Mathematics in Independent Component Analysis

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262 Chapter 20. Signal Processing 86(3):603-623, 2006 On the use of sparse signal decomposition in the analysis of multi-channel surface electromyograms Fabian J. Theis a,∗ , Gonzalo A. García b a Institute of Biophysics, University of Regensburg, 93040 Regensburg, Germany b Department of Bioinformatic Engineering, Osaka University, Osaka, Japan Abstract The decomposition of surface electromyogram data sets (s-EMG) is studied using blind source separation techniques based on sparseness; namely independent component analysis, sparse nonnegative matrix factorization, and sparse component analysis. When applied to artificial signals we find noticeable differences of algorithm performance depending on the source assumptions. In particular, sparse nonnegative matrix factorization outperforms the other methods with regards to increasing additive noise. However, in the case of real s-EMG signals we show that despite the fundamental differences in the various models, the methods yield rather similar results and can successfully separate the source signal. This can be explained by the fact that the different sparseness assumptions (super-Gaussianity, positivity together with minimal 1-norm and fixed number of zeros respectively) are all only approximately fulfilled thus apparently forcing the algorithms to reach similar results, but from different initial conditions. Key words: surface EMG, blind source separation, sparse component analysis, independent component analysis, sparse nonnegative matrix factorization 1 Introduction A basic question in data analysis, signal processing, data mining as well as neuroscience is how to represent a large data set X (observed as an (m × ∗ Corresponding author. Email addresses: fabian@theis.name (Fabian J. Theis), garciaga@ieee.org (Gonzalo A. García). Preprint submitted to Elsevier Science 8 April 2005
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. 2
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viii CONTENTS 3 Signal Processing 8
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Chapter 1 Statistical machine learn
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1.1. Introduction 5 auditory cortex
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S U M M A R Y T his �le contains
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1.3. Dependent component analysis 1
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Table 1.1: BSS algorithms based on
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