Measuring association with Wasserstein distances

Abstract

Let π ∈ Π(µ, ν) be a coupling between two probability measures µ and ν on a Polish space. In this article we propose and study a class of nonparametric measures of association between µ and ν. The analysis is based on the Wasserstein distance between ν and the disintegration πx 1 of π with respect to the first coordinate. We also establish basic statistical properties of this new class of measures: we develop a statistical theory for strongly consistent estimators and determine their convergence rate. Throughout our analysis we make use of the so-called adapted/causal Wasserstein distance, in particular we rely on results established in [Backhoff, Bartl, Beiglböck, Wiesel. Estimating processes in adapted Wasserstein distance. 2020]. Our class of measures offers on alternative to the correlation coefficient proposed by [Dette, Siburg and Stoimenov (2013). A copula-based non-parametric measure of regression dependence. Scandinavian Journal of Statistics 40(1), 21-41] and [Chatterjee (2020). A new coefficient of correlation. Journal of the American Statistical Association, 1-21]. In contrast to these works, our approach also applies to probability laws on general Polish spaces.

Transport Dependency: Optimal Transport Based Dependency Measures

“…To conclude, we want to emphasize that this article should only be regarded as one step amongst many towards the goal of fully exploring the use of optimal transport as a statistical association measure between random variables. Indeed, as witnessed by the work of Wiesel (2021) and Mordant and Segers (2021) conducted in parallel to ours, this topic currently experiences a surge of a ention, with many questions le open. In particular, the framework of transport dependency has yet to be ve ed in real world applications to make sure that its promising performance is not limited to arti cial se ings.…”

“…Indeed, the more chaotic the relation becomes, the further away one is from an isometric dependency, and the lower the value of the distance correlation will typically be (see Figure 2). is draws a sharp distinction to other (non-parametric) concepts of dependency, like the mutual information or several recently proposed coe cients of association (Cha erjee 2020; Wiesel 2021;Deb et al 2020), which assume maximal values for any measurable deterministic relation -not only for structured ones.…”

“…Independently from our work, another close relative of the transport dependency 𝜏 has recently been explored by Wiesel (2021), who proposes the association measure…”

“…• 𝜌 ∞ (𝛾) = 1 i there is a measurable function 𝜑 : 𝑋 → 𝑌 with 𝜁 = 𝜑 (𝜉), where the equality 𝜁 = 𝜑 (𝜉) only has to hold 𝜇-almost surely. In fact, the marginal transport correlation 𝜌 ∞ is equal to the measure of association introduced by Wiesel (2021), who also recognized the above properties as essential.…”

“…where (𝛾 𝑥 ) 𝑥 ∈𝑋 denotes the disintegration of 𝛾 with respect to the rst coordinate (i.e., 𝛾 𝑥 is the law of 𝜁 given 𝜉 = 𝑥), and 𝑐 𝑌 is (the power of) a metric on the space 𝑌 . For normalization to a coe cient in [0, 1], Wiesel (2021) also uses the diameter of 𝜈, which does not only upper bound 𝜏 (𝛾), but 𝜏 𝑌 (𝛾) as well. He further shows that this coe cient exhibits a list of desirable properties advanced by the work of Cha erjee (2020), and suggests a method by which it can be estimated consistently.…”