def multigammaln(a, d):
"""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::
\Gamma_d(a) = \int_{A>0}{e^{-tr(A)\cdot{|A|}^{a - (m+1)/2}dA}}
with the condition ``a > (d-1)/2``, and ``A > 0`` being the set of all the
positive definite matrices of dimension s. 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::
\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).
"""
a = np.asarray(a)
if not np.isscalar(d) or (np.floor(d) != d):
raise ValueError("d should be a positive integer (dimension)")
if np.any(a <= 0.5 * (d - 1)):
raise ValueError("condition a (%f) > 0.5 * (d-1) (%f) not met" \
% (a, 0.5 * (d-1)))
res = (d * (d-1) * 0.25) * np.log(np.pi)
if a.size == 1:
axis = -1
else:
axis = 0
res += np.sum(loggam([(a - (j - 1.)/2) for j in range(1, d+1)]), axis)
return res
def multigammaln(a, d):
r"""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).
"""
a = np.asarray(a)
if not np.isscalar(d) or (np.floor(d) != d):
raise ValueError("d should be a positive integer (dimension)")
if np.any(a <= 0.5 * (d - 1)):
raise ValueError("condition a (%f) > 0.5 * (d-1) (%f) not met" %
(a, 0.5 * (d - 1)))
res = (d * (d - 1) * 0.25) * np.log(np.pi)
res += np.sum(loggam([(a - (j - 1.) / 2) for j in range(1, d + 1)]),
axis=0)
return res
def multigammaln(a, d):
"""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.
Note
----
The formal definition of the multivariate gamma of dimension d for a real a
is :
\Gamma_d(a) = \int_{A>0}{e^{-tr(A)\cdot{|A|}^{a - (m+1)/2}dA}}
with the condition a > (d-1)/2, and A>0 being the set of all the positive
definite matrices of dimension s. Note that a is a scalar: the integration
is multivariate, the argument is not.
This can be proven to be equal to the much friendler equation:
\Gamma_d(a) = \pi^{d(d-1)/4}\prod_{i=1}^{d}{\Gamma(a - (i-1)/2)}.
Reference:
----------
R. J. Muirhead, Aspects of multivariate statistical theory (Wiley Series in
probability and mathematical statistics). """
a = np.asarray(a)
if not np.isscalar(d) or (np.floor(d) != d):
raise ValueError("d should be a positive integer (dimension)")
if np.any(a <= 0.5 * (d - 1)):
raise ValueError("condition a (%f) > 0.5 * (d-1) (%f) not met" \
% (a, 0.5 * (d-1)))
res = (d * (d - 1) * 0.25) * np.log(np.pi)
if a.size == 1:
axis = -1
else:
axis = 0
res += np.sum(loggam([(a - (j - 1.) / 2) for j in range(1, d + 1)]), axis)
return res
