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

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.

Published at http://dx.doi.org/10.1214/15-AOS1363 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (5)

• Prediction in functional linear regression
• Methodology and convergence rates for functional linear regression
• Uniform convergence rates for nonparametric regression and principal component analysis in functional/longitudinal data
• Common functional principal components
• Robust functional principal components: A projection-pursuit approach

Cited by in corpus (35)

• LQD-RKHS-based distribution-to-distribution regression methodology for restoring the probability distributions of missing SHM data
• Glucodensities: a new representation of glucose profiles using distributional data analysis
• Distributional data analysis via quantile functions and its application to modelling digital biomarkers of gait in Alzheimer's Disease
• Scalar on time-by-distribution regression and its application for modelling associations between daily-living physical activity and cognitive functions in Alzheimer's Disease
• Cointegration in functional autoregressive processes
• Density Estimation with Distribution Element Trees
• A General Framework for Multivariate Functional Principal Component Analysis of Amplitude and Phase Variation
• Kernel Partial Correlation Coefficient -- a Measure of Conditional Dependence
• Penalized Barycenters in the Wasserstein Space
• Wasserstein Autoregressive Models for Density Time Series
• Principal Component Analysis for Functional Data on Riemannian Manifolds and Spheres
• Wasserstein Regression with Empirical Measures and Density Estimation for Sparse Data
• Latent Deformation Models for Multivariate Functional Data and Time Warping Separability
• Intrinsic Wasserstein Correlation Analysis
• Conditional Wasserstein Barycenters and Interpolation/Extrapolation of Distributions
• Additive Models for Symmetric Positive-Definite Matrices, Riemannian Manifolds and Lie groups
• Cointegrated Density-Valued Linear Processes
• Autoregressive Optimal Transport Models
• Probabilistic Models for Integration Error in the Assessment of Functional Cardiac Models
• Functional instrumental variable regression with an application to estimating the impact of immigration on native wages
• Minimax convergence rate for estimating the Wasserstein barycenter of random measures on the real line
• A characterization of the strong law of large numbers for Bernoulli sequences
• These Unprecedented Times: The Dynamic Pattern Of COVID-19 Deaths Around The World
• Functional outlier detection for density-valued data with application to robustify distribution to distribution regression
• Conformal inference for random objects
• Wasserstein multivariate auto-regressive models for modeling distributional time series
• Intrinsic Sliced Wasserstein Distances for Comparing Collections of Probability Distributions on Manifolds and Graphs
• Causal Inference on Distribution Functions
• Shape-Preserving Prediction for Stationary Functional Time Series
• Wasserstein Gradients for the Temporal Evolution of Probability Distributions
• Individual Heterogeneity Learning in Distributional Data Response Additive Models
• Analyzing Cause-Specific Mortality Trends using Compositional Functional Data Analysis
• Changing reference measure in Bayes spaces with applications to functional data analysis
• Wasserstein Regression
• Mean and Covariance Estimation for Discretely Observed High-Dimensional Functional Data: Rates of Convergence and Division of Observational Regimes