collesslikeI {treebalance}

Calculation of the Colless-like indices for rooted trees

Description

This function calculates the Colless-like index for a given rooted tree T according to the chosen weight function f and dissimilarity D. The Colless-like index CL(T) relative to D and f is the sum of the (D,f)-balance values over all inner vertices of the tree. More precisely,

CL(T)=\sum_{v\in V_{in}(T)} bal_{D,f}(v)

where V_{in}(T) is the set of inner vertices of T. The (D,f)-balance value of v with children v_1,...,v_k is computed as

bal_{D,f}(v)=D(fs(T_{v_1}),...,fs(T_{v_k}))

with D denoting the dissimilarity and fs denoting the f.size.
The f.size fs(T) of a tree T uses the function f, which maps any integer to a non-negative real number, to build a weighted sum of the out-degrees of all vertices in T. More precisely,

fs(T)=\sum_{v\in V(T)} f(deg+(v))

where V(T) is the set of all vertices of T and deg+(v) denotes the out-degree (i.e. the number of children) of the vertex v. The f-functions that are already implemented are f(x)=e^x and f(x)=ln(x+e).
The dissimilarity D(x_1,...,x_k) of a vector x_1,...,x_k assigns a non-negative value to the vector, is independent of the order of the vector entries and equals zero if and only if x_1=...=x_k. In this implementation the following dissimilarity functions are already built-in: mean deviation from the median (mdm), the sample variance (var) and the sample standard deviation (sd).
collesslikeI also allows the use of other functions for the weight function f and the dissimilarity D.

Special cases: For n=1 the function returns CL(T)=0 and a warning.

For details on the family of Colless-like indices, see also Chapter 16 in "Tree balance indices: a comprehensive survey" (https://doi.org/10.1007/978-3-031-39800-1_16).

Usage

collesslikeI(tree, f.size, dissim)

Arguments

tree

A rooted binary tree in phylo format.

f.size

A character string specifying the function f that shall be used to compute the f.size. It can be one of the following: "exp", "ln" or the name of a function as a string.

dissim

A character string specifying the dissimilarity that shall be used. It can be one of the following: "mdm", "var", "sd" or the name of a function as a string.

Value

collesslikeI returns the Colless-like index of the given tree according to the chosen function and dissimilarity.

Author(s)

Luise Kuehn, Sophie Kersting

References

A. Mir, L. Rotger, and F. Rossello. Sound Colless-like balance indices for multifurcating trees. PLOSONE, 13(9):e0203401, 2018. doi: 10.1371/journal.pone.0203401

Examples

tree <- ape::read.tree(text="((((,),),(,)),(((,),),(,)));")
collesslikeI(tree, f.size="exp", dissim="mdm")
collesslikeI(tree, f.size="exp", dissim="var")
collesslikeI(tree, f.size="ln", dissim="sd")
myfsize <- function(x) return(x+1)
mydissim <- function(x) return (var(x))
collesslikeI(tree, f.size="myfsize",dissim = "mydissim")