Hendy, M. D.; Penny, D.; Steel, M. A. A discrete Fourier analysis for evolutionary trees. (English) Zbl 0791.92017 Proc. Natl. Acad. Sci. USA 91, No. 8, 3339-3343 (1994). Summary: Discrete Fourier transformations have recently been developed to model the evolution of two-state characters (the Cavender/Farris model). We report here the extension of these transformations to provide invertible relationships between a phylogenetic tree \(T\) (with three probability parameters of nucleotide substitution on each edge corresponding to Kimura’s 3ST model) and the expected frequencies of the nucleotide patterns in the sequences. We refer to these relationships as spectral analysis. In either model with independent and identically distributed site substitutions, spectral analysis allows global correction for all multiple substitutions (second- and higher-order interactions), independent of any particular tree. From these corrected data we use a least-squares selection procedure, the closest tree algorithm, to infer an evolutionary tree. Other selection criteria such as parsimony or compatibility analysis could also be used; each of these criteria will be statistically consistent for these models. The closest tree algorithm selects a unique best-fit phylogenetic tree together with independent edge length parameters for each edge. The method is illustrated with an analysis of some primate hemoglobin sequences. Cited in 16 Documents MSC: 92D15 Problems related to evolution 42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type 42A99 Harmonic analysis in one variable Keywords:Hadamard conjugations; discrete Fourier transformations; Cavender/Farris model; invertible relationships; phylogenetic tree; Kimura’s 3ST model; expected frequencies; nucleotide patterns; spectral analysis; multiple substitutions; higher-order interactions; least-squares selection procedure; closest tree algorithm; evolutionary tree; parsimony; compatibility analysis; unique best-fit phylogenetic tree; primate hemoglobin sequences PDFBibTeX XMLCite \textit{M. D. Hendy} et al., Proc. Natl. Acad. Sci. USA 91, No. 8, 3339--3343 (1994; Zbl 0791.92017) Full Text: DOI