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<p>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 <i>nonlinear sufficient dimension reduction</i> framework for <i>conditional quantiles</i> 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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\sigma \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>σ</mi> </math></EquationSource> </InlineEquation>-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 <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\tau \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>τ</mi> </math></EquationSource> </InlineEquation>th functional generalized central quantile subspace (<InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\tau \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>τ</mi> </math></EquationSource> </InlineEquation>-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.</p>

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Nonlinear sufficient dimension reduction for Conditional quantiles in scalar-on-function single-index models

  • 期刊论文

  • 发表日期：

    2025年12月1日

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  • Shanshan Wang,
  • Eliana Christou,
  • Eftychia Solea,
  • Jun Song

摘要

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 \(\sigma \) σ -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 \(\tau \) τ th functional generalized central quantile subspace ( \(\tau \) τ -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.