ICA and ICAMM Methods - On Statistical Pattern Recognition in Independent Component Analysis Mixture Modelling - page 29

Chapter 2
ICA and ICAMM Methods

2.1 Introduction

The seminal work of the research in ICA was provided by Jutten in [ 1 - 4 ].
Independent component analysis (ICA) aims to separate hidden sources from their
observed linear mixtures without any prior knowledge. The only assumption about
the sources is that they are mutually independent [ 5 ]. Thus, the goal is blind source
estimation; although it has been recently alleviated by incorporating prior
knowledge about the sources into the ICA model in the so-called semi-blind source
separation (see for instance [ 6 - 8 ]). This technique has been widely used in many
fields of application such as telecommunications, bioengineering, and material
testing [ 5 ]. There is extensive literature that reviews and provides taxonomies and
comparisons about the large number of ICA algorithms that have been developed
during the last two decades (see for example [ 5 , 9 - 13 ]). Therefore, in this chapter,
instead of undertaking an exhaustive review of the methods, we will focus on
reviewing the following: the ICA basic concepts, some ICA algorithms that will be
used for comparison with those proposed in this work, and existing ICAMM
algorithms.
The standard noiseless instantaneous ICA formulates a Mx1 random vector x by
linear mixtures of M random variables that are mutually independent s 1 ; ... ; s M
whose distributions are totally unknown. That is, for s ¼ s 1 ; ... ; s M Þ T ð and some

matrix A

x ¼ As ð 2 : 1 Þ

The essential principle is to estimate the so-called mixing matrix A, or equiva-
lently B ¼ A 1 (the demixing matrix). The matrix A contains the coefficients of
the linear transformation that represents the transfer function from sources to
observations. Thus, given N i.i.d. observations

ð x 1 ; ... ; x N Þ from the distribution of

x,A 1 can be applied to separate each of the sources s i ¼ B i x ; where B i is the ith