On metric choice in dimension reduction for Fr\'echet regression

TL;DR

This paper investigates how the choice of metric impacts dimension reduction in Fréchet regression, emphasizing its importance in analyzing complex health data like brain networks and glucose monitoring.

Contribution

It reviews existing methods and explores the effect of different metrics on dimension reduction estimators in Fréchet regression, supported by numerical and real data studies.

Findings

Different metrics significantly influence the estimators of central and mean spaces.

Metric choice can alter the conclusions drawn from brain connectivity and glucose data.

Numerical studies demonstrate the impact of metric selection on estimation accuracy.

Abstract

Fr\'echet regression is becoming a mainstay in modern data analysis for analyzing non-traditional data types belonging to general metric spaces. This novel regression method is especially useful in the analysis of complex health data such as continuous monitoring and imaging data. Fr\'echet regression utilizes the pairwise distances between the random objects, which makes the choice of metric crucial in the estimation. In this paper, existing dimension reduction methods for Fr\'echet regression are reviewed, and the effect of metric choice on the estimation of the dimension reduction subspace is explored for the regression between random responses and Euclidean predictors. Extensive numerical studies illustrate how different metrics affect the central and central mean space estimators. Two real applications involving analysis of brain connectivity networks of subjects with and without Parkinson's disease and an analysis of the distributions of glycaemia based on continuous glucose monitoring data are provided, to demonstrate how metric choice can influence findings in real applications.