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Issue No.01 - January-March (2009 vol.6)
pp: 89-95
Frederick A. Matsen , UC Berkeley, Berkeley
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 $\leq 1$," each list characterizing the phylogenetically relevant semialgebraic subsets of the phylogenetic varieties.
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Frederick A. Matsen, "Fourier Transform Inequalities for Phylogenetic Trees", IEEE/ACM Transactions on Computational Biology and Bioinformatics, vol.6, no. 1, pp. 89-95, January-March 2009, doi:10.1109/TCBB.2008.68
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