Abstract
BackgroundContinuous time Markov chains (CTMCs) is a widely used model for describing the evolution of DNA sequences on the nucleotide, amino acid or codon level. The sufficient statistics for CTMCs are the time spent in a state and the number of changes between any two states. In applications past evolutionary events (exact times and types of changes) are unaccessible and the past must be inferred from DNA sequence data observed in the present.ResultsWe describe and implement three algorithms for computing linear combinations of expected values of the sufficient statistics, conditioned on the end-points of the chain, and compare their performance with respect to accuracy and running time. The first algorithm is based on an eigenvalue decomposition of the rate matrix (EVD), the second on uniformization (UNI), and the third on integrals of matrix exponentials (EXPM). The implementation in R of the algorithms is available at http://www.birc.au.dk/~paula/.ConclusionsWe use two different models to analyze the accuracy and eight experiments to investigate the speed of the three algorithms. We find that they have similar accuracy and that EXPM is the slowest method. Furthermore we find that UNI is usually faster than EVD.
Highlights
Continuous time Markov chains (CTMCs) is a widely used model for describing the evolution of DNA sequences on the nucleotide, amino acid or codon level
The case where the CTMC is only recorded at discretely observed time points arises in molecular evolution where DNA sequence data is extracted at present day and past evolutionary events are missing
The first class of applications is concerned with rate matrix estimation. [1] describes how the expectationmaximization (EM) algorithm can be applied to estimate the rate matrix from DNA sequence data observed in the leaves of an evolutionary tree
Summary
Continuous time Markov chains (CTMCs) is a widely used model for describing the evolution of DNA sequences on the nucleotide, amino acid or codon level. The case where the CTMC is only recorded at discretely observed time points arises in molecular evolution where DNA sequence data is extracted at present day and past evolutionary events are missing. In this situation, efficient methods for calculating these types of expectations are needed. The first class of applications is concerned with rate matrix estimation. [1] describes how the expectationmaximization (EM) algorithm can be applied to estimate the rate matrix from DNA sequence data observed in the leaves of an evolutionary tree. The EM algorithm is implemented in the software XRate [2] and has been
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