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scipy.stats.wasserstein_distance#

scipy.stats.wasserstein_distance(u_values, v_values, u_weights=None, v_weights=None)[source]#

Compute the Wasserstein-1 distance between two 1D discrete distributions.

The Wasserstein distance, also called the Earth mover’s distance or the optimal transport distance, is a similarity metric between two probability distributions [1]. In the discrete case, the Wasserstein distance can be understood as the cost of an optimal transport plan to convert one distribution into the other. The cost is calculated as the product of the amount of probability mass being moved and the distance it is being moved. A brief and intuitive introduction can be found at [2].

Added in version 1.0.0.

Parameters:
u_values1d array_like

A sample from a probability distribution or the support (set of all possible values) of a probability distribution. Each element is an observation or possible value.

v_values1d array_like

A sample from or the support of a second distribution.

u_weights, v_weights1d array_like, optional

Weights or counts corresponding with the sample or probability masses corresponding with the support values. Sum of elements must be positive and finite. If unspecified, each value is assigned the same weight.

Returns:
distancefloat

The computed distance between the distributions.

See also

wasserstein_distance_nd

Compute the Wasserstein-1 distance between two N-D discrete distributions.

Notes

Given two 1D probability mass functions, \(u\) and \(v\), the first Wasserstein distance between the distributions is:

\[l_1 (u, v) = \inf_{\pi \in \Gamma (u, v)} \int_{\mathbb{R} \times \mathbb{R}} |x-y| \mathrm{d} \pi (x, y)\]

where \(\Gamma (u, v)\) is the set of (probability) distributions on \(\mathbb{R} \times \mathbb{R}\) whose marginals are \(u\) and \(v\) on the first and second factors respectively. For a given value \(x\), \(u(x)\) gives the probabilty of \(u\) at position \(x\), and the same for \(v(x)\).

If \(U\) and \(V\) are the respective CDFs of \(u\) and \(v\), this distance also equals to:

\[l_1(u, v) = \int_{-\infty}^{+\infty} |U-V|\]

See [3] for a proof of the equivalence of both definitions.

The input distributions can be empirical, therefore coming from samples whose values are effectively inputs of the function, or they can be seen as generalized functions, in which case they are weighted sums of Dirac delta functions located at the specified values.

References

[2]

Lili Weng, “What is Wasserstein distance?”, Lil’log, https://lilianweng.github.io/posts/2017-08-20-gan/#what-is-wasserstein-distance.

[3]

Ramdas, Garcia, Cuturi “On Wasserstein Two Sample Testing and Related Families of Nonparametric Tests” (2015). arXiv:1509.02237.

Examples

>>> from scipy.stats import wasserstein_distance
>>> wasserstein_distance([0, 1, 3], [5, 6, 8])
5.0
>>> wasserstein_distance([0, 1], [0, 1], [3, 1], [2, 2])
0.25
>>> wasserstein_distance([3.4, 3.9, 7.5, 7.8], [4.5, 1.4],
...                      [1.4, 0.9, 3.1, 7.2], [3.2, 3.5])
4.0781331438047861
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