Chatterjee's "rank correlation"

Sourav Chatterjee proposed a new correlation coefficient. Different from the classical correlation coefficient (e.g., Pearson's) that can only capture linear relationship, the Chatterjee's coefficient can indicate if $Y$ is related to any function of $X$, whether it is linear or nonlinear. Suppose $Y_i = X_i^2$ for all $i$, then $Y_i$ is determined by $X_i$. Pearson's coefficient will yield very low coefficient since there was little to no linear correlation found. Meanwhile, Chatterjee's coefficient will still have high value because it knows that $Y_i$ depends on $X_i$ by some function (in this case, quadratic).

This coefficient value is between 0 and 1. Meaning, the relationship is weak if the coefficient value gets closer to 0 and strong as it gets closer to 1. It is 0 if and only if $X$ and $Y$ are independent, and is 1 if and only if $Y$ is a measurable function of $X$ almost surely.

given pairs of i.i.d. random variables $(X_i, Y_i)$ where $i=1, ..., n$. Chatterjee's correlation is obtained through following procedure:

• Sort the pairs according to $X$ as $(X_{[i]}, Y_{[i]})$ such that $X_{[i]} \leq ... \leq X_{[n]}$

• Calculate $r_i$, which is the rank for $Y_{[i]}$, for $i=1, ..., n$. The rank is basically the count of $j$ such that $Y_{[j]} \leq Y_{[i]}$. Formally, $r_i$ itself is obtained by $$ r_i = \sum_{j=1}^n \mathbb{1}\left(Y_{[j]} \leq Y_{[i]}\right) $$ where $\mathbb{1}\left(Y_{[j]} \leq Y_{[i]}\right)$ is an indicator function that evaluates to 1 if the expression in the parenthesis is true, and zero otherwise.

• And finally calculate the coefficient itself:

  $$ \xi_n(X, Y) = 1-\frac{3 \sum_{i=1}^{n-1} \lvert r_{i+1} - r_i \rvert}{n^2 - 1} $$

It can be implemented as a python function in few lines of code:

import numpy as np

def chatterjee(x, y):
    # sort x and get the sorted index
    idx = np.argsort(x.ravel())

    # rearrange y based on idx
    y_ = y.ravel()[idx]

    # calculate a list of ranks
    r = (y_[:, None] <= y_).sum(1)

    # the coefficient calculation
    xi = 1 - (3 * np.sum(np.abs(r[1:] - r[:-1]))) / (len(y_) ** 2 - 1)

    return xi

I compared the classical linear correlation coefficient, i.e., Pearson's r-score ($r$), with Chatterjee's xi-score ($\xi$). It is interesting to see in a glimpse that xi-score can capture relationship between $X$ and $Y$, where the function of $X$ is not necessarily linear. For example, the last example, i.e., quadratic function.

Linear relationship

Random scatter

Quadratic relationship

Note that although in the first example $Y=X$, the xi-score is less than one! This is one of the caveats of Chatterjee's xi-score. Chatterjee himself put a remark:

"(9) If there are no ties among the $Y_i$'s, the maximum possible value of $\xi_n(X, Y)$ is $(n -2) / (n + 1)$, which is attained if $Y_i = X_i$ for all $i$. This can be noticeably less than 1 for small $n$."

All in all, I think Chatterjee's coefficient is a good addition to our toolbox.