Metrics

Comparison Between Phylogenetic and Climatic Trees

The comparison between phylogenetic trees (i.e., trees based on genetic data) and climatic trees involves a phylogeography step using Robinson and Foulds distance (i.e., topology distance) and Least Square distance (i.e., branch length distance).

Least Squares Distance

\[LS(T_1, T_2) = \sum_{i=1}^{n-1} \sum_{j=i}^{n} \lvert \delta(i,j) - \xi(i,j) \rvert\]

Where \(T_1\) is phylogenetic tree 1, \(T_2\) is phylogenetic tree 2, \(i\) and \(j\) are two species, \(\delta(i,j)\) is the distance between species \(i\) and species \(j\) in \(T_1\), \(\xi(i,j)\) is the distance between species \(i\) and species \(j\) in \(T_2\), and \(n\) is the total number of species.

Robinson-Foulds Distance

The RF distance between the phylogenetic tree (\(T_1\)) and reference tree (\(T_2\)) is the number of non-trivial bipartitions of \(T_1\) that are not in \(T_2\) plus the number of non-trivial bipartitions of \(T_2\) that are not in \(T_1\). This distance RF between \(T_1\) and \(T_2\) is computed by the following formula:

\[RF(T_1, T_2) = \frac{|(Q \backslash P) \cup (P \backslash Q)|}{2n-6},\]

where \(Q\) is the set of all possible bipartitions in phylogenetic tree \(T_1\), \(P\) is the set of all possible bipartitions in reference tree \(T_2\), and \(n\) is the number of leaves in \(T_1\) (or \(T_2\)). It is often relevant to normalize this distance by the maximum possible value of RF (equal to \(2n-6\) for two binary trees with \(n\) common leaves).