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6.2.3 Biclustering Structures
The structure of biclustering is defined to be the relationships between biclusters
from
based on the data matrix
A
.
For the structures of biclustering, there are some properties which should be no-
ticed: exclusive, overlapping, and exhaustive, although some concepts or terms have
been used previously. For a data matrix
A
, and the corresponding set of biclusters
B
=
{
B
=
{
B
k
=(
S
k
,F
k
)
:
k
=
1
,
2
, ···,
K
}
B
k
=(
S
k
,F
k
)
:
k
=
1
,
2
, ···,
K
}
, we have the following formal definitions.
•
Exclusive (nonexclusive). A biclustering structure is said to be row exclusive
if
k
∈{
k
; to be column exclusive if
S
k
∩S
k
=
0 for any
k
,
1
, ···,
K
},
k
=
k
∈{
k
; to be exclusive if it is both row
F
k
∩F
k
=
0 for any
k
,
1
, ···,
K
},
k
=
exclusive and column exclusive.
•
Overlapping (nonoverlapping). A biclustering structure is said to be overlapping
if some entry
a
ij
belongs to two or more biclusters; otherwise, it is nonoverlap-
ping.
•
Exhaustive (nonexhaustive). A biclustering structure is said to be row exhaustive
if any row
S
i
belongs to at least one bicluster; to be column exhaustive if any
column
F
j
belongs to at least one bicluster; to be exhaustive if it is both row
and column exhaustive. Otherwise, it is said to be nonexhaustive if some row or
column does not belong to any bicluster.
Here, exclusive and overlapping are not opposite to each other, and it can found
from structure 7. The following biclustering structures are based on these three prop-
erties.
Still following the classification of Madeira and Oliveira in [37], the biclustering
structures are identified into following nine groups.
1. Single bicluster. In this single biclustering structure, only one submatrix is found,
i.e.,
k
,from
A
.
2. Exclusive row and column biclusters. Given a data matrix
A
, as Definition 1
in [5], the structure of exclusive row and column biclusters
=
1 and
B
=
{
B
1
=(
S
1
,F
1
)
}
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
(6.1)
⎩
k
=
k
,
0
,
k
,
1
, ···,
K
,
k
=
and for corresponding columns
⎧
⎨
F
k
⊆F,
(
)
,
F
1
∪F
2
∪···∪F
K
=
F,
F
k
∩F
k
=
k
=
1
, ··· ,
K
(6.2)
⎩
k
=
k
.
0
,
k
,
1
, ···,
K
,
k
=
In proper rearrangement of rows and columns of data matrix
A
, the biclusters are
the submatrices in a diagonal way without overlap between any two biclusters.
