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Module « scipy.stats »

Fonction wasserstein_distance - module scipy.stats

Signature de la fonction wasserstein_distance

def wasserstein_distance(u_values, v_values, u_weights=None, v_weights=None) 

Description

help(scipy.stats.wasserstein_distance)

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]_.

.. versionadded:: 1.0.0

Parameters
----------
u_values : 1d 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_values : 1d array_like
    A sample from or the support of a second distribution.

u_weights, v_weights : 1d 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
-------
distance : float
    The computed distance between the distributions.

Notes
-----
Given two 1D probability mass functions, :math:`u` and :math:`v`, the first
Wasserstein distance between the distributions is:

.. math::

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

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

If :math:`U` and :math:`V` are the respective CDFs of :math:`u` and
:math:`v`, this distance also equals to:

.. math::

    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
----------
.. [1] "Wasserstein metric", https://en.wikipedia.org/wiki/Wasserstein_metric
.. [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`.

See Also
--------
wasserstein_distance_nd: Compute the Wasserstein-1 distance between two N-D
    discrete distributions.

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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