Fourier transform inequalities for phylogenetic trees.
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 bona fide trees. Said in mathematical language, this paper explicitly presents two finite lists of inequalities in Fourier coordinates of the form "monomial < or = 1", each list characterizing the phylogenetically relevant semialgebraic subsets of the phylogenetic varieties.
Matsen
Frederick A
FA
Department of Statistics, University of California, Berkeley, 367 Evans Hall #429, Berkeley, CA 94720-3860, USA. matsen@berkeley.edu
D000465:Algorithms; D001499:Bayes Theorem; D019295:Computational Biology / Q000379:methods*; D005583:Fourier Analysis; D014644:Genetic Variation; D008390:Markov Chains; D008957:Models, Genetic; D015233:Models, Statistical; D009154:Mutation; D010802:Phylogeny