94 lines
3.4 KiB
Python
94 lines
3.4 KiB
Python
# Last Change: Sat Mar 21 02:00 PM 2009 J
|
|
|
|
# Copyright (c) 2001, 2002 Enthought, Inc.
|
|
#
|
|
# All rights reserved.
|
|
#
|
|
# Redistribution and use in source and binary forms, with or without
|
|
# modification, are permitted provided that the following conditions are met:
|
|
#
|
|
# a. Redistributions of source code must retain the above copyright notice,
|
|
# this list of conditions and the following disclaimer.
|
|
# b. Redistributions in binary form must reproduce the above copyright
|
|
# notice, this list of conditions and the following disclaimer in the
|
|
# documentation and/or other materials provided with the distribution.
|
|
# c. Neither the name of the Enthought nor the names of its contributors
|
|
# may be used to endorse or promote products derived from this software
|
|
# without specific prior written permission.
|
|
#
|
|
#
|
|
# THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
|
|
# AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
|
|
# IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
|
|
# ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE FOR
|
|
# ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
|
|
# DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR
|
|
# SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER
|
|
# CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
|
|
# LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
|
|
# OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH
|
|
# DAMAGE.
|
|
|
|
"""Some more special functions which may be useful for multivariate statistical
|
|
analysis."""
|
|
|
|
import numpy as np
|
|
from scipy.special import gammaln as loggam
|
|
|
|
|
|
__all__ = ['multigammaln']
|
|
|
|
|
|
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
|