On the epsilon-delta Structure Underlying Chatterjee's Rank Correlation

2025-12-13

AI Summary

Chatterjee’s rank correlation coefficient lacks a rigorous theoretical foundation clarifying its underlying probabilistic structure. Method: We rigorously analyze the local averaging process in the ε → 0 limit, revealing that the coefficient’s intrinsic ε–δ structure arises from local dependence between random variables. We reformulate it as a natural empirical L₁ residual measure and develop the first formal ε–δ–based interpretive framework. We prove that the probability integral transform serves only a normalization role and does not contribute to the core structural definition; we further propose an L₂ variant that recovers Pearson’s R² under Gaussianity. Contribution/Results: Our work decouples distribution-freeness from model interpretability, unifies multiple dependence measures—including Spearman, Kendall, and Pearson—under a common local-to-global dependency lens, and establishes a theoretical bridge from local dependence to global correlation. The framework provides both statistical robustness and transparent geometric intuition, enabling principled extensions to multivariate and conditional settings.

Technology Category

• Reasoning under Uncertainty: Relational Probabilistic Models
• Machine Learning: Calibration & Uncertainty Quantification
• Knowledge Representation and Reasoning: Preferences

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Security and Privacy: Large-scale security measurements

Abstract

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.

Problem

• Interpreting Chatterjee's rank correlation via epsilon-delta local dependence
• Deriving rank-based measures as limits of local averaging procedures
• Formulating a moment-based analogue independent of rank transformations

Innovation

• Epsilon-delta framework interprets Chatterjee's rank correlation
• Local averaging procedures yield rank-based dependence measures
• L2 formulation recovers Pearson's coefficient under Gaussian assumption