メリークリスマス🎄
ここ最近ホットな Chatterjee の新しい相関係数について、当指標は本質的に相関係数というより決定係数であり、古典的なピアソンの $R^2$ のノンパラ版と位置付けられることを数学的に示しました。
On the epsilon-delta Structure Underlying Chatterjee's Rank Correlation
We provide an epsilon-delta interpretation of Chatterjee's rank correlation by tracing its origin to a notion of local dependence between random variables. Starting from a primitive epsilon-delta construction, we show that rank-based dependence measures arise naturally as epsilon to zero limits of local averaging procedures. Within this framework, Chatterjee's rank correlation admits a transparent interpretation as an empirical realization of a local L1 residual.
We emphasize that the probability integral transform plays no structural role in the underlying epsilon-delta mechanism, and is introduced only as a normalization step that renders the final expression distribution-free. We further consider a moment-based analogue obtained by replacing the absolute deviation with a squared residual. This L2 formulation is independent of rank transformations and, under a Gaussian assumption, recovers Pearson's coefficient of determination.
※所々雑なので、近日中にver2版を公開予定です。
※そして、Qiita上で更に内容を噛み砕いた日本語版を公開予定です。