Biology Reference
In-Depth Information
5.4.1.
Model-based Method
Model-based method assumes that the dataset consists of samples from a mixture
of populations, and each population has a determined form of probability distri-
bution with unknown parameters. The pdf of the mixture model is:
K
f
(
x
)=
f
(
x
|
x
∈
C
j
)
p
(
x
∈
C
j
)
(5.23)
j
=1
where
f
(
x
C
j
) is the pdf of vector
x
conditioning on vector
x
is generated by
class
j
, denoted by
C
j
,
p
(
x
|
x
∈
C
j
) is the probability that vector
x
is generated by
C
j
,and
K
is the number of assumed mixed probability populations.
Model-based method to determine the number of clusters works as follows:
∈
(1) Assume the form of pdf
f
(
x
|
x
∈
C
j
). Usually,wetakethesameformof
|
∈
C
j
),
j
=1
,
2
,...,
K
. We denote the model parameters of
function for
f
(
x
x
=
θ
j
.
(2) Let
K
takes value from 1 to
K
,where
K
is a large integer. For each
K
,we
cluster the dataset into
K
clusters. For cluster
j
,
j
=1 to
K
, we calculate the
maximum likelihood estimates of
θ
j
, denoted as
θ
j
. Probability
p
(
x
f
(
x
|
x
∈
C
j
) as
θ
j
.For
f
(
x
|
x
∈
C
i
) and
f
(
x
|
x
∈
C
j
),if
i
=
j
,
θ
i
∈
C
j
) is
calculated by:
C
j
)=
i
=1
I
(
x
i
∈
C
j
)
p
(
x
∈
(5.24)
N
It is just the percent of the observations assigned into cluster
j
.
(3) For each
K
, calculate the adjusted log-likelihood value by:
N
K
C
j
, θ
j
)
p
(
x
i
∈
l
(
K
)=2
log
(
f
(
x
i
|
x
i
∈
C
j
))
−
g
(
K
)
(5.25)
i
=1
j
=1
C
j
,θ
j
) is the pdf of
x
i
with conditions that
x
i
is generated by
class
C
j
and model parameters are
θ
j
. Function
g
(
K
) is a penalty function. It
is a monotonously increasing function of
K
. Fraley and Raftery [10] suggest
g
(
K
)=
m
K
log
(
N
) ,where
m
K
is the number of independent parameters to be
estimated in the mixture model defined in Eq. 5.23.
(4) Choose the
K
corresponding to the largest
l
(
K
) as the number of clusters,
denoted as
K
∗
, i.e.,
where
f
(
x
i
|
x
i
∈
(
l
(
K
)
,
K
=1
,
2
,..., K
)
K
∗
=argmax
K
(5.26)
Let us review Eq. 5.25 in step (3). We can find that Eq. 5.25 is very similar
to the Bayesian Information Criterion (BIC) when we determine the best model
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