Independent Component Analysis and Blind Source Separation - Biomedical Signal Analysis: Contemporary Methods and Applications - page 102

(a) sources s ( t ) (b) mixtures x ( t ) (c) recoveries (d) WA

Figure 4.1
Two-dimensional example of ICA-based source separation. The observed mixture
signal (b) is composed of two unknown source signals (a), using a linear mapping.
Application of ICA (here: Hessian ICA) yields the recovered sources (c), which
coincide with the original sources up to permutation and scaling: s 1 ( t ) ≈ 1 . 5 s 2 ( t )
and s 2 ( t ) ≈− 1 . 5 s 1 ( t ). The composition of mixing matrix A and separating matrix
W equals a unit matrix (d) up to the unavoidable indeterminacies of scaling and
permutation.

of input-output pairs ( x ( t 1 ) , s ( t 1 )) ,..., ( x ( t T ) , s ( t T )). These training
samples can be used for interpolation and learning of the map f or
the basis A (regression). If the sources s are discrete, this leads to
a classification problem. The resulting map f can then be used for
prediction.
• Unsupervised models: Instead of samples, weak statistical assumptions

are made on either s ( t )or f / A . A common assumption, for example,
is that the source components s i ( t ) are mutually independent, which
results in an analysis methods called independent component analysis
(ICA) .

Here, we will focus mostly on the second situation. The unsuper-
vised analysis is often called blind source separation (BSS) , since nei-
ther features or “sources” s ( t ) nor mixing mapping f are assumed to
be known. The field of BSS has been rather intensively studied by the
community for more than a decade. Since the introduction of a neural-
network-based BSS solution by Herault and Jutten [112], various algo-
rithms have been proposed to solve the blind source separation problem
[25, 46, 59, 124, 259]. Good textbook-level introductions to the topic
are given by Hyvarinen et al. [123] and Cichocki and Amari [57]. Re-
cent research centers on generalizations and applications. The first part
of this volume deals with such extended models and algorithms; some
applications will be presented later.