One-ICA Version of the Mixca Algorithm - On Statistical Pattern Recognition in Independent Component Analysis Mixture Modelling - page 179

where a is a normalization constant and h is a constant that adjusts 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 equation ( A.3 ) with respect to the unknown matrix W:

dL ð W Þ
dW

¼ d log j det W j p ð y Þ
dW
¼ d log j det W j
dW

ð A : 6 Þ

þ d log p ð y Þ
dW

:

where

d log j det W j
dW

¼ð W T Þ 1 ð A : 7 Þ

Imposing independence on y and using equation ( A.4 ):

¼ X

M

¼ X

M

d log p ð y Þ
dW

d log p ð y m Þ
dW

1
p ð y m Þ

dp ð y m Þ
dW

m ¼ 1 m ¼ 1

ð A : 8 Þ

¼ X

M

1
p ð y m Þ

dp ð y m Þ
dy m

dy m
dW :

m ¼ 1

where using equation ( A.5 ):

! 1
h ; m ¼ 1...M

y m y ð n 0 Þ
m

¼ a X
n 0

y m y ð n 0 Þ

dp ð y m Þ
dy m

2

m h

e ð A : 9 Þ

2

Let us call w m the m-th row of W. Then y m ¼ w m x ; and

dy m
dW ¼ M m

ð A : 10 Þ

where M m ð l ; l 0 Þ¼ d ð l m Þ x l 0
Substituting equations ( A.5 ), ( A.9 ), and ( A.10 ) in equation ( A.8 ) we have:

¼ X

M

d log p ð y Þ
dW

f ð y m Þ M m ð A : 11 Þ

m ¼ 1

where

2
4

3
5

y m y ð n 0 Þ
m

2

P

h

y m e

2

f ð y m Þ¼ 1
h 2

n 0

y m ð A : 12 Þ

y m y ð n 0 Þ
m

P

2

h

e

2

n 0