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Module « scipy.stats »

Fonction invwishart - module scipy.stats

Signature de la fonction invwishart

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

Description

help(scipy.stats.invwishart)

An inverse 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 covariance matrix.

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 an inverse Wishart distribution.
entropy(df, scale)
    Differential entropy of the 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
--------
wishart

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 inverse Wishart distribution is often denoted

.. math::

    W_p^{-1}(\nu, \Psi)

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

The probability density function for `invwishart` has support over positive
definite matrices :math:`S`; if :math:`S \sim W^{-1}_p(\nu, \Sigma)`,
then its PDF is given by:

.. math::

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

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

If the scale matrix is 1-dimensional and equal to one, then the inverse
Wishart distribution :math:`W_1(\nu, 1)` collapses to the
inverse Gamma distribution with parameters shape = :math:`\frac{\nu}{2}`
and scale = :math:`\frac{1}{2}`.

Instead of inverting a randomly generated Wishart matrix as described in [2],
here the algorithm in [4] is used to directly generate a random inverse-Wishart
matrix without inversion.

.. versionadded:: 0.16.0

References
----------
.. [1] M.L. Eaton, "Multivariate Statistics: A Vector Space Approach",
       Wiley, 1983.
.. [2] M.C. Jones, "Generating Inverse Wishart Matrices", Communications
       in Statistics - Simulation and Computation, vol. 14.2, pp.511-514,
       1985.
.. [3] Gupta, M. and Srivastava, S. "Parametric Bayesian Estimation of
       Differential Entropy and Relative Entropy". Entropy 12, 818 - 843.
       2010.
.. [4] S.D. Axen, "Efficiently generating inverse-Wishart matrices and
       their Cholesky factors", :arXiv:`2310.15884v1`. 2023.

Examples
--------
>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> from scipy.stats import invwishart, invgamma
>>> x = np.linspace(0.01, 1, 100)
>>> iw = invwishart.pdf(x, df=6, scale=1)
>>> iw[:3]
array([  1.20546865e-15,   5.42497807e-06,   4.45813929e-03])
>>> ig = invgamma.pdf(x, 6/2., scale=1./2)
>>> ig[:3]
array([  1.20546865e-15,   5.42497807e-06,   4.45813929e-03])
>>> plt.plot(x, iw)
>>> 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" inverse Wishart
random variable:

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

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