TESI DOCTORAL - La Salle

⎛
OΛ = ΛT ⎜
Λ = ⎜
⎝
⎛
⎜
= ⎜
⎝
Chapter 6. Voting based consensus functions for soft cluster ensembles
0.054 0.921 0.025
0.026 0.932 0.042
0.057 0.905 0.038
0.969 0.025 0.006
0.976 0.019 0.005
0.959 0.030 0.011
0.009 0.014 0.976
0.016 0.055 0.929
0.010 0.017 0.972
⎞
⎟ ⎛
⎟ ⎝
⎟
⎠
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
0.852 0.861 0.838 0.076 0.070 0.080 0.038 0.075 0.040
0.861 0.871 0.847 0.049 0.043 0.053 0.054 0.091 0.057
0.838 0.847 0.824 0.078 0.073 0.082 0.050 0.086 0.053
0.076 0.049 0.078 0.940 0.946 0.930 0.015 0.022 0.016
0.070 0.043 0.073 0.946 0.953 0.937 0.014 0.021 0.015
0.080 0.053 0.082 0.930 0.937 0.921 0.020 0.027 0.021
0.038 0.054 0.050 0.015 0.014 0.020 0.953 0.908 0.949
0.075 0.091 0.086 0.022 0.021 0.027 0.908 0.866 0.904
0.040 0.057 0.053 0.016 0.015 0.021 0.949 0.904 0.945
⎞
⎟
⎠
⎞
⎠
(6.10)
Quite obviously, consensus functions based on object co-association matrices do not
operate on matrices Oλ or OΛ, as they are derived upon a single clustering solution. However,
the computation of object co-association matrices can easily be extended to a set of
clustering solutions compiled in either a hard or a soft cluster ensemble. We start with the
derivation of the object co-association matrix upon a hard cluster ensemble E containing l
clusterings, represented by means of a l × n integer valued matrix:
⎛
λ1
⎜
⎜λ2
⎟
E = ⎜ ⎟
⎝ . ⎠
λl
⎞
(6.11)
In this case, the incidence matrix of the hard cluster ensemble, denoted as IEλ ,isa
kl × n matrix resulting from the concatenation of the incidence matrices of the l clusterings
that make up the ensemble (see equation (6.12)).
IEλ =
⎛ ⎞
Iλ1
⎜Iλ2
⎟
⎜ ⎟
⎜ ⎟
⎝ . ⎠
Iλl
(6.12)
As in the case of a single clustering, the object co-association matrix of the hard cluster
ensemble, denoted as OEλ , can be derived upon IEλ by simple matrix multiplication, as
presented in equation (6.13).
OEλ
T
= IEλ
IEλ
169
(6.13)