Database Reference
In-Depth Information
The mixtures can be built with different types of components. However, it is
usually assumed that the components of the mixture have the same functional
form, such as Gaussian or Laplacian. In the following, the LMM is utilized
and the parameters defining the mixture model is estimated by the expectation-
maximization (EM) algorithm.
10.2.2
Laplacian Mixture Model and Parameter Estimation
The univariate Laplacian distribution is defined as:
2
b
exp
1
−
|
x
−
μ
|
b
p
(
x
)=
(10.3)
where x is an instance of the random variable,
is the location parameter, and
b
represents the width of the distribution. A Laplacian distribution with
μ
μ
=
0 and
b
1 is shown in Fig.
10.1
. Previous studies have shown that the density of the
wavelet coefficients is symmetrically centered at zero [
325
]. This is due to the fact
that most of the wavelet coefficients in a frequency sub-band have small magnitudes
close to zero. Therefore, we assume that the underlying distribution is a mixture of
M
Laplacian components centered at zero. Then the probability function of the
i
-th
data point,
p
=
x
(
i
)
|
ʸ
m
)
is simply a conditional probability of generating given that the
m
-th model is chosen:
(
2
b
m
exp
x
(
i
)
|
b
m
1
−
|
x
(
i
)
|
ʸ
m
)=
p
(
(10.4)
The EM algorithm is applied for the estimation of the parameters. We assume the
existence of a hidden variable
z
t
whose values are not known. This
M
-dimensional hidden variable
z
(
i
)
is associated with each data point
x
(
i
)
and
indicates which component of the mixture has generated
x
(
i
)
. For example, if a
data value has been generated by the
m
-th component of the mixture, then the
m
-th
component of this vector
z
(
i
m
=
=[
z
1
,...,
z
M
]
1 and all other component values will be 0. Here,
i
represents the index of the data point and
i
N
because the total number
of observations are taken to be
N
. In the presence of both
x
and
z
, the complete
likelihood can be written as follows:
=
1
,...,
m
=
1
z
(
i
m
ln
p
(
x
(
i
)
|
ʸ
m
)+
ln ʱ
m
N
i
=
1
M
L
c
(
ʘ
,
x
,
z
)=
(10.5)
The EM algorithm proceeds by alternatively applying the following two steps:
Search WWH ::
Custom Search