Fourier Transform Inequalities for Phylogenetic Trees

F.A. Matsen
2009 IEEE/ACM Transactions on Computational Biology & Bioinformatics  
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 nonnegative off-diagonal entries; this nonnegativity requirement implies nontrivial 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
more » ... uency 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 bona fide trees. Said in mathematical language, this paper explicitly presents two finite lists of inequalities in Fourier coordinates of the form "monomial 1," each list characterizing the phylogenetically relevant semialgebraic subsets of the phylogenetic varieties.
doi:10.1109/tcbb.2008.68 pmid:19179701 fatcat:aiys6dgpsbgzvan2mxdp73tlme