#! /usr/bin/env python
# Last Change: Sat Mar 21 02:00 PM 2009 J
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"""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']
[docs]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)
res += np.sum(loggam([(a - (j - 1.)/2) for j in range(1, d+1)]), axis=0)
return res