Title: Functional independent component analysis
Authors: Germain Van Bever - Universite de Namur (Belgium) [presenting]
Radka Sabolova - The Open University (United Kingdom)
Frank Critchley - Open University (United Kingdom)
Bing Li - The Pennsylvania State University (United States)
Hannu Oja - University of Turku (Finland)
Abstract: With the increase in measurement precision, functional data is becoming common practice. Relatively few techniques for analysing such data have been developed, however, and a first step often consists in reducing the dimension via Functional PCA, which amounts to diagonalising the covariance operator. Joint diagonalisation of a pair of scatter functionals has proved useful in many different setups, such has Independent Component Analysis (ICA), Invariant Coordinate Selection (ICS), etc. The Fourth Order Blind Identification procedure is extended to the case of data on a separable Hilbert space (with classical FDA setting being the go-to example). In the finite-dimensional setup, this procedure provides a matrix Gamma such that Gamma $X$ has independent components, if one assumes that the random vector $X$ satisfies $X = \Psi Z$, where $Z$ has independent marginals and $\Psi$ is an invertible mixing matrix. 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 its convergence rates are derived. The procedure is amply illustrated on simulated and real datasets.