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 utilizes the pairwise distances between the random objects, which makes the choice of metric crucial in the estimation. In this paper, the effect of metric choice on the estimation of the dimension reduction subspace for the regression between random responses and Euclidean predictors is investigated. Extensive numerical studies illustrate how different metrics affect the central and central mean space estimates for regression involving responses belonging to some popular metric spaces versus Euclidean predictors. An analysis of the distributions of glycaemia based on continuous glucose monitoring data demonstrate how metric choice can influence findings in real applications.

翻译：Fréchet回归正逐渐成为分析属于一般度量空间的非传统数据类型的现代数据分析主流方法。这种新颖的回归方法利用随机对象之间的成对距离，使得度量选择在估计中至关重要。本文研究了度量选择对随机响应与欧几里得预测变量之间回归的降维子空间估计的影响。广泛的数值研究说明了不同度量如何影响涉及属于某些常见度量空间的响应与欧几里得预测变量的回归的中心空间和中心均值空间估计。基于连续血糖监测数据的血糖分布分析展示了度量选择如何影响实际应用中的研究发现。