Module « scipy.stats »

Fonction multivariate_t - module scipy.stats

Signature de la fonction multivariate_t

def multivariate_t(loc=None, shape=1, df=1, allow_singular=False, seed=None) 

Description

multivariate_t.__doc__

A multivariate t-distributed random variable.

    The `loc` parameter specifies the location. The `shape` parameter specifies
    the positive semidefinite shape matrix. The `df` parameter specifies the
    degrees of freedom.

    In addition to calling the methods below, the object itself may be called
    as a function to fix the location, shape matrix, and degrees of freedom
    parameters, returning a "frozen" multivariate t-distribution random.

    Methods
    -------
    ``pdf(x, loc=None, shape=1, df=1, allow_singular=False)``
        Probability density function.
    ``logpdf(x, loc=None, shape=1, df=1, allow_singular=False)``
        Log of the probability density function.
    ``rvs(loc=None, shape=1, df=1, size=1, random_state=None)``
        Draw random samples from a multivariate t-distribution.

    Parameters
    ----------
    x : array_like
        Quantiles, with the last axis of `x` denoting the components.
    
    loc : array_like, optional
        Location of the distribution. (default ``0``)
    shape : array_like, optional
        Positive semidefinite matrix of the distribution. (default ``1``)
    df : float, optional
        Degrees of freedom of the distribution; must be greater than zero.
        If ``np.inf`` then results are multivariate normal. The default is ``1``.
    allow_singular : bool, optional
        Whether to allow a singular matrix. (default ``False``)
    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.

    Notes
    -----
    Setting the parameter `loc` to ``None`` is equivalent to having `loc`
    be the zero-vector. The parameter `shape` can be a scalar, in which case
    the shape matrix is the identity times that value, a vector of
    diagonal entries for the shape matrix, or a two-dimensional array_like.
    The matrix `shape` must be a (symmetric) positive semidefinite matrix. The
    determinant and inverse of `shape` are computed as the pseudo-determinant
    and pseudo-inverse, respectively, so that `shape` does not need to have
    full rank.

    The probability density function for `multivariate_t` is

    .. math::

        f(x) = \frac{\Gamma(\nu + p)/2}{\Gamma(\nu/2)\nu^{p/2}\pi^{p/2}|\Sigma|^{1/2}}
               \exp\left[1 + \frac{1}{\nu} (\mathbf{x} - \boldsymbol{\mu})^{\top}
               \boldsymbol{\Sigma}^{-1}
               (\mathbf{x} - \boldsymbol{\mu}) \right]^{-(\nu + p)/2},

    where :math:`p` is the dimension of :math:`\mathbf{x}`,
    :math:`\boldsymbol{\mu}` is the :math:`p`-dimensional location,
    :math:`\boldsymbol{\Sigma}` the :math:`p \times p`-dimensional shape
    matrix, and :math:`\nu` is the degrees of freedom.

    .. versionadded:: 1.6.0

    Examples
    --------
    >>> import matplotlib.pyplot as plt
    >>> from scipy.stats import multivariate_t
    >>> x, y = np.mgrid[-1:3:.01, -2:1.5:.01]
    >>> pos = np.dstack((x, y))
    >>> rv = multivariate_t([1.0, -0.5], [[2.1, 0.3], [0.3, 1.5]], df=2)
    >>> fig, ax = plt.subplots(1, 1)
    >>> ax.set_aspect('equal')
    >>> plt.contourf(x, y, rv.pdf(pos))