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Module « scipy.stats »

Fonction energy_distance - module scipy.stats

Signature de la fonction energy_distance

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

Description

help(scipy.stats.energy_distance)

Compute the energy distance between two 1D distributions.

.. 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 energy distance between two distributions :math:`u` and :math:`v`, whose
respective CDFs are :math:`U` and :math:`V`, equals to:

.. math::

    D(u, v) = \left( 2\mathbb E|X - Y| - \mathbb E|X - X'| -
    \mathbb E|Y - Y'| \right)^{1/2}

where :math:`X` and :math:`X'` (resp. :math:`Y` and :math:`Y'`) are
independent random variables whose probability distribution is :math:`u`
(resp. :math:`v`).

Sometimes the square of this quantity is referred to as the "energy
distance" (e.g. in [2]_, [4]_), but as noted in [1]_ and [3]_, only the
definition above satisfies the axioms of a distance function (metric).

As shown in [2]_, for one-dimensional real-valued variables, the energy
distance is linked to the non-distribution-free version of the Cramér-von
Mises distance:

.. math::

    D(u, v) = \sqrt{2} l_2(u, v) = \left( 2 \int_{-\infty}^{+\infty} (U-V)^2
    \right)^{1/2}

Note that the common Cramér-von Mises criterion uses the distribution-free
version of the distance. See [2]_ (section 2), for more details about both
versions of the distance.

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] Rizzo, Szekely "Energy distance." Wiley Interdisciplinary Reviews:
       Computational Statistics, 8(1):27-38 (2015).
.. [2] Szekely "E-statistics: The energy of statistical samples." Bowling
       Green State University, Department of Mathematics and Statistics,
       Technical Report 02-16 (2002).
.. [3] "Energy distance", https://en.wikipedia.org/wiki/Energy_distance
.. [4] Bellemare, Danihelka, Dabney, Mohamed, Lakshminarayanan, Hoyer,
       Munos "The Cramer Distance as a Solution to Biased Wasserstein
       Gradients" (2017). :arXiv:`1705.10743`.

Examples
--------
>>> from scipy.stats import energy_distance
>>> energy_distance([0], [2])
2.0000000000000004
>>> energy_distance([0, 8], [0, 8], [3, 1], [2, 2])
1.0000000000000002
>>> energy_distance([0.7, 7.4, 2.4, 6.8], [1.4, 8. ],
...                 [2.1, 4.2, 7.4, 8. ], [7.6, 8.8])
0.88003340976158217

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