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February 2016 Functional data analysis for density functions by transformation to a Hilbert space
Alexander Petersen, Hans-Georg Müller
Ann. Statist. 44(1): 183-218 (February 2016). DOI: 10.1214/15-AOS1363

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.

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Alexander Petersen. Hans-Georg Müller. "Functional data analysis for density functions by transformation to a Hilbert space." Ann. Statist. 44 (1) 183 - 218, February 2016. https://doi.org/10.1214/15-AOS1363

Information

Received: 1 December 2014; Revised: 1 July 2015; Published: February 2016
First available in Project Euclid: 10 December 2015

zbMATH: 1331.62203
MathSciNet: MR3449766
Digital Object Identifier: 10.1214/15-AOS1363

Subjects:
Primary: 62G05
Secondary: 62G07, 62G20

Rights: Copyright © 2016 Institute of Mathematical Statistics

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Vol.44 • No. 1 • February 2016
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