The embedding problem for Markov matrices
Characterizing whether a Markov process of discrete random variables has a homogeneous continuous-time realization is a hard problem. In practice, this problem reduces to deciding when a given Markov matrix can be written as the exponential of some rate matrix (a Markov generator). This is an old qu...
| Autores: | , , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2023 |
| País: | España |
| Institución: | Universitat Politècnica de Catalunya (UPC) |
| Repositorio: | UPCommons. Portal del coneixement obert de la UPC |
| Idioma: | inglés |
| OAI Identifier: | oai:upcommons.upc.edu:2117/384822 |
| Acceso en línea: | https://hdl.handle.net/2117/384822 https://dx.doi.org/10.5565/PUBLMAT6712308 |
| Access Level: | acceso abierto |
| Palabra clave: | Algebras, Linear Multilinear algebra Matrices Markov processes Markov matrix Markov generator Embedding problem Rate identifiability Àlgebra lineal Àlgebra multilineal Matrius (Àlgebra) Markov, Processos de Àrees temàtiques de la UPC::Matemàtiques i estadística::Àlgebra::Àlgebra lineal i multilineal Àrees temàtiques de la UPC::Matemàtiques i estadística::Anàlisi matemàtica |
| Sumario: | Characterizing whether a Markov process of discrete random variables has a homogeneous continuous-time realization is a hard problem. In practice, this problem reduces to deciding when a given Markov matrix can be written as the exponential of some rate matrix (a Markov generator). This is an old question known in the literature as the embedding problem [11], which has been solved only for matrices of size 2 × 2 or 3 × 3. In this paper, we address this problem and related questions and obtain results along two different lines. First, for matrices of any size, we give a bound on the number of Markov generators in terms of the spectrum of the Markov matrix. Based on this, we establish a criterion for deciding whether a generic (distinct eigenvalues) Markov matrix is embeddable and propose an algorithm that lists all its Markov generators. Then, motivated and inspired by recent results on substitution models of DNA, we focus on the 4 × 4 case and completely solve the embedding problem for any Markov matrix. The solution in this case is more concise as the embeddability is given in terms of a single condition. |
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