A direct extension of Azadkia & Chatterjee's rank correlation to multi-response vectors
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866913713232019456 |
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| author | Ansari, Jonathan Fuchs, Sebastian |
| author_facet | Ansari, Jonathan Fuchs, Sebastian |
| contents | Recently, Chatterjee (2023) recognized the lack of a direct generalization of his rank correlation $ξ$ in Azadkia and Chatterjee (2021) to a multi-dimensional response vector. As a natural solution to this problem, we here propose an extension of $ξ$ that is applicable to a set of $q \geq 1$ response variables, where our approach builds upon converting the original vector-valued problem into a univariate problem and then applying the rank correlation $ξ$ to it. Our novel measure $T$ quantifies the scale-invariant extent 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 perfect dependence of $\mathbf{Y}$ on $\mathbf{X}$ and hence fulfills all the characteristics of a measure of predictability. Aiming at maximum interpretability, we provide various invariance results for $T$ as well as a closed-form expression in multivariate normal models. Building upon the graph-based estimator for $ξ$ in Azadkia and Chatterjee (2021), we obtain a non-parametric, strongly consistent estimator for $T$ and show -- as a main contribution -- its asymptotic normality. Based on this estimator, we develop a model-free and rank-based feature ranking and forward feature selection for multiple-outcome data that works without any tuning parameters. Simulation results and real case studies illustrate $T$'s broad applicability. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2212_01621 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | A direct extension of Azadkia & Chatterjee's rank correlation to multi-response vectors Ansari, Jonathan Fuchs, Sebastian Statistics Theory Methodology Recently, Chatterjee (2023) recognized the lack of a direct generalization of his rank correlation $ξ$ in Azadkia and Chatterjee (2021) to a multi-dimensional response vector. As a natural solution to this problem, we here propose an extension of $ξ$ that is applicable to a set of $q \geq 1$ response variables, where our approach builds upon converting the original vector-valued problem into a univariate problem and then applying the rank correlation $ξ$ to it. Our novel measure $T$ quantifies the scale-invariant extent 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 perfect dependence of $\mathbf{Y}$ on $\mathbf{X}$ and hence fulfills all the characteristics of a measure of predictability. Aiming at maximum interpretability, we provide various invariance results for $T$ as well as a closed-form expression in multivariate normal models. Building upon the graph-based estimator for $ξ$ in Azadkia and Chatterjee (2021), we obtain a non-parametric, strongly consistent estimator for $T$ and show -- as a main contribution -- its asymptotic normality. Based on this estimator, we develop a model-free and rank-based feature ranking and forward feature selection for multiple-outcome data that works without any tuning parameters. Simulation results and real case studies illustrate $T$'s broad applicability. |
| title | A direct extension of Azadkia & Chatterjee's rank correlation to multi-response vectors |
| topic | Statistics Theory Methodology |
| url | https://arxiv.org/abs/2212.01621 |