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Fonction dirichlet - module scipy.stats

Signature de la fonction dirichlet

def dirichlet(alpha, seed=None) 

Description

dirichlet.__doc__

A Dirichlet random variable.

    The `alpha` keyword specifies the concentration parameters of the
    distribution.

    .. versionadded:: 0.15.0

    Methods
    -------
    ``pdf(x, alpha)``
        Probability density function.
    ``logpdf(x, alpha)``
        Log of the probability density function.
    ``rvs(alpha, size=1, random_state=None)``
        Draw random samples from a Dirichlet distribution.
    ``mean(alpha)``
        The mean of the Dirichlet distribution
    ``var(alpha)``
        The variance of the Dirichlet distribution
    ``entropy(alpha)``
        Compute the differential entropy of the Dirichlet distribution.

    Parameters
    ----------
    x : array_like
        Quantiles, with the last axis of `x` denoting the components.
    alpha : array_like
        The concentration parameters. The number of entries determines the
        dimensionality 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
    concentration parameters, returning a "frozen" Dirichlet
    random variable:

    rv = dirichlet(alpha)
        - Frozen object with the same methods but holding the given
          concentration parameters fixed.

    Notes
    -----
    Each :math:`\alpha` entry must be positive. The distribution has only
    support on the simplex defined by

    .. math::
        \sum_{i=1}^{K} x_i = 1

    where 0 < x_i < 1.

    If the quantiles don't lie within the simplex, a ValueError is raised.

    The probability density function for `dirichlet` is

    .. math::

        f(x) = \frac{1}{\mathrm{B}(\boldsymbol\alpha)} \prod_{i=1}^K x_i^{\alpha_i - 1}

    where

    .. math::

        \mathrm{B}(\boldsymbol\alpha) = \frac{\prod_{i=1}^K \Gamma(\alpha_i)}
                                     {\Gamma\bigl(\sum_{i=1}^K \alpha_i\bigr)}

    and :math:`\boldsymbol\alpha=(\alpha_1,\ldots,\alpha_K)`, the
    concentration parameters and :math:`K` is the dimension of the space
    where :math:`x` takes values.

    Note that the dirichlet interface is somewhat inconsistent.
    The array returned by the rvs function is transposed
    with respect to the format expected by the pdf and logpdf.

    Examples
    --------
    >>> from scipy.stats import dirichlet

    Generate a dirichlet random variable

    >>> quantiles = np.array([0.2, 0.2, 0.6])  # specify quantiles
    >>> alpha = np.array([0.4, 5, 15])  # specify concentration parameters
    >>> dirichlet.pdf(quantiles, alpha)
    0.2843831684937255

    The same PDF but following a log scale

    >>> dirichlet.logpdf(quantiles, alpha)
    -1.2574327653159187

    Once we specify the dirichlet distribution
    we can then calculate quantities of interest

    >>> dirichlet.mean(alpha)  # get the mean of the distribution
    array([0.01960784, 0.24509804, 0.73529412])
    >>> dirichlet.var(alpha) # get variance
    array([0.00089829, 0.00864603, 0.00909517])
    >>> dirichlet.entropy(alpha)  # calculate the differential entropy
    -4.3280162474082715

    We can also return random samples from the distribution

    >>> dirichlet.rvs(alpha, size=1, random_state=1)
    array([[0.00766178, 0.24670518, 0.74563305]])
    >>> dirichlet.rvs(alpha, size=2, random_state=2)
    array([[0.01639427, 0.1292273 , 0.85437844],
           [0.00156917, 0.19033695, 0.80809388]])