Information Technology Reference
In-Depth Information
Appendix
One-ICA Version of the Mixca Algorithm
ICA defines a generative model for the observed data. The conventional model is
given by:
x
¼
As
ð
A
:
1
Þ
where x is the observed data as a random vector whose elements are the mixtures
x
1
;
...
;
x
N
;
A is a NxM matrix, called the mixing matrix, and s is a random vector
with M elements s
1
;
...
;
s
M
;
called the source elements.
The goal of ICA is to find a linear transformation of the data such that the latent
variables are as statistically independent from each other as possible. Suppose y is
an estimate of the sources, so that:
y
¼
Wx
ð
A
:
2
Þ
When the sources are exactly recovered, W is the inverse of A.
The probability density function of the data x can be expressed as:
p
ð
x
Þ¼j
det W
j
p
ð
y
Þ
ð
A
:
3
Þ
where p(y) can be expressed as the product of the marginal distributions since it is
the estimate of the independent components:
p
ð
y
Þ¼
Y
M
p
i
ð
y
i
Þ
ð
A
:
4
Þ
i
¼
1
If we assume a non-parametric model for p(y) we can estimate the source pdf's
from a set of training samples obtained from the original dataset using equation
(
A.2
). The marginal distribution of a reconstructed component is approximated as
(kernel density estimation):
p
ð
y
m
Þ¼
a
X
n
0
y
m
y
ð
n
0
Þ
m
2
h
e
2
;
m
¼
1...M
ð
A
:
5
Þ
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