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Embedding of Markov matrices for d ⩽ 4
Michael Baake1, Jeremy Sumner2
1Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, 33501, Bielefeld, Germany. mbaake@math.uni-bielefeld.de.
Journal of Mathematical Biology
|July 2, 2024
Summary
This study addresses the Markov matrix embedding problem, crucial for phylogeny and population genetics. Researchers provide a simplified treatment for specific dimensions and explore time-inhomogeneous Markov chains for real-world applications.
Area of Science:
- Mathematics
- Probability Theory
- Computational Biology
Background:
- The Markov matrix embedding problem in Markov semigroups is a long-standing challenge.
- Recent advancements in phylogeny and population genetics have renewed interest in this problem.
Purpose of the Study:
- To provide a comprehensive account of Markov matrix embedding for dimensions up to 4.
- To offer a simplified and complete treatment for the case of 2x2 matrices.
- To extend the analysis to time-inhomogeneous Markov chains for practical applications.
Main Methods:
- Systematic derivation of embedding results for specified dimensions.
- Analysis of the embedding problem for time-inhomogeneous Markov chains.
- Review of existing literature and identification of future research directions.
Main Results:
- A complete and simplified treatment for the embedding of 2x2 Markov matrices.
- Derivation of embedding results for dimensions up to 4.
- Characterization of additional embedding cases for time-inhomogeneous Markov chains.
Conclusions:
- The study offers a systematic approach to the Markov matrix embedding problem.
- The findings have potential applications in evolutionary biology and genetics.
- Further research is outlined for more complex, time-inhomogeneous scenarios.
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