Functional sufficient dimension reduction through distance covariance

TL;DR

This paper introduces a new, flexible method for reducing the dimensionality of functional data using distance covariance, which is robust, model-free, and suitable for sparse longitudinal data, outperforming existing techniques.

Contribution

The paper presents a novel distance covariance-based approach for functional dimension reduction that relaxes traditional assumptions and is applicable to sparse longitudinal data.

Findings

Demonstrates improved performance over existing methods in simulations

Establishes statistical consistency of the estimator

Effective in real data analysis

Abstract

Our research proposes a novel method for reducing the dimensionality of functional data, specifically for the case where the response is a scalar and the predictor is a random function. Our method utilizes distance covariance, and has several advantages over existing methods. Unlike current techniques which require restrictive assumptions such as linear conditional mean and constant covariance, our method has mild requirements on the predictor. Additionally, our method does not involve the use of the unbounded inverse of the covariance operator. The link function between the response and predictor can be arbitrary, and our proposed method maintains the advantage of being model-free, without the need to estimate the link function. Furthermore, our method is naturally suited for sparse longitudinal data. We utilize functional principal component analysis with truncation as a regularization mechanism in the development of our method. We provide justification for the validity of our proposed method, and establish statistical consistency of the estimator under certain regularization conditions. To demonstrate the effectiveness of our proposed method, we conduct simulation studies and real data analysis. The results show improved performance compared to existing methods.