Independent Component Analysis (ICA) - one of the basic tools in data analysis - aims to find a coordinate system in which the components of the data are independent. Most popular ICA methods use kurtosis as a metric of non-Gaussianity to maximize, such as FastICA and JADE. However, their assumption of fourth-order moment (kurtosis) may not always be satisfied in practice. One of the possible solution is to use third-order moment (skewness) instead of kurtosis, which was applied in @math and EcoICA. In this paper we present a competitive approach to ICA based on the Split Generalized Gaussian distribution (SGGD), which is well adapted to heavy-tailed as well as asymmetric data. Consequently, we obtain a method which works better than the classical approaches, in both cases: heavy tails and non-symmetric data.
Lacoume and Ruiz @cite_55 sketched a mathematical approach to the problem using higher-order statistics, which can be interpreted as a measure of fitting independent components. Cardoso @cite_62 @cite_48 focused on the algebraic properties of the fourth-order cumulant (kurtosis), which is still a popular approach @cite_24 .
@cite_24 @cite_48 @cite_55 @cite_62
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This discussion presents a new perspective of subspace independent component analysis (ICA). The notion of a function of cumulants (kurtosis) is generalized to vector kurtosis. This vector kurtosis is utilized in the subspace ICA algorithm to estimate subspace independent components. One of the main advantages of the presented approach is its computational simplicity. The experiments have shown promising results in estimating subspace independent components. This article considers high-order measures of independence for the independent component analysis problem and discusses the class of Jacobi algorithms for their optimization. Several implementations are discussed. We compare the proposed approaches with gradient-based techniques from the algorithmic point of view and also on a set of biomedical data. Ideas for higher-order array processing are introduced, focusing on fourth-order cumulant statistics. They are expressed in an index-free formalism allowing the exploitation of all their symmetry properties. It is shown that, when dealing with 4-index quantities, symmetries are related to rank properties. The rich symmetry structure yields a class of identification algorithms. An algebraic technique for blind identification is included, and a few others are briefly indicated. >