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

Description

Quantile-Quantile (Q-Q) plots are commonly used to assess the distributional similarity between two datasets. Traditionally designed for univariate distributions, Q-Q plots are ineffective at capturing the complex dependencies present in multivariate data. In this study developed with S. Singha and S. Vadlamani, we propose a novel approach for constructing multivariate Q-Q plots that extends the traditional methodology to handle high-dimensional data, crucial in modern risk management. Our approach utilizes optimal transport (OT) and entropy-regularized optimal transport (EOT) to align the empirical quantiles of the two datasets. Additionally, we introduce a technique based on OT and EOT potentials that can effectively compare two multivariate datasets using a single bivariate plot, regardless of the distribution's dimension. 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 rigorously compare two distributions. Illustrations are provided in the context of risk management.