The embedding problem for Markov matrices

Casanellas, Marta; Fernández-Sánchez, Jesús; Roca-Lacostena, Jordi

doi:10.5565/publmat6712308

Cited by 5 publications

(27 citation statements)

References 36 publications

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“…We also compare the results on relative volumes obtained using our method with the algorithm suggested in Casanellas et al. ( 2023 ) to showcase the advantages of our method. In Sect.…”

Section: Introductionmentioning

confidence: 87%

“…(Casanellas et al. 2023 , Proposition 4.3). Let be an Markov matrix with and distinct eigenvalues for and for , all of them pairwise distinct.…”

Section: Preliminariesmentioning

confidence: 98%

“…Theorem 5.5 in Casanellas et al. ( 2023 ) provides boundaries for the values of k for which the corresponding inequalities may hold.…”

Section: Embeddability Of Centrosymmetric Matricesmentioning

confidence: 99%

“…For Markov matrices the embedding problem is completely settled in a series of papers (Casanellas et al. 2020a , 2023 ; Roca-Lacostena and Fernández-Sánchez 2018a ), where similarly to the case the full characterisation of embeddable matrices is distinguished into cases depending on the Jordan form of the Markov matrices.…”

Section: Introductionmentioning

confidence: 99%

“…This is achieved by providing an algorithm that outputs all Markov generators of such a Markov matrix (Casanellas et al. 2023 ; Roca-Lacostena 2021 ).…”

Section: Introductionmentioning

confidence: 99%

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Embeddability of centrosymmetric matrices capturing the double-helix structure in natural and synthetic DNA

2023

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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 substitution symmetries arising from the double helix structure of the DNA. Deciding whether a transition matrix is embeddable or not enables us to know if the observed substitution probabilities are consistent with a homogeneous continuous time substitution model, such as the Kimura models, the Jukes-Cantor model or the general time-reversible model. On the other hand, the generalization to higher order matrices is motivated by the setting of synthetic biology, which works with different sizes of genetic alphabets.

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“…We also compare the results on relative volumes obtained using our method with the algorithm suggested in Casanellas et al. ( 2023 ) to showcase the advantages of our method. In Sect.…”

Section: Introductionmentioning

confidence: 87%

“…(Casanellas et al. 2023 , Proposition 4.3). Let be an Markov matrix with and distinct eigenvalues for and for , all of them pairwise distinct.…”

Section: Preliminariesmentioning

confidence: 98%

“…Theorem 5.5 in Casanellas et al. ( 2023 ) provides boundaries for the values of k for which the corresponding inequalities may hold.…”

Section: Embeddability Of Centrosymmetric Matricesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…This is achieved by providing an algorithm that outputs all Markov generators of such a Markov matrix (Casanellas et al. 2023 ; Roca-Lacostena 2021 ).…”

Section: Introductionmentioning

confidence: 99%

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Embeddability of centrosymmetric matrices capturing the double-helix structure in natural and synthetic DNA

2023

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Phylogenetic Reconstruction Based on Algebra

Casanellas

2024

RSME Springer Series

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Embedding of Markov matrices for $$\varvec{d \leqslant 4}$$

Baake,

Sumner

2024

J. Math. Biol.

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The embedding problem of Markov matrices in Markov semigroups is a classic problem that regained a lot of impetus and activities through recent needs in phylogeny and population genetics. Here, we give an account for dimensions $$d\leqslant 4$$ d ⩽ 4 , including a complete and simplified treatment of the case $$d=3$$ d = 3 , and derive the results in a systematic fashion, with an eye on the potential applications. Further, we reconsider the setup of the corresponding problem for time-inhomogeneous Markov chains, which is needed for real-world applications because transition rates need not be constant over time. Additional cases of this more general embedding occur for any $$d\geqslant 3$$ d ⩾ 3 . We review the known case of $$d=3$$ d = 3 and describe the setting for future work on $$d=4$$ d = 4 .

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The embedding problem for Markov matrices

Cited by 5 publications

References 36 publications

Embeddability of centrosymmetric matrices capturing the double-helix structure in natural and synthetic DNA

Embeddability of centrosymmetric matrices capturing the double-helix structure in natural and synthetic DNA

Phylogenetic Reconstruction Based on Algebra

Embedding of Markov matrices for $$\varvec{d \leqslant 4}$$

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