Dynamic Partial Sufficient Dimension Reduction

TL;DR

This paper introduces a novel dynamic partial dimension reduction method that effectively handles the dependency of the reduced subspace of X on W, improving performance over existing approaches especially when X and W are related.

Contribution

The paper proposes a new dynamic partial dimension reduction technique that addresses the variability of the subspace of X with W and demonstrates its theoretical consistency and superior empirical performance.

Findings

Method is asymptotically consistent.

Outperforms existing methods in numerical studies.

Effective in real data analysis.

Abstract

Sufficient dimension reduction aims for reduction of dimensionality of a regression without loss of information by replacing the original predictor with its lower-dimensional subspace. Partial (sufficient) dimension reduction arises when the predictors naturally fall into two sets, X and W, and we seek dimension reduction on X alone while considering all predictors in the regression analysis. Though partial dimension reduction is a very general problem, only very few research results are available when W is continuous. To the best of our knowledge, these methods generally perform poorly when X and W are related, furthermore, none can deal with the situation where the reduced lower-dimensional subspace of X varies dynamically with W. In this paper, We develop a novel dynamic partial dimension reduction method, which could handle the dynamic dimension reduction issue and also allows the dependency of X on W. The asymptotic consistency of our method is investigated. Extensive numerical studies and real data analysis show that our {\it Dynamic Partial Dimension Reduction} method has superior performance comparing to the existing methods.