/
_spfun_stats.py
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/
_spfun_stats.py
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# 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).
Examples
--------
>>> 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
"""
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