Independent Component Analysis and Blind Source Separation - Biomedical Signal Analysis: Contemporary Methods and Applications - page 101

4

Independent Component Analysis and Blind Source
Separation

Biostatistics deals with the analysis of high-dimensional data sets origi-
nating from biological or biomedical problems. An important challenge
in this analysis is to identify underlying statistical patterns that facilitate
the interpretation of the data set using techniques from machine learn-
ing. A possible approach is to learn a more meaningful representation
of the data set, which maximizes certain statistical features. Such often
linear representations have several potential applications including the
decomposition of objects into “natural” components [150], redundancy
and dimensionality reduction [87], biomedical data analysis, microarray
data mining or enhancement, feature extraction of images in nuclear
medicine, etc. [6, 34, 57, 123, 163, 177].
In this chapter, we review a representation model based on the
statistical independence of the underlying sources. We show that in
contrast to the correlation-based approach in PCA (see chapter 3), we
are now able to uniquely identify the hidden sources.

4.1 Introduction

m ,where t
indexes time, space, or some other quantity. Data analysis can be defined
as finding a meaningful representation of x ( t )thatis,as x ( t )= f ( s ( t ))
with unknown features s ( t )

Assume the data is given by a multivariate time series x ( t ) ∈ R

m and mixing mapping f .Often, f is
assumed to be linear, so we are dealing with the situation

∈ R

x ( t )= As ( t )

(4.1)

m×n .Often,whitenoise n ( t ) is added to
the model, yielding x ( t )= As ( t )+ n ( t ); this can be included in s ( t )
by increasing its dimension. In equation (4.1), the analysis problem is
reformulated as the search for a (possibly overcomplete) basis, in which
the feature signal s ( t ) allows more insight into the data than x ( t )does.
This of course has to be specified within a statistical framework.
There are two general approaches to finding data representations or
models as in equation (4.1):

with a mixing matrix A ∈ R

• Supervised analysis: Additional information, for example in the form