Coverage for /pythoncovmergedfiles/medio/medio/usr/local/lib/python3.8/site-packages/scipy/special/_spfun_stats.py: 38%

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31 

32"""Some more special functions which may be useful for multivariate statistical 

33analysis.""" 

34 

35import numpy as np 

36from scipy.special import gammaln as loggam 

37 

38 

39__all__ = ['multigammaln'] 

40 

41 

42def multigammaln(a, d): 

43 r"""Returns the log of multivariate gamma, also sometimes called the 

44 generalized gamma. 

45 

46 Parameters 

47 ---------- 

48 a : ndarray 

49 The multivariate gamma is computed for each item of `a`. 

50 d : int 

51 The dimension of the space of integration. 

52 

53 Returns 

54 ------- 

55 res : ndarray 

56 The values of the log multivariate gamma at the given points `a`. 

57 

58 Notes 

59 ----- 

60 The formal definition of the multivariate gamma of dimension d for a real 

61 `a` is 

62 

63 .. math:: 

64 

65 \Gamma_d(a) = \int_{A>0} e^{-tr(A)} |A|^{a - (d+1)/2} dA 

66 

67 with the condition :math:`a > (d-1)/2`, and :math:`A > 0` being the set of 

68 all the positive definite matrices of dimension `d`. Note that `a` is a 

69 scalar: the integrand only is multivariate, the argument is not (the 

70 function is defined over a subset of the real set). 

71 

72 This can be proven to be equal to the much friendlier equation 

73 

74 .. math:: 

75 

76 \Gamma_d(a) = \pi^{d(d-1)/4} \prod_{i=1}^{d} \Gamma(a - (i-1)/2). 

77 

78 References 

79 ---------- 

80 R. J. Muirhead, Aspects of multivariate statistical theory (Wiley Series in 

81 probability and mathematical statistics). 

82 

83 Examples 

84 -------- 

85 >>> import numpy as np 

86 >>> from scipy.special import multigammaln, gammaln 

87 >>> a = 23.5 

88 >>> d = 10 

89 >>> multigammaln(a, d) 

90 454.1488605074416 

91 

92 Verify that the result agrees with the logarithm of the equation 

93 shown above: 

94 

95 >>> d*(d-1)/4*np.log(np.pi) + gammaln(a - 0.5*np.arange(0, d)).sum() 

96 454.1488605074416 

97 """ 

98 a = np.asarray(a) 

99 if not np.isscalar(d) or (np.floor(d) != d): 

100 raise ValueError("d should be a positive integer (dimension)") 

101 if np.any(a <= 0.5 * (d - 1)): 

102 raise ValueError("condition a (%f) > 0.5 * (d-1) (%f) not met" 

103 % (a, 0.5 * (d-1))) 

104 

105 res = (d * (d-1) * 0.25) * np.log(np.pi) 

106 res += np.sum(loggam([(a - (j - 1.)/2) for j in range(1, d+1)]), axis=0) 

107 return res