Fourier transform inequalities for phylogenetic trees
Abstract
Phylogenetic invariants are not the only constraints on site-pattern frequency vectors for phylogenetic trees. A mutation matrix, by its definition, is the exponential of a matrix with non-negative off-diagonal entries; this positivity requirement implies non-trivial constraints on the site-pattern frequency vectors. We call these additional constraints ``edge-parameter inequalities.'' In this paper, we first motivate the edge-parameter inequalities by considering a pathological site-pattern frequency vector corresponding to a quartet tree with a negative internal edge. This site-pattern frequency vector nevertheless satisfies all of the constraints described up to now in the literature. We next describe two complete sets of edge-parameter inequalities for the group-based models; these constraints are square-free monomial inequalities in the Fourier transformed coordinates. These inequalities, along with the phylogenetic invariants, form a complete description of the set of site-pattern frequency vectors corresponding to \emph{bona fide} trees. Said in mathematical language, this paper explicitly presents two finite lists of inequalities in Fourier coordinates of the form ``monomial ,'' each list characterizing the phylogenetically relevant semialgebraic subsets of the phylogenetic varieties.
Keywords
Cite
@article{arxiv.0711.3492,
title = {Fourier transform inequalities for phylogenetic trees},
author = {Frederick A. Matsen},
journal= {arXiv preprint arXiv:0711.3492},
year = {2008}
}