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) 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
def multigammaln(a, d): r"""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 .. 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). """ 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
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 integration is multivariate, the argument is not. 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)}. 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