle

Chapter 6. Voting based consensus functions for soft cluster ensembles
I πλ
⎛
0
1 ,λ2 λ = ⎝0
1
1
1
0
0
⎞ ⎛
0 0
1⎠⎝1
0 0
0
1
0
0
1
0
1
0
0
1
0
0
1
0
0
0
0
1
0
0
1
⎞ ⎛
0 1
0⎠
= ⎝0
1 0
1
0
0
1
0
0
0
0
1
0
0
1
0
0
1
0
1
0
0
1
0
⎞
0
1⎠
0
(6.22)
Therefore, assigning each object to the cluster it is most strongly associated to transforms
the cluster permuted incidence matrix I πλ1 ,λ2 λ into the disambiguated crisp clustering
1
λ πλ1 ,λ2 1 . In equation (6.23), this clustering is presented alongside λ2 —the clustering that
has been taken as the reference of the cluster disambiguation process.
λ πλ 1 ,λ 2
1 =[111333222] (6.23)
λ2 =[113333322] (6.24)
In the context of our voting-based soft consensus functions, though, cluster disambiguation
is conducted on pairs of soft clustering solutions. In order to illustrate how to proceed
in this case, we use a toy example that is the fuzzy version of the one just reported. For
brevity, we will only consider the case in which object-to-cluster associations are expressed
in terms of membership probabilities, although an analog procedure could be devised in
the case these were expressed by means of other metrics. Therefore, given the two fuzzy
partitions Λ1 and Λ2 of equation (6.25):
⎛
0.054 0.026 0.057 0.969 0.976 0.959 0.009 0.016
⎞
0.010
Λ1 = ⎝0.921
0.932 0.905 0.025 0.019 0.030 0.014 0.055 0.017⎠
0.025
⎛
0.932
0.042
0.921
0.038
0.019
0.006
0.030
0.005
0.014
0.011
0.025
0.976
0.057
0.929
0.017
0.972
⎞
0.055
Λ2 = ⎝0.042
0.025 0.005 0.011 0.009 0.006 0.038 0.972 0.929⎠
(6.25)
0.026 0.054 0.976 0.959 0.976 0.969 0.905 0.010 0.016
The cluster similarity matrix S Λ1 ,Λ 2 is computed upon the soft clustering matrices themselves,
as described by equation (6.26).
SΛ 1 ,Λ 2 = Λ1Λ T 2 =
⎛
= ⎝
⎛
= ⎝
0.054 0.026 0.057 0.969 0.976 0.959 0.009 0.016 0.010
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
0.143 0.054 2.878
1.738 0.136 1.042
0.188 1.845 0.969
⎞
⎛
⎜
⎞ ⎜
⎠ ⎜
⎝
0.932 0.042 0.026
0.921 0.025 0.054
0.019 0.005 0.976
0.030 0.011 0.959
0.014 0.009 0.976
0.025 0.006 0.969
0.057 0.038 0.905
0.017 0.972 0.010
0.055 0.929 0.016
⎠ (6.26)
175
⎞
⎟
⎠