Mathematics in Independent Component Analysis

Chapter 20. Signal Processing 86(3):603-623, 2006 263
N)−matrix) in different ways. One of the simplest approaches lies in a linear
decomposition, X = AS, with A an (m × N)−matrix (mixing matrix) and S
an (n × N)−matrix storing the sources. Both A and S are unknown, hence
this problem is often described as blind source separation (BSS). To get a
well-defined problem, A and S have to satisfy additional properties such as:
• the source components Si (rows of S) are assumed to be realizations of
stochastically independent random variables — this method is called independent
component analysis (ICA) [1,2],
• the sources S are required to contain as many zeros as possible — we then
speak of sparse component analysis (SCA) [3,4],
• A and S are assumed to be nonnegative, which is denoted by nonnegative
matrix factorization (NMF) [5].
The above-mentioned models as well as their interplay have recently been in
the focus of many researchers, for instance concerning the question of how
ICA and sparseness are related to each other and how they can be integrated
into BSS algorithms [6–10], how to deal with nonnegativity in the ICA case
[11,12] or how to extend NMF in order to include sparseness [13,14]. Much
work has already been devoted to these subjects, and their applications to
various fields are currently emerging. Indeed, linear representations such as the
above have several potential applications including decomposition of objects
into ‘natural’ components [5], redundancy and dimensionality reduction [2],
biomedical data analysis, micro-array data mining or enhancement, feature
extraction of images in nuclear medicine, etc. [1,2].
In this study, we will analyze and compare the above models, not from a theoretical
point of view but rather from a concrete, real-world example, namely
the analysis of surface electromyogram (s-EMG) data sets. An electromyogram
(EMG) denotes the electric signal generated by a contracting muscle
[15]; its study is relevant to the diagnosis of motoneuron diseases [16] as well
as neurophysiological research [17]. In general, EMG measurements make use
of invasive, painful needle electrodes. An alternative is to use s-EMG, which
is measured using non-invasive, painless surface electrodes. However, in this
case the signals are rather more difficult to interpret due to noise and overlap
of several source signals as shown in Fig 1(a). We have already applied ICA in
order to solve the s-EMG decomposition problem [18], however performance
in real-world noisy s-EMG is still problematic, and it is yet unknown if the
assumption of independent sources holds very well in the setting of s-EMG.
In the present work, we apply sparse BSS methods based on various model
assumptions to s-EMG signals. We first outline each of those methods and
the corresponding performance indices used for their comparison. We then
present the decompositions obtained with each method, and finally discuss
these results in section 4.
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