Package jline.api.moment

Class Moment_joint_cumulant_from_raw

java.lang.Object
jline.api.moment.Moment_joint_cumulant_from_raw
public final class Moment_joint_cumulant_from_raw extends Object
Joint cumulants from joint power (raw) moments.

The joint cumulants of a random vector (N_1,...,N_d) are the coefficients of the joint cumulant generating function 

   log E[exp(s_1 N_1 + ... + s_d N_d)] = sum_(a != 0) kappa_a prod_j
   s_j^(a_j) / a_j!

They obey the multivariate exponential formula, equivalently the Leonov-Shiryaev partition formula. With j the first dimension in which the multi-index a is nonzero, 

   m_a = sum_(0<b<=a) prod_l nchoosek(a_l-[l=j], b_l-[l=j]) kappa_b m_(a-b)

which isolates kappa_a because the b = a term has unit coefficient and m_0 = 1. Unlike every other conversion in the house, this one does not factor into a product of univariate transforms: the cumulant of multi-order (1,1) is the covariance, which mixes the dimensions. Multi-indices are swept in row-major (lexicographic) order, under which every b <= a precedes a.

Reference: V. P. Leonov and A. N. Shiryaev. On a method of calculation of semi-invariants. Theory of Probability and its Applications, 4(3):319-329, 1959.

Since:
LINE 3.0

  • Method Details

    • moment_joint_cumulant_from_raw

      public static double[] moment_joint_cumulant_from_raw(double[] m, int[] dims)
      Converts joint power moments into joint cumulants.
      Parameters:
      m - flattened joint power moments in row-major order, element 0 being 1
      dims - extents of the array, dims[j] = n_j+1
      Returns:
      flattened joint cumulants, same layout, element 0 being 0