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Module « scipy.special »

Fonction multigammaln - module scipy.special

Signature de la fonction multigammaln

def multigammaln(a, d) 

Description

help(scipy.special.multigammaln)

Returns the log of multivariate gamma, also sometimes called the
generalized gamma.

Parameters
----------
a : ndarray
    The multivariate gamma is computed for each item of `a`.
d : int
    The dimension of the space of integration.

Returns
-------
res : ndarray
    The values of the log multivariate gamma at the given points `a`.

Notes
-----
The formal definition of the multivariate gamma of dimension d for a real
`a` is

.. math::

    \Gamma_d(a) = \int_{A>0} e^{-tr(A)} |A|^{a - (d+1)/2} dA

with the condition :math:`a > (d-1)/2`, and :math:`A > 0` being the set of
all the positive definite matrices of dimension `d`.  Note that `a` is a
scalar: the integrand only is multivariate, the argument is not (the
function is defined over a subset of the real set).

This can be proven to be equal to the much friendlier equation

.. math::

    \Gamma_d(a) = \pi^{d(d-1)/4} \prod_{i=1}^{d} \Gamma(a - (i-1)/2).

References
----------
R. J. Muirhead, Aspects of multivariate statistical theory (Wiley Series in
probability and mathematical statistics).

Examples
--------
>>> import numpy as np
>>> from scipy.special import multigammaln, gammaln
>>> a = 23.5
>>> d = 10
>>> multigammaln(a, d)
454.1488605074416

Verify that the result agrees with the logarithm of the equation
shown above:

>>> d*(d-1)/4*np.log(np.pi) + gammaln(a - 0.5*np.arange(0, d)).sum()
454.1488605074416

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