Phylogenetic Stochastic Mapping without Matrix Exponentiation

TL;DR

This paper introduces a new MCMC-based method for phylogenetic stochastic mapping that avoids matrix exponentiation, significantly improving computational efficiency for large and sparse state spaces like codons.

Contribution

The authors develop a novel uniformization-based MCMC algorithm that reduces computational complexity and leverages sparsity, outperforming traditional matrix exponentiation methods for large state spaces.

Findings

The new method is faster than existing approaches for large state spaces.

It efficiently exploits sparsity in rate matrices to improve performance.

The approach is effective even for the moderately large codon state space.

Abstract

Phylogenetic stochastic mapping is a method for reconstructing the history of trait changes on a phylogenetic tree relating species/organisms carrying the trait. State-of-the-art methods assume that the trait evolves according to a continuous-time Markov chain (CTMC) and work well for small state spaces. The computations slow down considerably for larger state spaces (e.g. space of codons), because current methodology relies on exponentiating CTMC infinitesimal rate matrices -- an operation whose computational complexity grows as the size of the CTMC state space cubed. In this work, we introduce a new approach, based on a CTMC technique called uniformization, that does not use matrix exponentiation for phylogenetic stochastic mapping. Our method is based on a new Markov chain Monte Carlo (MCMC) algorithm that targets the distribution of trait histories conditional on the trait data observed at the tips of the tree. The computational complexity of our MCMC method grows as the size of the CTMC state space squared. Moreover, in contrast to competing matrix exponentiation methods, if the rate matrix is sparse, we can leverage this sparsity and increase the computational efficiency of our algorithm further. Using simulated data, we illustrate advantages of our MCMC algorithm and investigate how large the state space needs to be for our method to outperform matrix exponentiation approaches. We show that even on the moderately large state space of codons our MCMC method can be significantly faster than currently used matrix exponentiation methods.