An introduction of independent component analysis ica
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An Introduction of Independent Component Analysis (ICA). Xiaoling Wang Jan. 28, 2003. What Is ICA?. Application: blind source separation (BSS) and deconvolution Motivation: “cocktail party problem” Assumption: two people speaking simultaneously, two microphones in different locations.

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An Introduction of Independent Component Analysis (ICA)

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An introduction of independent component analysis ica

An Introduction of Independent Component Analysis (ICA)

Xiaoling Wang

Jan. 28, 2003


What is ica

What Is ICA?

  • Application: blind source separation (BSS) and deconvolution

  • Motivation: “cocktail party problem”

    • Assumption: two people speaking simultaneously, two microphones in different locations


Principles of ica algorithm

Principles of ICA Algorithm

  • Assumption: sources are statistically independent

  • Goal: it seeks a transformation to coordinates in which the data are maximally statistically independent

  • Definition:

Mixing process

Demixing process

– mixing matrix, – separation matrix


Hierarchy of ica models

Hierarchy of ICA Models

Nonlinear mixing

Non-stationary

mixing

Linear mixing

Non-stationary

sources

Non-Gaussian sources

Gaussian sources

No noise

Independent

Factor analysis

Classical ICA

Factor Analysis

R diagonal

Approximations to

mutual information

Cumulant based

methods

Flexible

Source model

Switching

source model

Probabilistic

PCA

Fixed

source model

Kurtosis

minimization

No noise

FastICA

Infomax

PCA

orthogonal mixing


Independence of sources

Independence of Sources

  • Independence: the pdf of sources can be factorized

  • Nongaussian is independent

  • Seek the separation matrix W which maximize the nongaussianity of the estimated sources


Measures of nongaussianity

Measures of Nongaussianity

  • Kurtosis (4th order cumulant):

    • Subgaussian: negative kurtosis

    • Supergaussian: positive kurtosis

  • Negentropy:

entropy

differential

entropy

negentropy


Measures of nongaussianity cont

Measures of Nongaussianity (Cont.)

  • Mutual information:

For ,


Fastica algorithm

FastICA Algorithm

  • Basic form:

    • Choose an initial (e.g. Random) weight vector

    • Let

    • Let

    • If not converged, go back to step 2

  • For several units: decorrelation

    • Let

    • Let


Nonlinear ica

Nonlinear ICA

  • Model:

  • Existence and uniqueness of solutions

    • There always exists an infinity of solutions if the space of the nonlinear mixing functions is not limited

    • Post-nonlinear problem

mixing

demixing


Algorithms for nonlinear ica

Algorithms for Nonlinear ICA

  • Burel’s approach: neural solution, known nonlinearities on unknown parameters

  • Krob & Benidir: high order moments, polynomial mixtures

  • Pajunen et al.: SOMs, locally factorable pdf

  • Pajunen et al.: GTM(generative topographic mapping), output distribution matches the known source distributions

  • Post nonlinear mixtures:

    • Taleb & Jutten: adaptive componentwise separation

    • Yang et al.: two-layer neural network

    • Puntonet et al.: nonlinearities are a power function, geometrical considerations


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