Advances in E-learning-Experiences and Methodologies

Knowledge Discovery from E-Learning Activities
e-Learning
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Annex 1
This annex contains the formulation of the parametric
ICA algorithm applied in the chapter.
The probability density function of the data x
can be expressed as:
p(x) = | det W|p(y) (3)
where p(y) can be expressed as the product of the
marginal distributions since it is the estimate of
the independent components:
M
p( y) = ∏ pi
( yi
)
i=
1
(4)
If we assume a nonparametric model for p(y), we
can estimate the source pdf’s from a set of training
samples obtained from the original dataset
using (2). We propose a kernel density estimation
technique (Scott & Sain, 2004; Duda, Hart, &
Stork, 2000), where the marginal distribution of
a reconstructed component is approximated as:
( m)
∑
n'
( ')
1⎛
n
ym
− y ⎞
m
− 2 ⎜ h ⎟
⎝ ⎠2
p y = a ⋅ e , m = 1... M
(5)
where a is a normalization constant and h is a
constant that adjust the degree of smoothing
of the estimated pdf. The learning algorithm
can be derived using the maximum-likelihood
estimation. In a probability context it is usual to
maximize the log-likelihood of (3) with respect
to the unknown matrix W:
( W) log det W . p( y)
L
=
W
log det W
= +
W
where
W
log p
W
log det W
T
=
−
W
( W ) 1
( y)
.
(6)
(7)
Imposing independence on y and using (4):
M
∑
log
p( y
M
) log p( ym)
= ∑
=
W m=
1 W
1 ( )
M
1
= ∑
( m)
W p( ym)
p y
( )
m= 1 m=
1
m
where using (5):
( )
(8)
p ym p ym ym
.
y W
( ')
1 ⎛ n
ym
− y ⎞
m
− n '
m 2 ⎜ h ⎟ ⎛
⎝ ⎠2
m
−
m
(9)
p y ( )
y y ⎞ 1
= − a∑e
, m = 1... M
y
⎜
m
n '
h ⎟
⎝ ⎠ h