Functional data analysis for density functions by transformation to a Hilbert space

TL;DR

This paper introduces a transformation-based approach to analyze density functions within a Hilbert space framework, enabling the use of functional data analysis methods on nonnegative, constrained density data.

Contribution

It proposes a novel invertible transformation to map densities into a Hilbert space, facilitating functional analysis techniques for density data with proven convergence rates.

Findings

Transformations like log quantile density enable linear analysis of densities.

Convergence rates are established for the density representations.

Applications demonstrate effectiveness in brain imaging data.

Abstract

Functional data that are nonnegative and have a constrained integral can be considered as samples of one-dimensional density functions. Such data are ubiquitous. Due to the inherent constraints, densities do not live in a vector space and, therefore, commonly used Hilbert space based methods of functional data analysis are not applicable. To address this problem, we introduce a transformation approach, mapping probability densities to a Hilbert space of functions through a continuous and invertible map. Basic methods of functional data analysis, such as the construction of functional modes of variation, functional regression or classification, are then implemented by using representations of the densities in this linear space. Representations of the densities themselves are obtained by applying the inverse map from the linear functional space to the density space. Transformations of interest include log quantile density and log hazard transformations, among others. Rates of convergence are derived for the representations that are obtained for a general class of transformations under certain structural properties. If the subject-specific densities need to be estimated from data, these rates correspond to the optimal rates of convergence for density estimation. The proposed methods are illustrated through simulations and applications in brain imaging.