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

wasserstein_distance.__doc__

    Compute the first Wasserstein distance between two 1D distributions.

    This distance is also known as the earth mover's distance, since it can be
    seen as the minimum amount of "work" required to transform :math:`u` into
    :math:`v`, where "work" is measured as the amount of distribution weight
    that must be moved, multiplied by the distance it has to be moved.

    .. versionadded:: 1.0.0

    Parameters
    ----------
    u_values, v_values : array_like
        Values observed in the (empirical) distribution.
    u_weights, v_weights : array_like, optional
        Weight for each value. If unspecified, each value is assigned the same
        weight.
        `u_weights` (resp. `v_weights`) must have the same length as
        `u_values` (resp. `v_values`). If the weight sum differs from 1, it
        must still be positive and finite so that the weights can be normalized
        to sum to 1.

    Returns
    -------
    distance : float
        The computed distance between the distributions.

    Notes
    -----
    The first Wasserstein distance between the distributions :math:`u` and
    :math:`v` 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.

    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 [2]_ 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] 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