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Dergiler / Turkish Journal of Mathematics and Computer Science / 2021 / Cilt: 13 - Sayı: 1

Geometric Interpretation and Manifold Structure of Markov Matrices

Bülent Karakaş
Şenay Baydaş
Sayfa
14–18
Yayın Tarihi
2022-11-02
DOI
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⚡ TL;DR(1 Cümlede Hızlı Çıkarım)

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.

📌 3 Maddede Bu Makale

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Amaç & Hipotez:In probability theory and statistics, the term Markov property refers to the memoryless property of a stochastic process.
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Temel Bulgular & Veri:It is named after the Russian mathematician AndreyAndreyevichMarkov.
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Sonuç & Katkı: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.
🔬Tür:Araştırma Makalesi
📍Kapsam:Genişletilmiş Akademik Örneklem
⚙️Değişkenler:Alan Parametreleri

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 AndreyAndreyevichMarkov. 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.

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Bu Makaleye Atıf Yap

Karakaş, B., Baydaş, Ş. (2022). Geometric Interpretation and Manifold Structure of Markov Matrices. Turkish Journal of Mathematics and Computer Science, Cilt: 13 - Sayı: 1, 14–18. https://ajindex.com/articles/609096

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Kaynakça (8)

  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.