Module « scipy.stats »

Fonction invwishart - module scipy.stats

Signature de la fonction invwishart

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

Description

invwishart.__doc__

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.

    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" 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.

    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.

    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}`.

    .. 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.

    Examples
    --------
    >>> 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)

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