Comparing Multivariate Distributions: A Novel Approach Using Optimal Transport-based Plots

TL;DR

This paper introduces a new method for multivariate Q-Q plots using optimal transport techniques, enabling better comparison of high-dimensional datasets and capturing complex dependencies.

Contribution

It extends traditional univariate Q-Q plots to multivariate data using OT and EOT, providing a novel visualization and testing framework for high-dimensional distribution comparison.

Findings

Effectively captures multivariate dependencies

Identifies distributional differences like tail behavior

Provides new test statistics for distribution comparison

Abstract

Quantile-Quantile (Q-Q) plots are widely used for assessing the distributional similarity between two datasets. Traditionally, Q-Q plots are constructed for univariate distributions, making them less effective in capturing complex dependencies present in multivariate data. In this paper, we propose a novel approach for constructing multivariate Q-Q plots, which extend the traditional Q-Q plot methodology to handle high-dimensional data. Our approach utilizes optimal transport (OT) and entropy-regularized optimal transport (EOT) to align the empirical quantiles of the two datasets. Additionally, we introduce another technique based on OT and EOT potentials which can effectively compare two multivariate datasets. Through extensive simulations and real data examples, we demonstrate the effectiveness of our proposed approach in capturing multivariate dependencies and identifying distributional differences such as tail behaviour. We also propose two test statistics based on the Q-Q and potential plots to compare two distributions rigorously.