On function-on-function linear quantile regression

2025-10-12

AI Summary

This paper addresses the challenge of quantile regression modeling between functional responses and functional predictors. We propose two novel functional partial quantile regression (FPQR) algorithms. Methodologically, we pioneer the integration of partial quantile regression into the function-on-function regression framework, combining functional principal component analysis (FPCA) for dimension reduction with basis function expansions to transform the infinite-dimensional quantile coefficient function estimation into a finite-dimensional multivariate quantile regression problem. Theoretical analysis and Monte Carlo simulations demonstrate that the proposed methods substantially improve estimation accuracy and robustness, particularly in small-sample settings. Empirical studies further confirm their practical effectiveness. The algorithms are implemented in the R package *ffpqr*, ensuring reproducibility and facilitating broad applicability in functional data analysis.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Relational Probabilistic Models
• Knowledge Representation and Reasoning: Qualitative Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

We present two innovative functional partial quantile regression algorithms designed to accurately and efficiently estimate the regression coefficient function within the function-on-function linear quantile regression model. Our algorithms utilize functional partial quantile regression decomposition to effectively project the infinite-dimensional response and predictor variables onto a finite-dimensional space. Within this framework, the partial quantile regression components are approximated using a basis expansion approach. Consequently, we approximate the infinite-dimensional function-on-function linear quantile regression model using a multivariate quantile regression model constructed from these partial quantile regression components. To evaluate the efficacy of our proposed techniques, we conduct a series of Monte Carlo experiments and analyze an empirical dataset, demonstrating superior performance compared to existing methods in finite-sample scenarios. Our techniques have been implemented in the ffpqr package in R.

Problem

• Developing functional partial quantile regression algorithms for accurate estimation
• Projecting infinite-dimensional variables onto finite-dimensional space efficiently
• Approximating function-on-function quantile regression using basis expansion methods

Innovation

• Functional partial quantile regression algorithms for coefficient estimation
• Basis expansion approach to approximate regression components
• Multivariate quantile regression model from partial components