1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
|
#! /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
# 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
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.
Note
----
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 friendler equation:
\Gamma_d(a) = \pi^{d(d-1)/4}\prod_{i=1}^{d}{\Gamma(a - (i-1)/2)}.
Notes
-----
Reference:
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
|
