CpjSrSSKrSSKJs Jr SSKJr SSK J r S/r Sr Sr \ "SS S /\ \RS /S 9rS r\ "SS S /\\R S /S 9rg)zVSome more special functions which may be useful for multivariate statistical analysis.N)gammaln)_with_cache_optimization multigammalnc R[R"U5n[R"U5Sn[R"U5(a[R"U5U:wa [ S5e[R "USUS- -:*5(a[ SUSSUS- -S35eXS- -S-[R "[R5-nU[R"[[SUS-5Vs/sH o0US - S - - PM sn5S S 9- nU$s snf) ajReturns the log of multivariate gamma, also sometimes called the generalized gamma. Parameters ---------- a : ndarray The multivariate gamma is computed for each item of `a`. d : int The dimension of the space of integration. Returns ------- res : ndarray The values of the log multivariate gamma at the given points `a`. Notes ----- The formal definition of the multivariate gamma of dimension d for a real `a` is .. math:: \Gamma_d(a) = \int_{A>0} e^{-tr(A)} |A|^{a - (d+1)/2} dA with the condition :math:`a > (d-1)/2`, and :math:`A > 0` being the set of all the positive definite matrices of dimension `d`. Note that `a` is a scalar: the integrand only is multivariate, the argument is not (the function is defined over a subset of the real set). This can be proven to be equal to the much friendlier equation .. math:: \Gamma_d(a) = \pi^{d(d-1)/4} \prod_{i=1}^{d} \Gamma(a - (i-1)/2). References ---------- R. J. Muirhead, Aspects of multivariate statistical theory (Wiley Series in probability and mathematical statistics). Examples -------- >>> import numpy as np >>> from scipy.special import multigammaln, gammaln >>> a = 23.5 >>> d = 10 >>> multigammaln(a, d) 454.1488605074416 Verify that the result agrees with the logarithm of the equation shown above: >>> d*(d-1)/4*np.log(np.pi) + gammaln(a - 0.5*np.arange(0, d)).sum() 454.1488605074416 z*d should be a positive integer (dimension)g?z condition a (z) > 0.5 * (d-1) (z ) not metg?g?r)axis) npasarrayisscalarfloor ValueErroranylogpisumloggamrange)adresjs V/var/www/html/pdf-tiff/venv/lib/python3.13/site-packages/scipy/special/_spfun_stats.pyrr-sp 1 A 1 bA ;;q>>bhhqkQ.EFF vva3!a%= !!=+MBC! LLC JCsD$a:Returns pmf of Poisson Binomial distribution. Parameters ---------- k : array Number of successes at which to evaluate pmf. p : array Success probabilities of independent Bernoulli trials. Notes ----- This is equivalent to a gufunc with signature ``()(i)->()``. The last dimension of `p` contains success probabilities and the preceding dimensions are batch dimensions. The batch dimensions are broadcast against ``k``. The output will be C contiguous regardless of the contiguity of `k` and `p`. _poisson_binom_pmfkpr)name arg_names docstringufunccache_arg_indicesa:Returns cdf of Poisson Binomial distribution. Parameters ---------- k : array Number of successes at which to evaluate cdf. p : array Success probabilities of independent Bernoulli trials. Notes ----- This is equivalent to a gufunc with signature ``()(i)->()``. The last dimension of `p` contains success probabilities and the preceding dimensions are batch dimensions. The batch dimensions are broadcast against ``k``. The output will be C contiguous regardless of the contiguity of `k` and `p`. _poisson_binom_cdf)__doc__numpyr scipy.special._gufuncsspecial_gufuncs scipy.specialrrscipy.special._ufunc_toolsr__all__r_poisson_binom_pmf_docr_poisson_binom_cdf_docr#rrr/s@ ))+?  BL2. Cj$  % %c 0. Cj$  % %c r.