64
Views
2
CrossRef citations to date
0
Altmetric
Research Article

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

, &
Pages 3768-3779 | Received 12 Jun 2020, Accepted 18 Nov 2020, Published online: 17 Dec 2020

References

  • Elfving G. Zur theorie der markoffschen ketten. Acta Soc Sci Finn. 01 1937;2:155–160.
  • Carette P. Characterizations of embeddable 33 stochastic matrices with a negative eigenvalue. New York J Math [electronic Only]. 01 1995;1:129.
  • Cuthbert JR. The logarithm function for finite-state Markov semi-groups. J Lond Math Soc. 1973;2(3):524–532.
  • Johansen S. Some results on the imbedding problem for finite Markov chains. J Lond Math Soc. 07 1974;s2–8(2):345–351.
  • Kingman JFC. The imbedding problem for finite Markov chains. Zeitschrift Für Wahrscheinlichkeitstheorie Und Verwandte Gebiete. 1962;1(1):14–24.
  • Casanellas M, Fernández-Sánchez J, Roca-Lacostena J. The embedding problem for Markov matrices; 2020. Submitted
  • Culver WJ. On the existence and uniqueness of the real logarithm of a matrix. Proc Amer Math Soc. 1966;17:1146–1151.
  • Cuthbert JR. On uniqueness of the logarithm for Markov semi-groups. J Lond Math Soc. 1972;2(4):623–630.
  • Israel RB, Rosenthal JS, Wei JZ. Finding generators for Markov chains via empirical transition matrices, with applications to credit ratings. Math Financ. 2001;11(2):245–265.
  • Runnenburg JT. On Elfving's problem of imbedding a time-discrete Markov chain in a time-continuous one for finitely many states. Proc KNAW – Ser A, Math Sci. 1962;65:536–548.
  • Jia C. A solution to the reversible embedding problem for finite Markov chains. Statist Probab Lett. 2016;116:122–130.
  • Roca-Lacostena J, Fernández-Sánchez J. Embeddability of Kimura 3st Markov matrices. J Theor Biol. 2018;445:128–135.
  • Geweke J, Marshall RC, Zarkin GA. Mobility indices in continuous time Markov chains. Econometrica. 1986;54(6):1407–1423.
  • Verbyla KL, Yap VB, Pahwa A, et al. The embedding problem for Markov models of nucleotide substitution. PLoS ONE. 7 2013;8:e69187.
  • Higham NJ. Functions of matrices – theory and computation. Philadelphia, PA, USA: SIAM; 2008.
  • Casanellas M, Sullivant S. The strand symmetric model. Algebraic statistics for computational biology. New York: Cambridge Univ. Press; 2005. p. 305–321.
  • Casanellas M, Kedzierska AM. Generating Markov evolutionary matrices for a given branch length. Linear Algebra Appl. 2013;438(5):2484–2499.
  • Casanellas M, Fernández-Sánchez J, Roca-Lacostena J. Embeddability and rate identifiability of Kimura 2-parameter matrices. J Math Biol. 2020;80:995–1019.
  • Roca-Lacostena J. Generating embeddable matrices whose principal logarithm is not a Markov generator. volume Extended Conference Abstracts – GEOMVAP; Birkahauser; 2020.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.