Hybrid Partial Least Squares Regression with Multiple Functional and Scalar Predictors

TL;DR

This paper introduces Hybrid PLS, a novel method for simultaneous dimension reduction and regression with multiple functional and scalar predictors, validated through synthetic and real renal imaging data.

Contribution

It extends nonlinear iterative PLS to a hybrid Hilbert space, incorporating roughness penalties and providing a computationally efficient solution with theoretical guarantees.

Findings

Successfully recovers predictive structure in multicollinear data

Produces low-dimensional, interpretable representations

Validated on synthetic and renal imaging datasets

Abstract

Motivated by renal imaging studies that combine renogram curves with pharmacokinetic and demographic covariates, we propose Hybrid partial least squares (Hybrid PLS) for simultaneous supervised dimension reduction and regression in the presence of cross-modality correlations. The proposed approach embeds multiple functional and scalar predictors into a unified hybrid Hilbert space and rigorously extends the nonlinear iterative PLS (NIPALS) algorithm. This theoretical development is complemented by a sample-level algorithm that incorporates roughness penalties to control smoothness. By exploiting the rank-one structure of the resulting optimization problem, the algorithm admits a computationally efficient closed-form solution that requires solving only linear systems at each iteration. We establish fundamental geometric properties of the proposed framework, including orthogonality of the latent scores and PLS directions. Extensive numerical studies on synthetic data, together with an application to a renal imaging study, validate these theoretical results and demonstrate the method's ability to recover predictive structure under intermodal multicollinearity, yielding parsimonious low-dimensional representations.