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Geometric Interpretation and Manifold Structure of Markov Matrices

Year 2021, , 14 - 18, 30.06.2021
https://doi.org/10.47000/tjmcs.797556

Abstract

In probability theory and statistics, the term Markov property refers to the memoryless property of a stochastic process. It is named after the Russian mathematician Andrey Andreyevich Markov. Every Markov matrix gives a linear equation system, and the solution of this equation system gives us a subset of $\mathbb{R}_{n}^{n}$. This paper presents the new manifold structure on the set of the Markov matrices. In addition, this paper presents the set of Markov matrices is drawable, and this gives geometrical interpretation to Markov matrices. For the proof, we use the one-to-one corresponding among $n \times n$ Markov matrices, the solution of linear equation system from derived Markov property, and the set of $(n-1)$-polytopes.

References

  • [1] Basharin, G.P., Langville, A.N., Naumov, V.A., The life and work of A.A. Markov, Linear Algebra Appl. 386(2004), 3-26.
  • [2] Bernhard, J., The geometry of Markov chain limit theorems, Markov Processes. Related Fields. 19(1)(2003), 99-124.
  • [3] Boothby, W.M., An Introduction to Differentiable Manifolds and Riemannian Geometry, Academic Press California, 1975.
  • [4] Helgason, S., Differential Geometry, Lie Groups and Symmetric Space. AMS Edition, Providence, 2001.
  • [5] Privault, N., Understanding Markov Chains, Springer, Singapore, 2013.
  • [6] Pullman, N., The geometry of finite Markov chains, Canad. Math. Bull. 8(3)(1965), 345-358.
  • [7] Sumner, J.G., Fernandez-Sanchez, J., Jarvis, P.D., Lie Markov models, J. Theoret. Biol. 2012(2012), 298.
  • [8] Sumner, J.G., Lie geometry of 2 x 2 Markov matrices, J. Theoret. Biol. 327(2013), 88-90.
Year 2021, , 14 - 18, 30.06.2021
https://doi.org/10.47000/tjmcs.797556

Abstract

References

  • [1] Basharin, G.P., Langville, A.N., Naumov, V.A., The life and work of A.A. Markov, Linear Algebra Appl. 386(2004), 3-26.
  • [2] Bernhard, J., The geometry of Markov chain limit theorems, Markov Processes. Related Fields. 19(1)(2003), 99-124.
  • [3] Boothby, W.M., An Introduction to Differentiable Manifolds and Riemannian Geometry, Academic Press California, 1975.
  • [4] Helgason, S., Differential Geometry, Lie Groups and Symmetric Space. AMS Edition, Providence, 2001.
  • [5] Privault, N., Understanding Markov Chains, Springer, Singapore, 2013.
  • [6] Pullman, N., The geometry of finite Markov chains, Canad. Math. Bull. 8(3)(1965), 345-358.
  • [7] Sumner, J.G., Fernandez-Sanchez, J., Jarvis, P.D., Lie Markov models, J. Theoret. Biol. 2012(2012), 298.
  • [8] Sumner, J.G., Lie geometry of 2 x 2 Markov matrices, J. Theoret. Biol. 327(2013), 88-90.
There are 8 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Bülent Karakaş This is me 0000-0002-3915-6526

Şenay Baydaş 0000-0002-1026-9471

Publication Date June 30, 2021
Published in Issue Year 2021

Cite

APA Karakaş, B., & Baydaş, Ş. (2021). Geometric Interpretation and Manifold Structure of Markov Matrices. Turkish Journal of Mathematics and Computer Science, 13(1), 14-18. https://doi.org/10.47000/tjmcs.797556
AMA Karakaş B, Baydaş Ş. Geometric Interpretation and Manifold Structure of Markov Matrices. TJMCS. June 2021;13(1):14-18. doi:10.47000/tjmcs.797556
Chicago Karakaş, Bülent, and Şenay Baydaş. “Geometric Interpretation and Manifold Structure of Markov Matrices”. Turkish Journal of Mathematics and Computer Science 13, no. 1 (June 2021): 14-18. https://doi.org/10.47000/tjmcs.797556.
EndNote Karakaş B, Baydaş Ş (June 1, 2021) Geometric Interpretation and Manifold Structure of Markov Matrices. Turkish Journal of Mathematics and Computer Science 13 1 14–18.
IEEE B. Karakaş and Ş. Baydaş, “Geometric Interpretation and Manifold Structure of Markov Matrices”, TJMCS, vol. 13, no. 1, pp. 14–18, 2021, doi: 10.47000/tjmcs.797556.
ISNAD Karakaş, Bülent - Baydaş, Şenay. “Geometric Interpretation and Manifold Structure of Markov Matrices”. Turkish Journal of Mathematics and Computer Science 13/1 (June 2021), 14-18. https://doi.org/10.47000/tjmcs.797556.
JAMA Karakaş B, Baydaş Ş. Geometric Interpretation and Manifold Structure of Markov Matrices. TJMCS. 2021;13:14–18.
MLA Karakaş, Bülent and Şenay Baydaş. “Geometric Interpretation and Manifold Structure of Markov Matrices”. Turkish Journal of Mathematics and Computer Science, vol. 13, no. 1, 2021, pp. 14-18, doi:10.47000/tjmcs.797556.
Vancouver Karakaş B, Baydaş Ş. Geometric Interpretation and Manifold Structure of Markov Matrices. TJMCS. 2021;13(1):14-8.