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#! /usr/bin/env 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
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#
# THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
# AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
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# ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
# DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR
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# 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__ = ['multigammln']


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
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