Dimension reduction based on constrained canonical correlation and variable filtering

TL;DR

This paper introduces a constrained canonical correlation ($C^3$) method combined with variable filtering to improve interpretability and maintain predictive power in high-dimensional data analysis, outperforming existing methods like SSIR.

Contribution

The paper proposes a novel $C^3$ method that constrains canonical correlation estimates and incorporates variable filtering for better interpretability without losing accuracy.

Findings

$C^3$ outperforms SSIR in simulation studies.

The method produces more interpretable composite directions.

Applications demonstrate practical usefulness.

Abstract

The ``curse of dimensionality'' has remained a challenge for high-dimensional data analysis in statistics. The sliced inverse regression (SIR) and canonical correlation (CANCOR) methods aim to reduce the dimensionality of data by replacing the explanatory variables with a small number of composite directions without losing much information. However, the estimated composite directions generally involve all of the variables, making their interpretation difficult. To simplify the direction estimates, Ni, Cook and Tsai [Biometrika 92 (2005) 242--247] proposed the shrinkage sliced inverse regression (SSIR) based on SIR. In this paper, we propose the constrained canonical correlation ($C^3$) method based on CANCOR, followed by a simple variable filtering method. As a result, each composite direction consists of a subset of the variables for interpretability as well as predictive power. The proposed method aims to identify simple structures without sacrificing the desirable properties of the unconstrained CANCOR estimates. The simulation studies demonstrate the performance advantage of the proposed $C^3$ method over the SSIR method. We also use the proposed method in two examples for illustration.