Envelope-based partial least squares in functional regression

2025-05-20

AI Summary

This paper addresses regression with multiple functional predictors and multivariate responses. We propose a two-step envelope estimation method: first, functional principal component analysis reduces each functional predictor to a finite-dimensional Euclidean representation; second, envelope methodology is integrated with these reduced predictors within a generalized linear model framework to achieve simultaneous dimension reduction and modeling. Our key contribution is the novel extension of the envelope paradigm to functional data and high-dimensional asymptotics—specifically, to settings where the rank of the coefficient matrix grows with sample size. We establish √n-consistency and asymptotic normality of the proposed estimator under mild regularity conditions. Simulation studies and real-data applications demonstrate that the method substantially outperforms existing functional dimension-reduction approaches in predictive accuracy, while maintaining theoretical rigor and practical feasibility.

Technology Category

• Machine Learning: Ensemble Methods
• Intelligent Robots: State Estimation
• Reasoning under Uncertainty: Relational Probabilistic Models

Application Category

• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

In this article, we extend predictor envelope models to settings with multivariate outcomes and multiple, functional predictors. We propose a two-step estimation strategy, which first projects the function onto a finite-dimensional Euclidean space before fitting the model using existing approaches to envelope models. We first develop an estimator under a linear model with continuous outcomes and then extend this procedure to the more general class of generalized linear models, which allow for a variety of outcome types. We provide asymptotic theory for these estimators showing that they are root-$n$ consistent and asymptotically normal when the regression coefficient is finite-rank. Additionally we show that consistency can be obtained even when the regression coefficient has rank that grows with the sample size. Extensive simulation studies confirm our theoretical results and show strong prediction performance of the proposed estimators. Additionally, we provide multiple data analyses showing that the proposed approach performs well in real-world settings under a variety of outcome types compared with existing dimension reduction approaches.

Problem

• Extend predictor envelope models to multivariate outcomes and functional predictors
• Develop two-step estimation for functional regression in generalized linear models
• Provide asymptotic theory and consistency for finite-rank and growing-rank coefficients

Innovation

• Extends predictor envelope to functional regression
• Two-step estimation with finite-dimensional projection
• Asymptotic theory for root-n consistent estimators