Stochastic Bifurcation Processes and Distributions of Fractions

Abstract

The stochastic bifurcation process, which partitions a unit into its positive fractions z ₁, …, Z _I ; such that z ₁, + … + z^I , = 1, is characterized by a topology and probability law governing the (I – 1) bifurcations. Aside from describing many natural phenomena, such as flows in dendritic networks, the stochastic bifurcation process also offers a device for generating families of multivariate distributions on the (I – 1)-dimensional simplex. In particular, the process with independent bifurcations governed by beta laws gives rise to the generalized Dirichlet family. Every member of this family is identified by the three-tuple: (1) the bifurcation topology, (2) the permutation of fractions, and (3) the parameters of the beta laws. Successively smaller subsets of bifurcation topologies, containing all possible topologies, only double-cascaded topologies, and only cascaded topologies, lead to progressively narrower distribution families of Types A, B, and C. Type C is the Connor-Mosimann distribution, which generalizes the Standard Dirichlet distribution, whose place in the hierarchy of our models makes it Type D. The Type A distribution family constitutes the ultimate generalization of the Standard Dirichlet. Because each bifurcation topology leads to a different sign structure of the correlation matrix for fractions, Type A distribution allows one to adapt its form to an empirical correlation structure by optimizing the bifurcation topology and the permutation of fractions.

KEY WORDS:

• Bifurcation process
• Dendritic network
• Dirichlet distribution
• Fraction
• Partition