On metric choice in dimension reduction for Fréchet regression

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.

A selective review of sufficient dimension reduction for multivariate response regression

We review sufficient dimension reduction (SDR) estimators with multivariate response in this paper. A wide range of SDR methods are characterized as inverse regression SDR estimators or forward regression SDR estimators. The inverse regression family include pooled marginal estimators, projective resampling estimators, and distance-based estimators. Ordinary least squares, partial least squares, and semiparametric SDR estimators, on the other hand, are discussed as estimators from the forward regression family.