2-Wasserstein distance calculation

Background

The 2-Wasserstein distance W is a metric to describe the distance between two distributions, representing e.g. two different conditions A and B.

For continuous distributions, it is given by

$$W := W(F_A, F_B) = \bigg( \int_0^1 \big|F_A^{-1}(u) - F_B^{-1}(u) \big|^2 du \bigg)^\frac{1}{2},$$

where FA and FB are the corresponding cumulative distribution functions (CDFs) and FA−1 and FB−1 the respective quantile functions.

We specifically consider the squared 2-Wasserstein distance d := W2 which offers the following decomposition into location, size, and shape terms: $$d := d(F_A, F_B) = \int_0^1 \big|F^{-1}(u) - F^{-1}(u) \big|^2 du = \underbrace{\big(\mu_A - \mu_B\big)^2}_{\text{location}} + \underbrace{\big(\sigma_A - \sigma_B\big)^2}_{\text{size}} + \underbrace{2\sigma_A \sigma_B \big(1 - \rho^{A,B}\big)}_{\text{shape}},$$

where μA and μB are the respective means, σA and σB are the respective standard deviations, and ρA, B is the Pearson correlation of the points in the quantile-quantile plot of FA and FB.

Usage in two-sample setting

In case the distributions FA and FB are not explicitly given and information is only availbale in the form of samples from FA and FB, respectively, we use the corresponding empirical CDFs A and B. Then, the 2-Wasserstein distance is computed by

$$d(\hat{F}_A, \hat{F}_B) \approx \frac{1}{K} \sum_{k=1}^K \big(Q_A^{\alpha_k} - Q_B^{\alpha_k} \big) \approx \big(\hat{\mu}_A - \hat{\mu}_B\big)^2 + \big(\hat{\sigma}_A - \hat{\sigma}_B\big)^2 + 2\hat{\sigma}_A \hat{\sigma}_B \big(1 - \hat{\rho}^{A,B}\big).$$

Here, QA and QB denote equidistant quantiles of FA and FB, respectively, at the levels $\alpha_k := \frac{k-0.5}{K}, k = 1, \dots , K$, where we use K := 1000 in our implementation. Moreover, μ̂A, μ̂B, σ̂A, σ̂B and ρ̂A, B denote the empirical versions of the corresponding quantities.

Three implementations

The waddR package offers three functions to compute the 2-Wasserstein distance in two-sample settings.

We will use samples from normal distributions to illustrate them.

library(waddR)

set.seed(24)
x <- rnorm(100,mean=0,sd=1)
y <- rnorm(100,mean=2,sd=1)

The first function, wasserstein_metric, offers a faster reimplementation in C++ of the wasserstein1d function from the R package transport, which is able to compute general p-Wasserstein distances. For p = 2, we obtain the 2-Wasserstein distance W.

wasserstein_metric(x,y,p=2)
#> [1] 2.044457

The corresponding value of the squared 2-Wasserstein distance d is then computed as:

wasserstein_metric(x,y,p=2)^2
#> [1] 4.179803

The second function, squared_wass_approx, computes the squared 2-Wasserstein distance by calculating the mean squared difference of the equidistant quantiles (first approximation in the previous formula). This function is currently used to compute the 2-Wasserstein distances in the testing procedures.

squared_wass_approx(x,y)
#> [1] 4.179803

The third function, squared_wass_decomp, approximates the squared 2-Wasserstein distance using the location, size, and shape terms from the above decomposition (second apporximation in the previous formula). It also returns the respective decomposition values.

squared_wass_decomp(x,y)
#> $distance
#> [1] 4.180458
#> 
#> $location
#> [1] 4.114983
#> 
#> $size
#> [1] 0.002307
#> 
#> $shape
#> [1] 0.06316766

In the considered example, the decomposition results suggest that the two distributions differ with respect to location (mean), but not in terms of size and shape, thus confirming the underlying normal model.

See also

Session Info

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