# Embeddability of Kimura 3ST Markov matrices.

@article{RocaLacostena2018EmbeddabilityOK, title={Embeddability of Kimura 3ST Markov matrices.}, author={Jordi Roca-Lacostena and Jes{\'u}s Fern{\'a}ndez-S{\'a}nchez}, journal={Journal of theoretical biology}, year={2018}, volume={445}, pages={ 128-135 } }

## 10 Citations

Embeddability of centrosymmetric matrices

- Mathematics
- 2022

In this paper, we discuss the embedding problem for centrosymmetric matrices, which are higher order generalizations of the matrices occurring in Strand Symmetric Models. These models capture the…

An open set of 4×4 embeddable matrices whose principal logarithm is not a Markov generator

- MathematicsLinear and Multilinear Algebra
- 2020

A Markov matrix is embeddable if it can represent a homogeneous continuous-time Markov process. It is well known that if a Markov matrix has real and pairwise-different eigenvalues, then the…

Model embeddability for symmetric group-based models

- Mathematics, Computer Science
- 2020

This work provides a characterisation of model embeddable Markov matrices corresponding to symmetric group-based phylogenetic models, and provides necessary and sufficient conditions in terms of the eigenvalues of symmetric Group-based matrices.

Embeddability and rate identifiability of Kimura 2-parameter matrices

- Mathematics, Computer ScienceJournal of mathematical biology
- 2019

This study concludes the embedding problem and rate identifiability for the K80 model of nucleotide substitution and its submodels and describes an open subset of embeddable matrices with non-identifiable rates.

The model-specific Markov embedding problem for symmetric group-based models

- Mathematics, Computer ScienceJournal of mathematical biology
- 2021

This work provides a characterisation of model embeddable Markov matrices corresponding to symmetric group-based phylogenetic models and provides necessary and sufficient conditions in terms of the eigenvalues of symmetric Group-based matrices.

Geometry of symmetric group-based models

- Mathematics
- 2017

This work explores an example where the maximum likelihood estimate does not exist, which would be difficult to discover without using algebraic methods, and identifies which mutation matrices are matrix exponentials of rate matrices that are invariant under a group action.

The embedding problem for Markov matrices

- Mathematics, Computer Science
- 2020

Characterizing whether a Markov process of discrete random variables has an homogeneous continuous-time realization is a hard problem. In practice, this problem reduces to deciding when a given…

Geometry of time-reversible group-based models

- Mathematics
- 2017

Phylogenetic models have polynomial parametrization maps. For time-reversible group-based models, Matsen studied the polynomial inequalities that characterize the joint probabilities in the image of…

Algebraic Statistics in Practice: Applications to Networks

- Mathematics, Computer Science
- 2019

This survey illustrates this on three problems related to networks, namely network models for relational data, causal structure discovery and phylogenetics, with emphasis on the statistical achievements made possible by these tools and their practical relevance for applications to other scientific disciplines.

## References

SHOWING 1-10 OF 32 REFERENCES

Embeddable Markov Matrices

- Mathematics
- 2010

We give an account of some results, both old and new, about any $n\times n$ Markov matrix that is embeddable in a one-parameter Markov semigroup. These include the fact that its eigenvalues must lie…

Multiplicatively closed Markov models must form Lie algebras

- Mathematics
- 2017

The key original contribution is to overcome an obstruction, due to the presence of inequalities that are unavoidable in the probabilistic application, which prevents free manipulation of terms in the Baker–Campbell–Haursdorff formula.

On the ideals of equivariant tree models

- Mathematics, Computer Science
- 2007

Equivariant tree models in algebraic statistics are introduced, which unify and generalise existing tree models such as the general Markov model, the strand symmetric model, and group-based modelssuch as the Jukes–Cantor and Kimura models and yield generators of the full ideal.

Toric Ideals of Phylogenetic Invariants

- MathematicsJ. Comput. Biol.
- 2005

Generators and Gröbner bases are determined for the Jukes-Cantor and Kimura models on a binary tree and for several widely used models for biological sequences that have transition matrices that can be diagonalized by means of the Fourier transform of an abelian group.

On the existence and uniqueness of the real logarithm of a matrix

- Mathematics
- 1966

We then take the natural logarithm of both sides of (2.2) and invert the similarity transformation to obtain the desired solution(s) X. As we will show rigorously, a real solution exists provided C…