107 lines
3.7 KiB
Python
107 lines
3.7 KiB
Python
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# Last Change: Sat Mar 21 02:00 PM 2009 J
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# Copyright (c) 2001, 2002 Enthought, Inc.
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#
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# All rights reserved.
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#
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# Redistribution and use in source and binary forms, with or without
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# modification, are permitted provided that the following conditions are met:
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#
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# a. Redistributions of source code must retain the above copyright notice,
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# this list of conditions and the following disclaimer.
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# b. Redistributions in binary form must reproduce the above copyright
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# notice, this list of conditions and the following disclaimer in the
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# documentation and/or other materials provided with the distribution.
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# c. Neither the name of the Enthought nor the names of its contributors
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# may be used to endorse or promote products derived from this software
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# without specific prior written permission.
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#
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#
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# THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
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# AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
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# IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
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# ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE FOR
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# ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
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# DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR
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# SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER
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# CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
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# LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
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# OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH
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# DAMAGE.
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"""Some more special functions which may be useful for multivariate statistical
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analysis."""
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import numpy as np
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from scipy.special import gammaln as loggam
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__all__ = ['multigammaln']
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def multigammaln(a, d):
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r"""Returns the log of multivariate gamma, also sometimes called the
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generalized gamma.
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Parameters
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----------
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a : ndarray
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The multivariate gamma is computed for each item of `a`.
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d : int
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The dimension of the space of integration.
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Returns
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-------
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res : ndarray
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The values of the log multivariate gamma at the given points `a`.
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Notes
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-----
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The formal definition of the multivariate gamma of dimension d for a real
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`a` is
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.. math::
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\Gamma_d(a) = \int_{A>0} e^{-tr(A)} |A|^{a - (d+1)/2} dA
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with the condition :math:`a > (d-1)/2`, and :math:`A > 0` being the set of
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all the positive definite matrices of dimension `d`. Note that `a` is a
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scalar: the integrand only is multivariate, the argument is not (the
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function is defined over a subset of the real set).
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This can be proven to be equal to the much friendlier equation
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.. math::
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\Gamma_d(a) = \pi^{d(d-1)/4} \prod_{i=1}^{d} \Gamma(a - (i-1)/2).
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References
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----------
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R. J. Muirhead, Aspects of multivariate statistical theory (Wiley Series in
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probability and mathematical statistics).
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Examples
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--------
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>>> import numpy as np
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>>> from scipy.special import multigammaln, gammaln
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>>> a = 23.5
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>>> d = 10
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>>> multigammaln(a, d)
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454.1488605074416
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Verify that the result agrees with the logarithm of the equation
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shown above:
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>>> d*(d-1)/4*np.log(np.pi) + gammaln(a - 0.5*np.arange(0, d)).sum()
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454.1488605074416
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"""
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a = np.asarray(a)
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if not np.isscalar(d) or (np.floor(d) != d):
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raise ValueError("d should be a positive integer (dimension)")
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if np.any(a <= 0.5 * (d - 1)):
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raise ValueError(f"condition a ({a:f}) > 0.5 * (d-1) ({0.5 * (d-1):f}) not met")
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res = (d * (d-1) * 0.25) * np.log(np.pi)
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res += np.sum(loggam([(a - (j - 1.)/2) for j in range(1, d+1)]), axis=0)
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return res
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