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Independent Component Analysis and Bayesian Separation Methods

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Nonlinear Blind Source Separation and Blind Mixture Identification

Part of the book series: SpringerBriefs in Electrical and Computer Engineering ((BRIEFSELECTRIC))

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Abstract

This chapter presents the first major class of blind source separation and blind mixture identification methods intended for linear-quadratic mixtures (including their bilinear and purely quadratic restricted versions), namely methods strongly based on a probabilistic framework. This mainly includes not only nonlinear versions of Independent Component Analysis (ICA) but also Bayesian methods. The description of these methods is split into two parts, depending whether the source signals are independent and identically distributed (i.i.d) or not. Various statistical approaches are considered, namely moment- and cumulant-based methods, the maximum likelihood approach, and methods based on information theory (especially using mutual information), whose connection with the maximum likelihood approach is discussed.

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Notes

  1. 1.

    This system was originally [56, 68] the linear recurrent, or feedback, neural network described in Sect. 3.4.1. Most methods then used the system defined by (3.1), which may be considered as a direct, or feedforward, network.

  2. 2.

    For this very specific mixing model, the analytical expression of the inverse of the mixing model is known, as explained in Sect. 3.2, and might therefore be used instead.

  3. 3.

    The mixing equations (4.14)–(4.15) obtained here are similar to the Eqs. (3.17)–(3.18) derived above for the bilinear mixing model, thanks to the use of the notations (4.12)–(4.13). However, it should be clear that this is here obtained by using the notations L ij for coefficients associated with quadratic (auto-)terms, whereas in the bilinear model (3.17)–(3.18) they were used for coefficients associated with linear terms, which was more coherent with these notations L ij.

  4. 4.

    In the general framework of BSS/BMI, Bayesian methods may also be used to estimate only one of the above types of unknowns: see, e.g., p. 471 of [85].

  5. 5.

    For linear-quadratic mixing models, and especially bilinear ones, various other investigations were also reported concerning the use of Bayesian methods, but for the non-blind (or supervised) configuration, i.e., when the mixing model parameters are known and only the sources are to be estimated (see, e.g., [54]), which is out of the scope of this book. Other types of non-blind methods for such mixtures are reported, e.g., in [76].

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Authors and Affiliations

  1. Paul Sabatier University, Toulouse, France

    Yannick Deville & Shahram Hosseini

  2. State University of Campinas, Campinas, Brazil

    Leonardo Tomazeli Duarte

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Deville, Y., Tomazeli Duarte, L., Hosseini, S. (2021). Independent Component Analysis and Bayesian Separation Methods. In: Nonlinear Blind Source Separation and Blind Mixture Identification. SpringerBriefs in Electrical and Computer Engineering. Springer, Cham. https://doi.org/10.1007/978-3-030-64977-7_4

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  • DOI: https://doi.org/10.1007/978-3-030-64977-7_4

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