Sparse dimension reduction based on energy and ball statistics

TL;DR

This paper develops sparse sufficient dimension reduction methods using energy and ball statistics, enabling variable selection in nonlinear models, with practical applications demonstrated.

Contribution

It introduces three novel sparse SDR estimators based on different association measures, expanding the toolkit for variable selection in nonlinear contexts.

Findings

All estimators achieve correct variable selection in simulations.

Estimators are sensitive to outliers and computationally intensive.

Martingale difference divergence-based estimator performs well in bioinformatics example.

Abstract

As its name suggests, sufficient dimension reduction (SDR) targets to estimate a subspace from data that contains all information sufficient to explain a dependent variable. Ample approaches exist to SDR, some of the most recent of which rely on minimal to no model assumptions. These are defined according to an optimization criterion that maximizes a nonparametric measure of association. The original estimators are nonsparse, which means that all variables contribute to the model. However, in many practical applications, an SDR technique may be called for that is sparse and as such, intrinsically performs sufficient variable selection (SVS). This paper examines how such a sparse SDR estimator can be constructed. Three variants are investigated, depending on different measures of association: distance covariance, martingale difference divergence and ball covariance. A simulation study shows that each of these estimators can achieve correct variable selection in highly nonlinear contexts, yet are sensitive to outliers and computationally intensive. The study sheds light on the subtle differences between the methods. Two examples illustrate how these new estimators can be applied in practice, with a slight preference for the option based on martingale difference divergence in the bioinformatics example.