Principal points and elliptical distributions from the multivariate setting to the functional case

TL;DR

This paper extends the concept of principal points and elliptical distributions from finite-dimensional vectors to infinite-dimensional functional data in a Hilbert space, providing new theoretical insights and formulas.

Contribution

It generalizes principal points, self-consistent points, and elliptical distributions to the functional setting, broadening their applicability to Gaussian processes and other random elements.

Findings

Established links between self-consistency and covariance operator eigenvectors.

Provided an explicit formula for the case k=2.

Extended elliptical distribution concepts to functional data.

Abstract

The $k$ principal points of a random vector $\mathbf{X}$ are defined as a set of points which minimize the expected squared distance between $\mathbf{X}$ and the nearest point in the set. They are thoroughly studied in Flury (1990, 1993), Tarpey (1995) and Tarpey, Li and Flury (1995). For their treatment, the examination is usually restricted to the family of elliptical distributions. In this paper, we present an extension of the previous results to the functional elliptical distribution case, i.e., when dealing with random elements over a separable Hilbert space ${\cal H}$. Principal points for gaussian processes were defined in Tarpey and Kinateder (2003). In this paper, we generalize the concepts of principal points, self-consistent points and elliptical distributions so as to fit them in this functional framework. Results linking self-consistency and the eigenvectors of the covariance operator are re-obtained in this new setting as well as an explicit formula for the $k=2$ case so as to include elliptically distributed random elements in ${\cal H}$.