ICA and ICAMM Methods - On Statistical Pattern Recognition in Independent Component Analysis Mixture Modelling - page 40

particular underlying distributions of the mixed signals (any information about the
sources is completely unknown), (see for instance [ 45 ]); the second consists of
including the maximum number of priors available in the cost function in order to
guide the algorithm to find particular sources (blind source extraction, semi-blind
source separation, etc.), (see for instance [ 46 , 47 ]). Some new methods that use
non-parametric (NP) density estimation have been recently developed from the
first direction in ICA research.
The new non-parametric ICA methods use techniques such as: minimization of
a kernel canonical correlation or a kernel generalized variance among recovered
sources (the so-called Kernel-ICA) [ 48 ]; maximum likelihood estimation (MLE)
by using spline-based density approximations [ 49 ]; MLE by using Gaussian kernel
density estimates (the so-called Npica) [ 45 ]; and minimization of the entropy of
the marginals by estimating their order statistics (the so-called Radical) [ 50 ].
These methods have shown good performance in simulations, but there are no
references about their performance in real applications. Theoretical analyses
(convergence, consistency, and other issues) of non-parametric density estimation
in the framework of ICA are found in [ 29 , 26 , 51 ]. We include a review of the
Npica, Radical, and Kernel-ICA algorithms in the following section.

2.3.1 Npica

The Npica algorithm [ 45 ] is a maximum loglikelihood ICA method that solves the
Eq. ( 2.9 ). It uses a non-parametric estimation for the probability density function p i ,
which is directly estimated from the data using a kernel density estimation tech-
nique [ 52 ].
Given a batch of sample data of size N, the marginal distribution of an arbitrary
reconstructed signal is approximated as follows:

;

X

N

p i s ðÞ¼ 1
Nh

s i s il
h

j i ¼ 1 ; ... ; M ð 2 : 24 Þ

I ¼ 1

1
2p

p e u 2 = 2 :

where h is the kernel bandwidth and j is the Gaussian kernel j ð u Þ ,

The kernel centroids s il are equal to s il ¼ w i x ð l Þ ¼ P

N

w il X li ; where x ð l Þ

is the lth

l ¼ 1

column of the mixture matrix X.
The expectation of the maximum loglikelihood solution is approximated by the
following cost function

L ð W Þ¼ L 0 ð W Þ log ð det W Þ ð 2 : 25 Þ

where L 0 ð W Þ is obtained by replacing the marginal pdf's p i with their kernel
density estimates