Adapted Chatterjee correlation coefficient

The need to accurately quantify dependence between random variables is a growing concern across various academic disciplines. Current correlation coefficients are typically intended for one of two purposes: testing independence or measuring relationship strength. Despite some attempts to address both aspects, the performance of these measures is still easily affected by oscillation and local noise. To address these limitations, we propose a new coefficient of correlation called the Adapted Chatterjee Correlation Coefficient $\left(A{C}^3\right)$. $A{C}^3$ is designed to accurately identify both independence and functional dependence between variables, even in the presence of noise. We establish the consistency and asymptotic theories of $\left(A{C}^3\right)$. Additionally, we present a novel method, called Iterative Signal Detection Procedure (ISDP), for local signal identification. Our numerical studies and real data application demonstrate that $A{C}^3$ outperforms state-of-the-art methods in terms of general performance and detecting local signals.