Structured functional additive regression in reproducing kernel Hilbert spaces.

Area of Science:

• Statistics
• Machine Learning
• Functional Data Analysis

Background:

• Functional additive models (FAMs) offer a flexible regression framework for functional predictors.
• Extending classical functional linear models, FAMs utilize data-driven bases in an additive structure.
• Selecting nonlinear additive components in FAMs remains a significant challenge.

Purpose of the Study:

• To propose a novel regularization framework for structure estimation in FAMs.
• To address the under-studied problem of selecting nonlinear additive components.
• To enhance the flexibility and applicability of functional regression models.

Main Methods:

• Development of a regularization framework within Reproducing Kernel Hilbert Spaces.
• Leveraging functional principal components for simplified implementation and analysis.
• Employing penalized least squares with a sparsity-inducing penalty for component selection and estimation.

Main Results:

• Theoretical properties, including convergence rates, are rigorously investigated.
• The proposed method demonstrates effective selection and estimation of additive components.
• Empirical validation through simulation studies and a real-world data application.

Conclusions:

• The proposed regularization framework offers a robust solution for structure estimation in FAMs.
• The approach facilitates the identification of key nonlinear functional predictors.
• This work advances the field of functional data analysis by providing a practical and theoretically sound method.