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Module « scipy.stats »

Fonction wishart - module scipy.stats

Signature de la fonction wishart

def wishart(df=None, scale=None, seed=None) 

Description

help(scipy.stats.wishart)

A Wishart random variable.

The `df` keyword specifies the degrees of freedom. The `scale` keyword
specifies the scale matrix, which must be symmetric and positive definite.
In this context, the scale matrix is often interpreted in terms of a
multivariate normal precision matrix (the inverse of the covariance
matrix). These arguments must satisfy the relationship
``df > scale.ndim - 1``, but see notes on using the `rvs` method with
``df < scale.ndim``.

Methods
-------
pdf(x, df, scale)
    Probability density function.
logpdf(x, df, scale)
    Log of the probability density function.
rvs(df, scale, size=1, random_state=None)
    Draw random samples from a Wishart distribution.
entropy()
    Compute the differential entropy of the Wishart distribution.

Parameters
----------
df : int
    Degrees of freedom, must be greater than or equal to dimension of the
    scale matrix
scale : array_like
    Symmetric positive definite scale matrix of the distribution
seed : {None, int, np.random.RandomState, np.random.Generator}, optional
    Used for drawing random variates.
    If `seed` is `None`, the `~np.random.RandomState` singleton is used.
    If `seed` is an int, a new ``RandomState`` instance is used, seeded
    with seed.
    If `seed` is already a ``RandomState`` or ``Generator`` instance,
    then that object is used.
    Default is `None`.

Raises
------
scipy.linalg.LinAlgError
    If the scale matrix `scale` is not positive definite.

See Also
--------
invwishart, chi2

Notes
-----


The scale matrix `scale` must be a symmetric positive definite
matrix. Singular matrices, including the symmetric positive semi-definite
case, are not supported. Symmetry is not checked; only the lower triangular
portion is used.

The Wishart distribution is often denoted

.. math::

    W_p(\nu, \Sigma)

where :math:`\nu` is the degrees of freedom and :math:`\Sigma` is the
:math:`p \times p` scale matrix.

The probability density function for `wishart` has support over positive
definite matrices :math:`S`; if :math:`S \sim W_p(\nu, \Sigma)`, then
its PDF is given by:

.. math::

    f(S) = \frac{|S|^{\frac{\nu - p - 1}{2}}}{2^{ \frac{\nu p}{2} }
           |\Sigma|^\frac{\nu}{2} \Gamma_p \left ( \frac{\nu}{2} \right )}
           \exp\left( -tr(\Sigma^{-1} S) / 2 \right)

If :math:`S \sim W_p(\nu, \Sigma)` (Wishart) then
:math:`S^{-1} \sim W_p^{-1}(\nu, \Sigma^{-1})` (inverse Wishart).

If the scale matrix is 1-dimensional and equal to one, then the Wishart
distribution :math:`W_1(\nu, 1)` collapses to the :math:`\chi^2(\nu)`
distribution.

The algorithm [2]_ implemented by the `rvs` method may
produce numerically singular matrices with :math:`p - 1 < \nu < p`; the
user may wish to check for this condition and generate replacement samples
as necessary.


.. versionadded:: 0.16.0

References
----------
.. [1] M.L. Eaton, "Multivariate Statistics: A Vector Space Approach",
       Wiley, 1983.
.. [2] W.B. Smith and R.R. Hocking, "Algorithm AS 53: Wishart Variate
       Generator", Applied Statistics, vol. 21, pp. 341-345, 1972.

Examples
--------
>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> from scipy.stats import wishart, chi2
>>> x = np.linspace(1e-5, 8, 100)
>>> w = wishart.pdf(x, df=3, scale=1); w[:5]
array([ 0.00126156,  0.10892176,  0.14793434,  0.17400548,  0.1929669 ])
>>> c = chi2.pdf(x, 3); c[:5]
array([ 0.00126156,  0.10892176,  0.14793434,  0.17400548,  0.1929669 ])
>>> plt.plot(x, w)
>>> plt.show()

The input quantiles can be any shape of array, as long as the last
axis labels the components.

Alternatively, the object may be called (as a function) to fix the degrees
of freedom and scale parameters, returning a "frozen" Wishart random
variable:

>>> rv = wishart(df=1, scale=1)
>>> # Frozen object with the same methods but holding the given
>>> # degrees of freedom and scale fixed.

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