Information Technology Reference
In-Depth Information
By proper transformation, the data matrix
A
is to be a joint probability distri-
bution matrix
p
(
S,F
)
between two discrete random variables
S,F
.Let
K
be the
number of disjoint clusters of samples and
K
the number of disjoint features. The
B
=(
S
,F
)=(
{S
k
,
:
k
=
K
}
)
set of biclusters is
:
k
=
1
, ··· ,
K
}, {F
k
1
, ··· ,
.The
mappings of
C
S
,
C
F
are objectives to find in this biclustering algorithm such that
C
S
:
{
S
1
,
S
2
, ··· ,
S
n
}→{S
1
, ··· ,S
K
},
C
F
:
{
F
1
,
F
2
, ··· ,
F
m
}→{F
1
, ··· ,F
K
}.
is the amount of
information shared between these two variables and is defined as in information
theory
The mutual information
I
(
S,F
)
of two random variables
S,F
n
i
=
1
m
j
=
1
p
(
S
i
,
F
j
)
log
p
(
S
i
,
F
j
)
I
(
S,F
)=
F
j
)
=
D
(
p
(
S
,
F
)
||
p
(
S
)
p
(
F
))
,
p
(
S
i
)
p
(
where
p
(
S
i
,
F
j
)
,
p
(
S
i
)
,
p
(
F
j
)
are probabilities from distribution matrix
p
(
S,F
)
, and
log
p
1
(
x
)
p
2
(
x
)
(
||
)=
∑
x
p
1
(
)
D
p
1
p
2
x
is the relative entropy between two probability dis-
tributions
p
1
.
The objective of this biclustering is to find optimal biclusters of
A
such that the
loss in mutual information is minimized, i.e.,
(
x
)
and
p
2
(
x
)
(
S
,F
)
.
min
I
(
S,F
)
−
I
x
,
y
)
x
,
y
)
x
)
y
)
In order to solve this objective function,
q
(
x
,
y
)=
p
(
p
(
p
(
x
|
p
(
y
|
is defined so that the objective function can be written as
(
S
,F
)=
min
I
(
S,F
)
−
I
D
(
p
(
S,F
)
||
q
(
S,F
))
.
For proof of this result, we refer to [18]. Then an iterative way is used to solve
by transformed the objective function [18].
6.3.5 Based on Probability
The following two biclustering algorithms (named as BBC and cMonkey) use the
theory of probability.
BBC.
Gu and Liu [26] proposed a Bayesian biclustering model (BBC) and im-
plemented a Gibbs sampling [34] procedure for its statistical inference. This model
can also consider an implementation of plain model [50] of biclustering.
Given data matrix
A
, assume the entry
k
=
1
((
μ
k
+
α
ik
+
β
jk
+
ε
ijk
)
δ
ik
κ
jk
)+
e
ij
1
−
k
=
1
δ
ik
κ
jk
K
K
a
ij
=
,
