B1278
Title: Hilbertian fourth order blind identification
Authors: Germain Van Bever - Universite de Namur (Belgium) [presenting]
Hannu Oja - University of Turku (Finland)
Frank Critchley - Open University (United Kingdom)
Bing Li - The Pennsylvania State University (United States)
Radka Sabolova - The Open University (United Kingdom)
Abstract: In the classical Independent Component (IC) model, the random vector $X$ is assumed to satisfy $X=\Psi Z$, where $Z$ has independent marginals and $\Psi$ is an invertible mixing matrix. Independent component analysis (ICA) encompasses all methods aiming at unmixing $X$, that is estimating a (non unique) unmixing matrix $\Gamma$ such that $\Gamma X$ has independent components. The celebrated Fourth Order Blind Identification (FOBI) procedure provides such a $\Gamma$ based on the regular covariance matrix and a scatter matrix based on fourth moments. Nowadays, functional data (FD) are occurring more and more often in practice, and relatively few statistical techniques have been developed to analyze this type of data. Functional PCA is one such technique which focuses on dimension reduction. We propose an extension of the FOBI methodology to the case of Hilbertian data, FD being the go-to example used throughout. When dealing with distributions on Hilbert spaces, two major problems arise: (i) the notion of ``marginals'' is not naturally defined and (ii) the covariance operator is, in general, non invertible. These limitations are tackled by reformulating the problem in a coordinate-free manner and by imposing natural restrictions on the mixing model. The proposed procedure is shown to be Fisher consistent and affine invariant. A sample estimator is provided and illustrated on simulated and real datasets.
