Hadamard Phylogenetic Methods and the n-taxon process

TL;DR

This paper explores the Hadamard transform in phylogenetics, demonstrating its relation to Markov chain models and showing that tree-based models are multivariate exponential distributions, simplifying evolutionary analysis.

Contribution

It expresses the Hadamard transform in terms of Markov chain formulas and introduces the $n$-taxon process, linking phylogenetic models to exponential distributions.

Findings

Hadamard transform can be expressed via Markov chain formulas.

Tree-based models are shown to be multivariate exponential distributions.

Derived transition probabilities for the $n$-taxon process.

Abstract

The Hadamard transform of \cite{Hendy89a, Hendy89} provides a way to work with stochastic models for sequence evolution without having to deal with the complications of tree space and the graphical structure of trees. Here we demonstrate that the transform can be expressed in terms of the familiar $\bP[\bt] = e^{\bQ[\bt]}$ formula for Markov chains. The key idea is to study the evolution of vectors of states, one vector entry for each taxa; we call this the $n$-taxon process. We derive transition probabilities for the process. Significantly, the findings show that tree-based models are indeed in the family of (multi-variate) exponential distributions.