Discrete Fourier analysis for phylogenetic trees

Discrete Fourier transformations (DFTs) provide a useful tool to assign a phylogenetic tree (PGT) to an observed frequency of nucleotide patterns in DNA sequences of species. The advantage of this sort of spectral analysis is that it allows global correction for multi-substitution processes [1]. SPECTRAL ANALYSIS OF PGTS Two spectras characterize a PGT, the probability spectrum p(T), and the expected sequence spectrum s(T). After labelling the edges of the tree in an appropriate way, they are can be related by two steps of transforms using vector functions called Hadamard conjugations. The intermediate vector is called the edge length spectrum. The transformation scheme is given in Fig. 1. This scheme can be used in two ways. Starting with a probability distribution we can calculate the edge length spectrum and the expected sequence spectrum. On the other hand, given a data set D, we can take the observed sequence spectrum s(D) (the relative frequencies of character patterns) as an estimate for s(T). From this we calculate a conjugate spectrum γ(D) (the ‘corrected partition frequencies’) [1, 4]. This will correct for all parallel, multiple, and higher order substitutions. We find the corresponding tree, that is the tree for which | γ(D) – q(T) | is minimal, using a fitting algorithm (e.g. least-squares best fit or ‘closest tree algorithm’). Having found the correct tree one is able to reconstruct the probability spectrum and expected sequence spectrum. HADAMARD CONJUGATION A conjugation consists of three transformations that are successively applied. The third transformation is the inverse of the first. The m ‰ m – Hardamard matrix Ht is defined as (Hendy et al., Proc. Natl. Acad. Sci. USA 91 (1994)) Fig.1. Scheme of transformations.