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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.""" 

 

from __future__ import division, print_function, absolute_import 

 

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