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Fonction wishart - module scipy.stats

Signature de la fonction wishart

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

Description

wishart.__doc__

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
    ----------
    x : array_like
        Quantiles, with the last axis of `x` denoting the components.
    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
    random_state : {None, int, `numpy.random.Generator`,
                    `numpy.random.RandomState`}, optional
    
        If `seed` is None (or `np.random`), the `numpy.random.RandomState`
        singleton is used.
        If `seed` is an int, a new ``RandomState`` instance is used,
        seeded with `seed`.
        If `seed` is already a ``Generator`` or ``RandomState`` instance then
        that instance is used.

    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.

    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.

    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 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)

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