Independent component analysis, A new concept?

Abstract

The independent component analysis (ICA) of a random vector consists of searching for a linear transformation that minimizes the statistical dependence between its components. In order to define suitable search criteria, the expansion of mutual information is utilized as a function of cumulants of increasing orders. An efficient algorithm is proposed, which allows the computation of the ICA of a data matrix within a polynomial time. The concept oflCA may actually be seen as an extension of the principal component analysis (PCA), which can only impose independence up to the second order and, consequently, defines directions that are orthogonal. Potential applications of ICA include data analysis and compression, Bayesian detection, localization of sources, and blind identification and deconvolution. Zusammenfassung Die Analyse unabhfingiger Komponenten (ICA) eines Vektors beruht auf der Suche nach einer linearen Transformation, die die statistische Abh~ingigkeit zwischen den Komponenten minimiert. Zur Definition geeigneter Such-Kriterien wird die Entwicklung gemeinsamer Information als Funktion von Kumulanten steigender Ordnung genutzt. Es wird ein effizienter Algorithmus vorgeschlagen, der die Berechnung der ICA ffir Datenmatrizen innerhalb einer polynomischen Zeit erlaubt. Das Konzept der ICA kann eigentlich als Erweiterung der 'Principal Component Analysis' (PCA) betrachtet werden, die nur die Unabh~ingigkeit bis zur zweiten Ordnung erzwingen kann und deshalb Richtungen definiert, die orthogonal sind. Potentielle Anwendungen der ICA beinhalten Daten-Analyse und Kompression, Bayes-Detektion, Quellenlokalisierung und blinde Identifikation und Entfaltung. R~sum~ L'Analyse en Composantes Ind6pendantes (ICA) d'un vecteur al6atoire consiste en la recherche d'une transformation lin6aire qui minimise la d6pendance statistique entre ses composantes. Afin de d6finir des crit6res d'optimisation appropribs, on utilise un d6veloppement en s6rie de l'information mutuelle en fonction de cumulants d'ordre croissant. On propose ensuite un algorithme pratique permettant le calcul de I'ICA d'une matrice de donn6es en un temps polynomial. Le concept d'ICA peut 6tre vu en r~alitb comme une extension de l'Analyse en Composantes Principales (PCA) qui, elle, ne peut imposer l'ind6pendance qu'au second ordre et d6finit par cons6quent des directions orthogonales. Les applications potentielles de I'ICA incluent l'analyse et la compression de donn6es, la d&ection bayesienne, la localisation de sources, et l'identification et la d6convolution aveugles.

Extraction of multiple pure component 1H and 13C NMR spectra from two mixtures: Novel solution obtained by sparse component analysis-based blind decomposition

“…Provided that available number of linearly independent mixtures is equal or greater than the number of components, it is possible to separate mixture's spectra into component spectra using only the measurements of the mixture's spectra. This problem is generally known as blind source separation (BSS) and is for described case (more measured mixture's spectra than component spectra) solved by algorithms of independent component analysis (ICA), [11][12][13][14][15][16][17][18]. ICA assumes that pure components are statistically independent and that at most one is normally distributed.…”

“…ICA algorithms solve the BSS problem provided that source signals or pure components are statistically independent and non-Gaussian, as well as that NM, [14][15][16][17][18][19][20].…”

A Bayesian inverse solution using independent component analysis

“…Here m max is the assumed number of independent sources. An ICA algorithm takes the observed x as input, and gives as output a demixing matrix w ∈ R mmax×nmax such that the rows of wx are estimates of the rows of z [3] [4]. Our original inverse problem, defined in Equation (1), is not precisely of this form, but is clearly related.…”

“…The idea of using ICA as a tool to solve inverse problems of the form (1) has already received wide attention in the EEG/MEG context [5] [4]. One strategy has been to first decompose the measurement data x into a sum of components x = x 1 + .…”

Using ICA for removal of ocular artifacts in EEG recorded from blind subjects

“…Independent Component Analysis (ICA) [Comon 1994] is one of a group of algorithms to achieve blind separation of sources [Jutten & Herault 1991]. ICA has already been used successfully for blind source separation of electro encephalogram (EEG) data.…”