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Module « scipy.stats »

Fonction dirichlet - module scipy.stats

Signature de la fonction dirichlet

def dirichlet(alpha, seed=None) 

Description

help(scipy.stats.dirichlet)

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
cov(alpha)
    The covariance of the Dirichlet distribution
entropy(alpha)
    Compute the differential entropy of the Dirichlet distribution.

Parameters
----------
alpha : array_like
    The concentration parameters. The number of entries determines the
    dimensionality 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`.

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 :math:`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
--------
>>> import numpy as np
>>> 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]])

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.

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