Mathematics in Independent Component Analysis

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140 Chapter 9. Neurocomputing (in press), 2007 A Robust Model for Spatiotemporal Dependencies Fabian J. Theis a,b,∗ , Peter Gruber b , Ingo R. Keck b , Elmar W. Lang b a Bernstein Center for Computational Neuroscience Max-Planck-Institute for Dynamics and Self-Organisation, Göttingen, Germany b Institute of Biophysics, University of Regensburg, Regensburg, Germany Abstract Real-world data sets such as recordings from functional magnetic resonance imaging often possess both spatial and temporal structure. Here, we propose an algorithm including such spatiotemporal information into the analysis, and reduce the problem to the joint approximate diagonalization of a set of autocorrelation matrices. We demonstrate the feasibility of the algorithm by applying it to functional MRI analysis, where previous approaches are outperformed considerably. Key words: blind source separation, independent component analysis, functional magnetic resonance imaging, autodecorrelation PACS: 07.05.Kf, 87.61.–c, 05.40.–a, 05.45.Tp 1 Introduction Blind source separation (BSS) describes the task of recovering an unknown mixing process and underlying sources of an observed data set. It has numerous applications in fields ranging from signal and image processing to the separation of speech and radar signals to financial data analysis. Many BSS algorithms assume either independence (independent component analysis, ICA) or diagonal autocorrelations of the sources [1,2]. Here we extend BSS algorithms based on time-decorrelation [3–8]. They rely on the fact that the data ∗ corresponding author Email address: fabian@theis.name (Fabian J. Theis). URL: http://fabian.theis.name (Fabian J. Theis). Preprint submitted to Elsevier 15 May 2007
Chapter 9. Neurocomputing (in press), 2007 141 sets have non-trivial autocorrelations so that the unknown mixing matrix can be recovered by generalized eigenvalue decomposition. Spatiotemporal BSS in contrast to the more common spatial or temporal BSS tries to achieve both spatial and temporal separation by optimizing a joint energy function. First proposed by Stone et al. [9], it is a promising method, which has potential applications in areas where data contains an inherent spatiotemporal structure, such as data from biomedicine or geophysics including oceanography and climate dynamics. Stone’s algorithm is based on the Infomax ICA algorithm [10], which due to its online nature involves some rather intricate choices of parameters, specifically in the spatiotemporal version, where online updates are being performed both in space and time. Commonly, the spatiotemporal data sets are recorded in advance, so we can easily replace spatiotemporal online learning by batch optimization. This has the advantage of greatly reducing the number of parameters in the system, and leads to more stable optimization algorithms. We focus on so-called algebraic BSS algorithms [3,5,6,11], reviewed for example in [12], which employ generalized eigenvalue decomposition and joint diagonalization for the factorization. The corresponding learning rules are essentially parameter-free and are known to be robust and efficient [13]. In this contribution, we extend Stone’s approach by generalizing the timedecorrelation algorithms to the spatiotemporal case, thereby allowing us to use the inherent spatiotemporal structures of the data. In the experiments presented, we observe good performance of the proposed algorithm when applied to noisy, high-dimensional data sets acquired from functional magnetic resonance imaging (fMRI). We concentrate on fMRI as it is well fit for spatiotemporal decomposition due to the fact that spatial activation networks are mixed with functional and structural temporal components. 2 Blind source separation We consider the following temporal BSS problem: Let x(t) be a second-order stationary, zero-mean, m-dimensional stochastic process and A a full rank matrix such that x(t) = As(t) + n(t). The n-dimensional source signals s(t) are assumed to have diagonal autocorrelations Rτ(s) := 〈s(t + τ)s(t) ⊤ 〉 for all τ, and the additive noise n(t) is modeled by a stationary, temporally and spatially white zero-mean process with variance σ 2 . x(t) is observed, and the goal is to recover A and s(t). Having found A, s(t) can be estimated by A † x(t), which is optimal in the maximum-likelihood sense, where A † denotes the pseudo-inverse of A and m ≥ n. So the BSS task reduces to the estimation of the mixing matrix A. 2
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vi Preface
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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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1.2. Uniqueness issues in independe
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1.2. Uniqueness issues in independe
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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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1.4. Sparseness 25 1.4 Sparseness O
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1.4. Sparseness 27 Theorem 1.4.3 (S
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1.4. Sparseness 29 R 3 R 3 A BSRA R
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1.4. Sparseness 31 Sparse projectio
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1.7. Outlook 51 noise data signal i
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Part II Papers 53
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298 BIBLIOGRAPHY Barber, C., Dobkin
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300 BIBLIOGRAPHY Georgiev, P. (2001
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302 BIBLIOGRAPHY Keck, I., Theis, F
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304 BIBLIOGRAPHY Schießl, I., Sch
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306 BIBLIOGRAPHY Theis, F. and Inou
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