Biology Reference
In-Depth Information
calculated as:
N
log
(
p
ij
([
x
ki
,
x
kj
])
I
(
x
i
,
x
j
)=
1
N
p
i
(
x
ki
)
p
j
(
x
kj
)
)
(5.8)
k
=1
where
p
ij
([
x
ki
,
x
kj
]) is the estimated likelihood of the sample [
x
ki
,
x
kj
] under the
joint distribution of sequences
x
i
and
x
j
,and
p
i
(
x
ki
) is the estimated likelihoods of
sample
x
ki
under the marginal distribution of
x
i
. The estimated likelihood value is
calculated as follows:
Nh
r
k
=1
g
(
[
x
i
,
x
j
]
−
[
x
ki
,
x
kj
]
1
p
ij
([
x
i
,
x
j
]) =
)
p
i
(
x
)=
Nh
r
N
(5.9)
h
k
=1
g
(
x
−
x
ki
)
h
where
g
(
) is a kernel function. A frequent choice for the kernel function is the
standard normal density, i.e.,
·
1
2
x
x
)
g
(
x
)=(2
π
)
−
r
/
2
exp
(
−
(5.10)
where
r
is the dimension of
x
. In the first equation in Eq. 5.8,
r
=2. Parameter
h
in
Eq. 5.9 is given by:
4
(2
r
+1)
N
]
1
/
(
r
+4)
h
=[
(5.11)
The mutual information
I
(
x
i
,
x
j
) between sequences
x
i
and
x
j
is always non-
negative, and is zero if and only if these two sequences are stochastically indepen-
dent. Usually we use the normalized mutual information as the similarity measure
which is ranged from 0 to 1. The transformation of mutual information to the sim-
ilarity measure is as follows:
s
(
x
i
,
x
j
)=
1
−
exp
(
−
2
I
(
x
i
,
x
j
))
(5.12)
Kojadinovic [20] gives the details of mutual information and how to use mu-
tual information as the similarity measure to cluster variables. Mutual informa-
tion has also been used in neuroscience to identify the associations between neu-
rons [2]. Readers should be noted that the mutual information introduced here
is for continuous variables. For discrete variables, such as the sequence of spike
train data of neurons, the calculation may be different.
There are some other measures to capture the association between variables,
such as the cross-intensity and coherence for associations between point pro-
cesses [3]. These two measures are commonly used for sequences of spike train
data from neurons.
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