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Module « scipy.stats »

Fonction wasserstein_distance_nd - module scipy.stats

Signature de la fonction wasserstein_distance_nd

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

Description

help(scipy.stats.wasserstein_distance_nd)

Compute the Wasserstein-1 distance between two N-D 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:: 1.13.0

Parameters
----------
u_values : 2d array_like
    A sample from a probability distribution or the support (set of all
    possible values) of a probability distribution. Each element along
    axis 0 is an observation or possible value, and axis 1 represents the
    dimensionality of the distribution; i.e., each row is a vector
    observation or possible value.

v_values : 2d 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 probability mass functions, :math:`u`
and :math:`v`, the first Wasserstein distance between the distributions
using the Euclidean norm is:

.. math::

    l_1 (u, v) = \inf_{\pi \in \Gamma (u, v)} \int \| x-y \|_2 \mathrm{d} \pi (x, y)

where :math:`\Gamma (u, v)` is the set of (probability) distributions on
:math:`\mathbb{R}^n \times \mathbb{R}^n` 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)`.

This is also called the optimal transport problem or the Monge problem.
Let the finite point sets :math:`\{x_i\}` and :math:`\{y_j\}` denote
the support set of probability mass function :math:`u` and :math:`v`
respectively. The Monge problem can be expressed as follows,

Let :math:`\Gamma` denote the transport plan, :math:`D` denote the
distance matrix and,

.. math::

    x = \text{vec}(\Gamma)          \\
    c = \text{vec}(D)               \\
    b = \begin{bmatrix}
            u\\
            v\\
        \end{bmatrix}

The :math:`\text{vec}()` function denotes the Vectorization function
that transforms a matrix into a column vector by vertically stacking
the columns of the matrix.
The transport plan :math:`\Gamma` is a matrix :math:`[\gamma_{ij}]` in
which :math:`\gamma_{ij}` is a positive value representing the amount of
probability mass transported from :math:`u(x_i)` to :math:`v(y_i)`.
Summing over the rows of :math:`\Gamma` should give the source distribution
:math:`u` : :math:`\sum_j \gamma_{ij} = u(x_i)` holds for all :math:`i`
and summing over the columns of :math:`\Gamma` should give the target
distribution :math:`v`: :math:`\sum_i \gamma_{ij} = v(y_j)` holds for all
:math:`j`.
The distance matrix :math:`D` is a matrix :math:`[d_{ij}]`, in which
:math:`d_{ij} = d(x_i, y_j)`.

Given :math:`\Gamma`, :math:`D`, :math:`b`, the Monge problem can be
transformed into a linear programming problem by
taking :math:`A x = b` as constraints and :math:`z = c^T x` as minimization
target (sum of costs) , where matrix :math:`A` has the form

.. math::

    \begin{array} {rrrr|rrrr|r|rrrr}
        1 & 1 & \dots & 1 & 0 & 0 & \dots & 0 & \dots & 0 & 0 & \dots &
            0 \cr
        0 & 0 & \dots & 0 & 1 & 1 & \dots & 1 & \dots & 0 & 0 &\dots &
            0 \cr
        \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \ddots
            & \vdots & \vdots & \vdots & \vdots & \ddots & \vdots  \cr
        0 & 0 & \dots & 0 & 0 & 0 & \dots & 0 & \dots & 1 & 1 & \dots &
            1 \cr \hline

        1 & 0 & \dots & 0 & 1 & 0 & \dots & \dots & \dots & 1 & 0 & \dots &
            0 \cr
        0 & 1 & \dots & 0 & 0 & 1 & \dots & \dots & \dots & 0 & 1 & \dots &
            0 \cr
        \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \ddots &
            \vdots & \vdots & \vdots & \vdots & \ddots & \vdots \cr
        0 & 0 & \dots & 1 & 0 & 0 & \dots & 1 & \dots & 0 & 0 & \dots & 1
    \end{array}

By solving the dual form of the above linear programming problem (with
solution :math:`y^*`), the Wasserstein distance :math:`l_1 (u, v)` can
be computed as :math:`b^T y^*`.

The above solution is inspired by Vincent Herrmann's blog [3]_ . For a
more thorough explanation, see [4]_ .

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] Hermann, Vincent. "Wasserstein GAN and the Kantorovich-Rubinstein
       Duality". https://vincentherrmann.github.io/blog/wasserstein/.
.. [4] Peyré, Gabriel, and Marco Cuturi. "Computational optimal
       transport." Center for Research in Economics and Statistics
       Working Papers 2017-86 (2017).

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

Examples
--------
Compute the Wasserstein distance between two three-dimensional samples,
each with two observations.

>>> from scipy.stats import wasserstein_distance_nd
>>> wasserstein_distance_nd([[0, 2, 3], [1, 2, 5]], [[3, 2, 3], [4, 2, 5]])
3.0

Compute the Wasserstein distance between two two-dimensional distributions
with three and two weighted observations, respectively.

>>> wasserstein_distance_nd([[0, 2.75], [2, 209.3], [0, 0]],
...                      [[0.2, 0.322], [4.5, 25.1808]],
...                      [0.4, 5.2, 0.114], [0.8, 1.5])
174.15840245217169

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