wasserstein1d: Compute the Wasserstein Distance Between Two Univariate...

In transport: Computation of Optimal Transport Plans and Wasserstein Distances

Compute the Wasserstein Distance Between Two Univariate Samples

Description

Given two vectors a and b, compute the Wasserstein distance of order p between their empirical distributions.

Usage

wasserstein1d(a, b, p = 1, wa = NULL, wb = NULL)

Arguments

a, b	two vectors.
p	a positive number. The order of the Wasserstein distance.
wa, wb	optional vectors of non-negative weights for a and b.

Details

The Wasserstein distance of order p is defined as the p-th root of the total cost incurred when transporting a pile of mass into another pile of mass in an optimal way, where the cost of transporting a unit of mass from x to y is given as the p-th power \|x-y\|^p of the Euclidean distance.

In the present function the vector a represents the locations on the real line of m deposits of mass 1/m and the vector b the locations of n deposits of mass 1/n. If the user specifies weights wa and wb, these default masses are replaced by wa/sum(wa) and wb/sum(wb), respectively.

In terms of the empirical distribution function F(t) = \sum_{i=1}^m w^{(a)}_i 1\{a_i \leq t\} of locations a_i with normalized weights w^{(a)}_i, and the corresponding function G(t) = \sum_{j=1}^n w^{(b)}_j 1\{b_j \leq t\} for b, the Wasserstein distance in 1-d is given as

W_p(F,G) = \left(\int_0^1 |F^{-1}(u)-G^{-1}(u)|^p \; du \right)^{1/p},

where F^{-1} and G^{-1} are generalized inverses. If p=1, we also have

W_1(F,G) = \int_{-\infty}^{\infty} |F(x)-G(x)| \; dx.

Value

A single number, the Wasserstein distance for the specified data.

Author(s)

Dominic Schuhmacher [e-mail]

See Also

wasserstein

Examples

  x <- rnorm(200)
  y <- rnorm(150,2)
  wasserstein1d(x,y)

transport documentation built on Sept. 17, 2024, 1:08 a.m.