Information Technology Reference
In-Depth Information
3. Checkerboard biclusters. The clusters
{S
k
:
k
=
1
, ···,
K
}
of samples
S
and the
clusters of
satisfy the same requirements (Equa-
tions (6.1) and (6.2)) as in structure 2. The set of checkerboard biclusters is
{F
k
:
k
=
1
, ···,
K
}
of features
F
k
=
B
=
{
B
kk
=(
S
k
,F
k
)
:
k
,
1
, ···,
K
},
i.e., any entry of
A
is in someone's biclusters.
Considering each bicluster as an entry, the proper rearrangement matrix of
A
is
a
K
K
matrix with entry
B
k
,
k
. In some cases, the number of samples' clusters
S
k
s do not need to be the same as that of features' clusters
×
F
k
s. This will imply
a rectangle not a square matrix.
4. Exclusive rows biclusters. Given a data matrix
A
, the structure of exclusive rows'
biclusters
B
=
{
B
k
=(
S
k
,F
k
)
:
k
=
1
,
2
, ···,
K
}
should satisfy the requirements
as follows: For rows
⎧
⎨
S
k
⊆S,
(
)
,
S
1
∪S
2
∪···∪S
K
=
S,
S
k
∩S
k
=
k
=
1
, ···,
K
(6.3)
⎩
k
=
k
,
0
,
k
,
1
, ···,
K
,
k
=
and for corresponding columns
F
k
⊆F,
(
)
,
F
1
∪F
2
∪···∪F
K
=
F.
k
=
1
, ···,
K
(6.4)
Comparing Equations (6.1) and (6.2) in structure 2, requirements for rows are
same, but for columns, Equation (6.4) has no disjoint requirement between
F
k
k
k
. In this structure, some features (columns) may belong to two or
more biclusters (submatrices), while any sample (row) should belong to exactly
one bicluster (submatrix).
5. Exclusive columns biclusters. Given a data matrix
A
, the structure of exclusive
columns' biclusters
and
F
k
,
=
B
=
{
B
k
=(
S
k
,F
k
)
:
k
=
1
,
2
, ···,
K
}
should satisfy the re-
quirements as follows: For rows
S
k
⊆S,
(
)
,
S
1
∪S
2
∪···∪S
K
=
S,
k
=
1
, ···,
K
(6.5)
and for corresponding columns
⎧
⎨
F
k
⊆F,
(
)
,
F
1
∪F
2
∪···∪F
K
=
F,
F
k
∩F
k
=
k
=
1
, ···,
K
(6.6)
⎩
k
=
k
.
0
,
k
,
1
, ···,
K
,
k
=
Comparing Equations (6.1) and (6.2) in structure 2, requirements for columns
are same, but for rows, Equation (6.5) has no disjoint requirement between
S
k
k
=
and
k
. In this structure, some samples (rows) may belong to two or more
biclusters (submatrices), while any feature (column) should belong to exactly one
bicluster (submatrix).
S
k
,
