Principal component-guided sparse reduced-rank regression

TL;DR

This paper introduces a novel reduced-rank regression method that biases coefficients toward principal components and incorporates group structures among explanatory variables, enhancing prediction and interpretability.

Contribution

The method integrates pcLasso into reduced-rank regression, effectively leveraging principal component directions and group structures for improved modeling.

Findings

Improves predictive accuracy by biasing toward principal components.

Effectively incorporates group structures among explanatory variables.

Demonstrates superior performance in simulations and real data applications.

Abstract

Reduced-rank regression estimates regression coefficients by imposing a low-rank constraint on the matrix of regression coefficients, thereby accounting for correlations among response variables. To further improve predictive accuracy and model interpretability, several regularized reduced-rank regression methods have been proposed. However, these existing methods cannot bias the regression coefficients toward the leading principal component directions while accounting for the correlation structure among explanatory variables. In addition, when the explanatory variables exhibit a group structure, the correlation structure within each group cannot be adequately incorporated. To overcome these limitations, we propose a new method that introduces pcLasso into the reduced-rank regression framework. The proposed method improves predictive accuracy by accounting for the correlation among response variables while strongly biasing the matrix of regression coefficients toward principal component directions with large variance. Furthermore, even in settings where the explanatory variables possess a group structure, the proposed method is capable of explicitly incorporating this structure into the estimation process. Finally, we illustrate the effectiveness of the proposed method through numerical simulations and real data application.