ftj*dZddlZddlmZdgZdZdS)zVSome more special functions which may be useful for multivariate statistical analysis.N)gammaln multigammalnc (tjtj|d}tj|rtj||krt dtjd|dz zkrt ddd|dz zd||dz zdztjtjz}|tjtfd td|dzDd z }|S) aReturns 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). Examples -------- >>> import numpy as np >>> from scipy.special import multigammaln, gammaln >>> a = 23.5 >>> d = 10 >>> multigammaln(a, d) 454.1488605074416 Verify that the result agrees with the logarithm of the equation shown above: >>> d*(d-1)/4*np.log(np.pi) + gammaln(a - 0.5*np.arange(0, d)).sum() 454.1488605074416 z*d should be a positive integer (dimension)g?z condition a (z) > 0.5 * (d-1) (z ) not metg?c&g|] }|dz dz z S)g?r).0jas X/var/www/html/CA-Chatbot/venv/lib/python3.11/site-packages/scipy/special/_spfun_stats.py z multigammaln..ks%BBBq1Bz>BBBr)axis) npasarrayisscalarfloor ValueErroranylogpisumloggamrange)r dress` r rr*s p 1 A 1 bA ;q>>GbhqkkQ..EFFF va3!a%= !!USSSSAaC[SSSTTT !9t rvbe}} ,C26&BBBBE!QqSMMBBBCC! L L LLC Jr)__doc__numpyr scipy.specialrr__all__rrrr r"sX@  ++++++  BBBBBr