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Independent component analysis: A statistical perspective

Overview

Hannu Oja, Klaus Nordhausen

Published Online: Jun 27 2018

DOI: 10.1002/wics.1440

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Independent component analysis (ICA) is a data analysis tool that can be seen as a refinement of principal component analysis or factor analysis. ICA recovers the structures in the data which stay hidden if only the covariance matrix is used in the analysis. The ICA problem is formulated as a latent variable model where the observed variables are linear combinations of unobserved mutually independent non‐Gaussian variables. The goal is to recover linear transformations back to these latent independent components (ICs). As a statistical tool, the unmixing procedure is expressed as a functional in a relevant semiparametric model which further allows a careful formulation of the inference problem and the comparison of competing estimation procedures. For most approaches, the ICs are found in two steps, (a) by standardizing the random vector and then (b) by rotating the standardized vector to the ICs. In the projection pursuit, the ICs can be found either one‐by‐one or simultaneously and this is discussed in detail when the convex combination of the squared third and fourth cumulants is used as a projection index. Alternative projection indices and their use are also explained. The classical fourth‐order blind identification (FOBI) and joint approximate diagonalization of eigenmatrices (JADE) are described as well. The statistical tools for the comparison of consistent and asymptotically multivariate normal unmixing matrix estimates are discussed. Finally, recent extensions for times series, matrix‐ and tensor‐valued and functional data are reviewed. This article is categorized under: Statistical and Graphical Methods of Data Analysis > Modeling Methods and Algorithms Statistical Models > Multivariate Models Statistical and Graphical Methods of Data Analysis > Dimension Reduction Statistical and Graphical Methods of Data Analysis > Information Theoretic Methods

Schematic representation of various location and scatter models. (Credits to Elina Vartiainen, 2017)

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Two estimated signal images and three noise images

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The ladle plot for the data with five images

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Five observed images of size 130 × 130

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The component signals estimated with a reloaded fastICA and g = tanh

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Observed eight signals in the electrocardiography (ECG) data of a pregnant women

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Estimated principal components, whitened components, and independent components (ICs) for bivariate Gaussian, elliptical, and IC datasets

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