Elsevier

Knowledge-Based Systems

Volume 173, 1 June 2019, Pages 113-116
Knowledge-Based Systems

Short communication
An overview of properties and extensions of FOBI

https://doi.org/10.1016/j.knosys.2019.02.026Get rights and content

Abstract

Recently, Spurek, Tabor, Struksi and Smieja (2018) suggested an independent component analysis method that is obtained via a generalized eigenvalue decomposition and stated that their estimator enjoys several useful properties. The estimator is however already known since 1989 as FOBI (Fourth order blind identification), and it indeed has many nice properties, even outside the independent component model. As there seems to be interest in the readers of Knowledge-Based Systems in the estimator, we shortly review it and various of its properties not mentioned by Spurek et al. (2018), along with some recent extensions of FOBI.

Section snippets

Independent component analysis

Independent component analysis (ICA) is a well-established multivariate method introduced in the mid-1980’s (for an account of its early history, see [1]). The first driving force of ICA was the computer science and signal processing community but in the recent years also statisticians have gotten more and more interested in ICA, i.e., deriving statistical properties of the existing ICA methods and suggesting new estimators. For some overviews, see for example [2], [3], [4].

The most basic model

Fourth order blind identification

The problem of finding the eigenvectors U of M(xst) can be written as UCOV(x)12COV4(x)COV(x)12U=Λ, where Λ is a diagonal matrix of eigenvalues in decreasing order and COV4(x)=E[xE(x)]COV(x)1[xE(x)][xE(x)][xE(x)] is the so-called matrix of fourth moments (note that in some references COV4(x) is divided by p+2 to make it consistent for COV at the multivariate normal model). COV4 can be seen as a member of the class of functionals of the form Egx̃COV(x)1x̃x̃x̃, where x̃=xE(x) and g:

Conclusion

Recently [5] rediscovered FOBI, a well-known ICA estimator with many nice properties established mainly in the statistics literature in the previous years. Despite being quite inefficient when compared to other ICA methods, the simplicity, ease of computation and numerous useful properties (both outside and inside the IC model) of FOBI help explain its persisting popularity. These aspects also make FOBI the perfect method for generalizing ICA to complex non-standard data structures, as the many

Acknowledgment

The work of KN was partly supported by CRoNoS COST Action (Austria) IC1408.

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