Dependence functions based on Chatterjee's rank correlation

We investigate a geometric and distributional reinterpretation of Chatterjee's $\xi$-coefficient, which measures functional dependence between a response variable $Y$ and a predictor vector $\mathbf{X}$. For this purpose, we analyze the Markov product $(Y,Y')$, where $Y'$ is a copy of $Y$ that is conditionally independent of $Y$ given $\mathbf{X}$. Based on this construction, we introduce and study two dependence functions, denoted by $\phi_{(Y,\mathbf{X})}$ and $\kappa_{(Y,\mathbf{X})}$. The proposed framework provides a geometric interpretation of the Markov product and extends Chatterjee's correlation coefficient to a richer and more interpretable object for the analysis of directed stochastic dependence. In particular, rather than only measuring how well $Y$ can be represented as a function of $\mathbf{X}$, the proposed dependence functions additionally quantify how strongly the corresponding Markov product is concentrated near the diagonal.