Independent component analysis


Independent component analysis

Independent component analysis (ICA) is a computational method for separating a multivariate signal into additive subcomponents supposing the mutual statistical independence of the non-Gaussian source signals. It is a special case of blind source separation.

Definition

When the independence assumption is correct, blind ICA separation of a mixed signal gives very good results. It is also used for signals that are not supposed to be generated by a mixing for analysis purposes. A simple application of ICA is the “cocktail party problem”, where the underlying speech signals are separated from a sample data consisting of people talking simultaneously in a room. Usually the problem is simplified by assuming no time delays and echoes. An important note to consider is that if "N" sources are present, at least "N" observations (i.e., microphones) are needed to get the original signals. This constitutes the square ("J" = "D", where "D" is the input dimension of the data and "J" is the dimension of the model). Other cases of underdetermined ("J" < "D") and overdetermined ("J" > "D") have been investigated.

The statistical method finds the independent components (aka factors, latent variables or sources) by maximizing the statistical independence of the estimated components. Non-Gaussianity, motivated by the central limit theorem, is one method for measuring the independence of the components. Non-Gaussianity can be measured, for instance, by kurtosis or approximations of negentropy. Mutual information is another popular criterion for measuring statistical independence of signals.

Typical algorithms for ICA use centering, whitening (usually with the eigenvalue decomposition), and dimensionality reduction as preprocessing steps in order to simplify and reduce the complexity of the problem for the actual iterative algorithm. Whitening and dimension reduction can be achieved with principal component analysis or singular value decomposition. Whitening ensures that all dimensions are treated equally "a priori" before the algorithm is run. Algorithms for ICA include infomax, FastICA, and JADE, but there are many others also.

Most ICA methods are not able to extract the actual number of source signals, the order of the source signals, nor the signs or the scales of the sources.

ICA is important to blind signal separation and has many practical applications. It is closely related to (or even a special case of) the search for a factorial code of the data, i.e., a new vector-valued representation of each data vector such that it gets uniquely encoded by the resulting code vector (loss-free coding), but the code components are statistically independent.

Mathematical definitions

Linear independent component analysis can be divided into noiseless and noisy cases,where noiseless ICA is a special case of noisy ICA. Nonlinear ICA should be considered as a separate case.

General definition

The data is represented by the random vector x=(x_1,ldots,x_m) and the components as the random vector s=(s_1,ldots,s_n). The task is to transform the observed data x, using a linear static transformation "W" as: s = W x ,,into maximally independent components s measured by some function F(s_1,ldots,s_n) of independence.

Generative model

Linear noiseless ICA

The components x_i of the observed random vector x=(x_1,ldots,x_m)^T are generated as a sum of the independent components s_k, k=1,ldots,n:

x_i = a_{i,1} s_1 + ldots + a_{i,k} s_k + cdots + a_{i,n} s_n

weighted by the mixing weights a_{i,k}.

The same generative model can be written in vectorial form as x=sum_{k=1}^{n} s_k a_k,where the observed random vector x is represented by the basis vectors a_k=(a_{1,k},ldots,a_{m,k})^T.The basis vectors a_k form the columns of the mixing matrix A=(a_1,ldots,a_n) and the generative formula can be writtenas x=As, where s=(s_1,ldots,s_n)^T.

Given the model and realizations (samples) x_1,ldots,x_N of the random vector x, the task is to estimate both the mixing matrix A and the sources s. This is done by adaptively calculating the w vectors and setting up a cost function which either maximizes the nongaussianity of the calculated s_k = (w^T*x) or minimizes the mutual information. In some cases, a priori knowledge of the probability distributions of the sources can be used in the cost function.

The original sources s can be recovered by multiplying the observed signals x with the inverse of the mixing matrix W=A^{-1}, also known as the unmixing matrix. Here it is assumed that the mixing matrix is square (n=m). If the number of basis vectors is greater than the dimensionality of the observed vectors, n, the task is overcomplete but is still solvable.

Linear noisy ICA

With the added assumption of zero-mean and uncorrelated Gaussian noise nsim N(0,operatorname{diag}(Sigma)), the ICA model takes the form x=As+n.

Nonlinear ICA

The mixing of the sources does not need to be linear. Using a nonlinear mixing function f(cdot| heta) with parameters heta the nonlinear ICA model is x=f(s| heta)+n.

Identifiability

The identifiability of independent component analysis requires that:

* At most one of the sources s_k can be Gaussian,
* The number of observed mixtures, m, must be at least as large as the number of estimated components n: m ge n. It is equivalent to say that the mixing matrix A must be of full rank, for inverse to exist.

