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Nonlinear sufficient dimension reduction for Conditional quantiles in scalar-on-function single-index models
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DOI：10.1007/s11222-025-10787-4.png)
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Abstract
En 中文
Functional data analysis is crucial in many applications, yet its high-dimensional nature necessitates effective dimension reduction techniques. While existing approaches primarily focus on linear reductions, we introduce a nonlinear sufficient dimension reduction framework for conditional quantiles of single-index models when the predictors are random functions. Our approach constructs two nested functional spaces: a Hilbert space representing the functional data and a reproducing kernel Hilbert space that captures nonlinearity. The kernel in the latter is determined by the inner product of the former, leading to a natural hierarchical structure. We begin by characterizing dimension reduction at the general level of sigma\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma $$\end{document}-fields and proceed to that of classes of functions, leading to the notion of the central quantile class. We introduce our proposed estimator, called the tau\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document}th functional generalized central quantile subspace (tau\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document}-fGCQS), and establish its convergence rate. Finally, we demonstrate the performance of our estimator through simulations and real-world applications to health studies, examining various health indicators, such as ADHD, Parkinson's disease, and BMI.
Keywords:
Dimension reduction
Functional data analysis
Quantile regression
Reproducing kernel Hilbert space
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1.6
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