Multidimensional Wilcoxon Mann Whitney U Test

This test aims to give a score that represents how well two sets of multidimensional points are separated. For each point $x$ of a given set, create the distance rankings to all other points and compute the statistic:

$$ U_x= \frac{\max(R_{1}-{n_{1}(n_{1}+1) \over 2}, R_{2}-{n_{2}(n_{2}+1) \over 2})}{n_1 n_2}$$

where $R_1$ is the sum of the rankings for the $n_1$ points of the same set as $x$, and $R_2$ is the sum of the rankings for the $n_2$ points of a different set than $x$. This statistic increases when the points of the same set as $x$ are closer to $x$ than the points the other set. Finally, we compute the mean of $U_x$ for all the points $x$, which we will call $AvgU$. This statistic lies in the range $0.5 - 1$, and the univariate WMW is equivalent to the AUC. This code was used in [1]. The computation of the presented statistic is a variant of the multivariate statistic presented in [2].

[1] Riera, P., Cerdeiro, M., Pepino, L., & Ferrer, L. (2023, June). Phone and speaker spatial organization in self-supervised speech representations. In 2023 IEEE International Conference on Acoustics, Speech, and Signal Processing Workshops (ICASSPW) (pp. 1-5). IEEE.

[2] Liu, J., Ma, S., Xu, W., & Zhu, L. (2022). A generalized Wilcoxon–Mann–Whitney type test for multivariate data through pairwise distance. Journal of Multivariate Analysis, 190, 104946.

Demo

Install

pip install multidimensional-wilcoxon-mann-whitney

Usage

from multidimensional_wilcoxon_mann_whitney import multidimensional_ranksum

U = multidimensional_ranksum(X,y)

Cite

@inproceedings{riera2023phone,
  title={Phone and speaker spatial organization in self-supervised speech representations},
  author={Riera, Pablo and Cerdeiro, Manuela and Pepino, Leonardo and Ferrer, Luciana},
  booktitle={2023 IEEE International Conference on Acoustics, Speech, and Signal Processing Workshops (ICASSPW)},
  pages={1--5},
  year={2023},
  organization={IEEE}
}