beta_statistic: Aldous' beta statistic.

Description

The Beta statistic fits a beta splitting model to each node, assuming that the number of extant descendents of each daughter branch is split following a beta distribution, such that the number of extant descendentants x and y at a node follows $q (x, y) = s_{n} (b e t a)^{-} 1 \frac{(g a m m a (x + 1 + b e t a) g a m m a (y + 1 + b e t a))}{g a m m a (x + 1) g a m m a (y + 1)}$ , where $s_{n} (b e t a)^{-} 1$ is a normalizing constant. When this model is fit to a tree, different values of beta correspond to the expectation following from different diversification models, such that a beta of 0 corresponds to a Yule tree, a beta of -3/2 to a tree following from a PDA model. In general, negative beta values correspond to trees more unbalanced than Yule trees, and beta values larger than zero indicate trees more balanced than Yule trees. The lower bound of the beta splitting parameter is -2.

Usage

beta_statistic(
  phy,
  upper_lim = 10,
  algorithm = "COBYLA",
  abs_tol = 1e-04,
  rel_tol = 1e-06
)

Value

Beta value

Arguments

phy: phylogeny or ltable
upper_lim: Upper limit for beta parameter, default = 10.
algorithm: optimization algorithm used, default is "COBYLA" (Constrained Optimization BY Linear Approximations), also available are "subplex" and "simplex". Subplex and Simplex seem to have difficulties with unbalanced trees, e.g. if beta < 0.
abs_tol: absolute stopping criterion of optimization. Default is 1e-4.
rel_tol: relative stopping criterion of optimization. Default is 1e-6.

References

Aldous, David. "Probability distributions on cladograms." Random discrete structures. Springer, New York, NY, 1996. 1-18. Jones, Graham R. "Tree models for macroevolution and phylogenetic analysis." Systematic biology 60.6 (2011): 735-746.

Examples

Run this code

simulated_tree <- ape::rphylo(n = 100, birth = 1, death = 0)
balanced_tree <- treestats::create_fully_balanced_tree(simulated_tree)
unbalanced_tree <- treestats::create_fully_unbalanced_tree(simulated_tree)
beta_statistic(balanced_tree) # should be approximately 10
beta_statistic(simulated_tree) # should be near 0
beta_statistic(unbalanced_tree) # should be approximately -2