See also

* Blind deconvolution
* Blind signal separation (BSS)
* Factor analysis
* Factorial codes
* Hilbert spectrum
* Image processing
* Non-negative matrix factorization (NMF)
* Nonlinear dimensionality reduction
* Principal component analysis (PCA)
* Projection pursuit
* Redundancy reduction
* Signal processing
* Singular value decomposition (SVD)
* Varimax rotation

References

*Pierre Comon (1994): " [http://www.i3s.unice.fr/~pcomon/SP94.html Independent Component Analysis: a new concept] ", Signal Processing, Elsevier, 36(3):287--314 (The original paper describing the concept of ICA)
*A. Hyvärinen, J. Karhunen, E. Oja (2001): " [http://www.cis.hut.fi/projects/ica/book/ Independent Component Analysis] ", New York: Wiley, ISBN 978-0-471-40540-5
** [http://www.cis.hut.fi/projects/ica/book/intro.pdf Introductory chapter]
*J.V. Stone, (2005): " [ftp://ftp.shef.ac.uk/pub/misc/personal/pc1jvs/papers/ica_encyc_jvs4everrit2005.pdf A Brief Introduction to Independent Component Analysis] " in Encyclopedia of Statistics in Behavioral Science, Volume 2, pp. 907–912, Editors Brian S. Everitt & David C. Howell, John Wiley & Sons, Ltd, Chichester, 2005 ISBN 978-0-470-86080-9
*T.-W. Lee (1998): "Independent component analysis: Theory and applications", Boston, Mass: Kluwer Academic Publishers, ISBN 0 7923 8261 7

External links

* [http://www.cs.helsinki.fi/u/ahyvarin/whatisica.shtml What is independent component analysis?] by Aapo Hyvärinen
* [http://www.idsia.ch/~juergen/ica.html Nonlinear ICA, Unsupervised Learning, Redundancy Reduction] by Jürgen Schmidhuber, with links to papers
* [http://www.cis.hut.fi/projects/ica/fastica/ FastICA as a package for Matlab, in R language, C++]
* [http://www.bsp.brain.riken.go.jp/ICALAB/ ICALAB Toolboxes] for Matlab, developed at RIKEN
* [http://www.nic.uoregon.edu/hipersat/index.php High Performance Signal Analysis Toolkit] provides C++ implementations of FastICA and Infomax
* [http://jim-stone.staff.shef.ac.uk/ Free software for ICA] by JV Stone.
* [http://mole.imm.dtu.dk/toolbox/ ICA toolbox] Matlab tools for ICA with Bell-Sejnowski, Molgedey-Schuster and mean field ICA. Developed at DTU.
* [http://www.cis.hut.fi/projects/ica/cocktail/cocktail_en.cgi Demonstration of the cocktail party problem]
* [http://sccn.ucsd.edu/eeglab/ EEGLAB Toolbox] ICA of EEG for Matlab, developed at UCSD.
* [http://sccn.ucsd.edu/fmrlab/ FMRLAB Toolbox] ICA of fMRI for Matlab, developed at UCSD
* [http://brandon-merkl.blogspot.com/2005/12/independent-component-analysis.html Discussion of ICA used in a biomedical shape-representation context]


Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Independent Component Analysis — (ICA) löst das Blind Source Separation Problem (BSS), indem angenommen wird, dass die Quellen einer Datenmischung statistisch unabhängig sind. Inhaltsverzeichnis 1 Problemstellung 2 Vorgehensweise 2.1 Normierung der Varianz 2.2 Bestimmung der… …   Deutsch Wikipedia

  • Component analysis — may refer to: Principal component analysis Kernel principal component analysis Independent component analysis Neighbourhood components analysis ANOVA simultaneous component analysis Connected Component Analysis This disambiguation pag …   Wikipedia

  • Principal component analysis — PCA of a multivariate Gaussian distribution centered at (1,3) with a standard deviation of 3 in roughly the (0.878, 0.478) direction and of 1 in the orthogonal direction. The vectors shown are the eigenvectors of the covariance matrix scaled by… …   Wikipedia

  • Analysis — (from Greek ἀνάλυσις , a breaking up ) is the process of breaking a complex topic or substance into smaller parts to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle,… …   Wikipedia

  • Analysis of variance — In statistics, analysis of variance (ANOVA) is a collection of statistical models, and their associated procedures, in which the observed variance in a particular variable is partitioned into components attributable to different sources of… …   Wikipedia

  • Component-based software engineering — A simple example of two components expressed in UML 2.0. The checkout component, responsible for facilitating the customer s order, requires the card processing component to charge the customer s credit/debit card (functionality that the latter… …   Wikipedia

  • Component-based usability testing — (CBUT) is a testing approach which aims at empirically testing the usability of an interaction component. The latter is defined as an elementary unit of an interactive system, on which behaviour based evaluation is possible. For this, a component …   Wikipedia

  • analysis — /euh nal euh sis/, n., pl. analyses / seez /. 1. the separating of any material or abstract entity into its constituent elements (opposed to synthesis). 2. this process as a method of studying the nature of something or of determining its… …   Universalium

  • Analysis — A psychology term for processes used to gain understanding of complex emotional or behavioral issues. * * * 1. The breaking up of a chemical compound or mixture into simpler elements; a process by which the composition of a substance is… …   Medical dictionary

  • Factor analysis — is a statistical method used to describe variability among observed, correlated variables in terms of a potentially lower number of unobserved, uncorrelated variables called factors. In other words, it is possible, for example, that variations in …   Wikipedia


Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”

We are using cookies for the best presentation of our site. Continuing to use this site, you agree with this.