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Recently, Chatterjee (2023) recognized the lack of a direct generalization ofhis rank correlation $\xi$ in Azadkia and Chatterjee (2021) to amulti-dimensional response vector. As a natural solution to this problem, wehere propose an extension of $\xi$ that is applicable to a set of $q \geq 1$response variables, where our approach builds upon converting the originalvector-valued problem into a univariate problem and then applying the rankcorrelation $\xi$ to it. Our novel measure $T$ quantifies the scale-invariantextent of functional dependence of a response vector $\mathbf{Y} =(Y_1,\dots,Y_q)$ on predictor variables $\mathbf{X} = (X_1, \dots,X_p)$,characterizes independence of $\mathbf{X}$ and $\mathbf{Y}$ as well as perfectdependence of $\mathbf{Y}$ on $\mathbf{X}$ and hence fulfills all thecharacteristics of a measure of predictability. Aiming at maximuminterpretability, we provide various invariance results for $T$ as well as aclosed-form expression in multivariate normal models. Building upon thegraph-based estimator for $\xi$ in Azadkia and Chatterjee (2021), we obtain anon-parametric, strongly consistent estimator for $T$ and show -- as a maincontribution -- its asymptotic normality. Based on this estimator, we develop amodel-free and rank-based feature ranking and forward feature selection formultiple-outcome data that works without any tuning parameters. Simulationresults and real case studies illustrate $T$'s broad applicability.

A direct extension of Azadkia & Chatterjee's rank correlation to multi-response vectors