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

By Alexander Petersen, H. Muller. 2016

214 citations recorded by the source corpus.

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.

Subfields: Geometry and Topology, Molecular Biology, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/c7121a418684a7f36ea233fcf3d3019cffbd4988

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

In Molecular Biology

In Statistics and Probability

Citations in the corpus

Wasserstein autoregressive models for density time series (2020, methodology): “This approach is based on the work of [27] and modified by [18].”
Additive Functional Regression for Densities as Responses (2020, methodology): “Such alternative transformations satisfying the required criteria are discussed in Petersen and Müller (2016), where specifically the log hazard transformation is investigated.”