Systems, Man & Cybernetics - April 2017 - 25
Cluster 1
Cocluster 2
Cocluster 3
Sample
Cluster 4
Cluster 3
Cluster 2
Sample
Sample
Cluster 2
Cluster 1
Cocluster 1
Cluster 3
Cluster 4
Cocluster 4
Feature
(a)
Feature
(b)
Feature
(c)
Figure 2. an illustration of the differences between clustering and coclustering. (a) Clustering involving data
classification along the sample direction only. (b) Clustering with data classification along the feature direction
only. (c) Coclustering involving data classification along both sample and feature directions.
condition in J, but the expression level is different for
a different condition.
3) Additive cocluster: A IJ = " a Tij = n i 1 J + a Tqj ; i, q ! I , . In an
additive cocluster, a column can be obtained by adding a
constant to another column. In gene expression analysis,
an additive cocluster means that the expression level of
each gene in I under a condition in J is always increased
or decreased by a constant under another condition.
Here, we have defined the additivity in terms of columns.
It is also possible to define the additivity in terms of rows,
but such a pattern has not been used in gene expression
analysis, to our knowledge.
4) Multiplicative cocluster: A IJ = " a Tij = n i a Tqj ; i, q ! I , . In
a multiplicative cocluster, a column can be obtained
by multiplying another column by a constant. In gene
expression analysis, a multiplicative cocluster means
that the expression level of each gene in I under a
condition in J is always scaled up or down by a con-
stant under another condition. Again, here we have
defined the scaling in terms of columns. It is also pos-
sible to define the scaling in terms of rows, but such a
pattern has not been used in gene expression analysis,
to our knowledge.
Several numerical examples of these coclusters are
shown in Figure 3.
Types of Coclusters
Let us consider 2-D data first. Assume that our input is a
matrix A = ^a ij h ! R M # M and that a 2-D cocluster AIJ contains
elements of A in NI rows I = " i 1, i 2, g, i N , and NJ columns
J = " j 1, j 2, g, j N ,. We will define several types of 2-D coclus-
ters [3], [16], [25]. These coclusters have been studied exten-
sively in gene expression analysis, and their biological
meanings will be explained.
1) Constant cocluster: A IJ = " a ij = n | i ! I, j ! J ,, where n is
a constant. In a constant cocluster, all elements take the
same value. In gene expression data analysis, a constant
cocluster means all genes in I have the same expression level
under all experiment conditions in J.
2) Constant-row cocluster: A IJ = " a Tij = n i 1 J | i ! I ,, where
1 J is a vector of N J ones. Similarly, we can define a
constant-column cocluster as A IJ = " a ij = n j 1 I ; j ! J ,,
where 1 I is a vector of N I ones. In a constant-row
cocluster, elements in each row have the same value,
while in a constant-column cocluster, elements in
each column have the same value. In gene expression
analysis, a constant-row cocluster means each gene
in I has the same expression level under all condi-
tions in J, but different genes have different expres-
sion levels. A constant- column cocluster means all
genes in I have the same expression level for each
1
2
I
J
8.2 8.2 8.2 8.2
1.5 1.5 1.5 1.5
1.5 5.2 2.8 9.3
6.8 5.7 7.7 4.8
2.2 3.3 1.1 4.4
8.2 8.2 8.2 8.2
5.2 5.2 5.2 5.2
1.5 5.2 2.8 9.3
5.6 4.5 6.5 3.6
3.4 5.1 1.7 6.8
8.2 8.2 8.2 8.2
2.8 2.8 2.8 2.8
1.5 5.2 2.8 9.3
3.3 2.2 4.2 1.3
4.6 6.9 2.3 9.6
8.2 8.2 8.2 8.2
9.3 9.3 9.3 9.3
1.5 5.2 2.8 9.3
8.5 7.4 9.4 6.5
5.6 8.4 2.8 11.2
(a)
(b)
(c)
(d)
(e)
Figure 3. Examples of coclusters: (a) a constant cocluster, (b) a constant-row cocluster, (c) a constant-column
cocluster, (d) an additive cocluster, and (e) a multiplicative cocluster.
Ap ri l 2017
IEEE SyStEmS, man, & CybErnEtICS magazInE
25
Table of Contents for the Digital Edition of Systems, Man & Cybernetics - April 2017
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