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...

Descripción completa

Detalles Bibliográficos
Autores: Casanellas i Rius, Marta|||0000-0002-1724-8358, Fernández-Sánchez, Jesus|||0000-0002-7062-8042, Roca-Lacostena, Jordi|||0000-0003-1651-9504
Tipo de recurso: artículo
Fecha de publicación:2023
País:España
Institución:Universitat Autònoma de Barcelona
Repositorio:Dipòsit Digital de Documents de la UAB
Idioma:inglés
OAI Identifier:oai:ddd.uab.cat:271770
Acceso en línea:https://ddd.uab.cat/record/271770
https://dx.doi.org/urn:doi:10.5565/PUBLMAT6712308
Access Level:acceso abierto
Palabra clave:Markov matrix
Markov generator
Embedding problem
Rate identifiability
id ES_a1bbf3a4229efd8745d7882fc2dcbeed
oai_identifier_str oai:ddd.uab.cat:271770
network_acronym_str ES
network_name_str España
repository_id_str
spelling The embedding problem for Markov matricesCasanellas i Rius, Marta|||0000-0002-1724-8358Fernández-Sánchez, Jesus|||0000-0002-7062-8042Roca-Lacostena, Jordi|||0000-0003-1651-9504Markov matrixMarkov generatorEmbedding problemRate identifiabilityCharacterizing 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.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 theembedding problem for any Markov matrix. The solution in this case is more concise as the embeddability is given in terms of a single condition. 22023-01-0120232023-01-01Articlehttp://purl.org/coar/resource_type/c_6501VoRhttp://purl.org/coar/version/c_970fb48d4fbd8a85info:eu-repo/semantics/articleapplication/pdfhttps://ddd.uab.cat/record/271770https://dx.doi.org/urn:doi:10.5565/PUBLMAT6712308reponame:Dipòsit Digital de Documents de la UABinstname:Universitat Autònoma de BarcelonaInglésengopen accesshttp://purl.org/coar/access_right/c_abf2Aquest material està protegit per drets d'autor i/o drets afins. Podeu utilitzar aquest material en funció del que permet la legislació de drets d'autor i drets afins d'aplicació al vostre cas. Per a d'altres usos heu d'obtenir permís del(s) titular(s) de drets.https://rightsstatements.org/vocab/InC/1.0/info:eu-repo/semantics/openAccessoai:ddd.uab.cat:2717702026-06-06T12:50:31Z
dc.title.none.fl_str_mv The embedding problem for Markov matrices
title The embedding problem for Markov matrices
spellingShingle The embedding problem for Markov matrices
Casanellas i Rius, Marta|||0000-0002-1724-8358
Markov matrix
Markov generator
Embedding problem
Rate identifiability
title_short The embedding problem for Markov matrices
title_full The embedding problem for Markov matrices
title_fullStr The embedding problem for Markov matrices
title_full_unstemmed The embedding problem for Markov matrices
title_sort The embedding problem for Markov matrices
dc.creator.none.fl_str_mv Casanellas i Rius, Marta|||0000-0002-1724-8358
Fernández-Sánchez, Jesus|||0000-0002-7062-8042
Roca-Lacostena, Jordi|||0000-0003-1651-9504
author Casanellas i Rius, Marta|||0000-0002-1724-8358
author_facet Casanellas i Rius, Marta|||0000-0002-1724-8358
Fernández-Sánchez, Jesus|||0000-0002-7062-8042
Roca-Lacostena, Jordi|||0000-0003-1651-9504
author_role author
author2 Fernández-Sánchez, Jesus|||0000-0002-7062-8042
Roca-Lacostena, Jordi|||0000-0003-1651-9504
author2_role author
author
dc.subject.none.fl_str_mv Markov matrix
Markov generator
Embedding problem
Rate identifiability
topic Markov matrix
Markov generator
Embedding problem
Rate identifiability
description 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.
publishDate 2023
dc.date.none.fl_str_mv 2
2023-01-01
2023
2023-01-01
dc.type.none.fl_str_mv Article
http://purl.org/coar/resource_type/c_6501
VoR
http://purl.org/coar/version/c_970fb48d4fbd8a85
dc.type.openaire.fl_str_mv info:eu-repo/semantics/article
format article
dc.identifier.none.fl_str_mv https://ddd.uab.cat/record/271770
https://dx.doi.org/urn:doi:10.5565/PUBLMAT6712308
url https://ddd.uab.cat/record/271770
https://dx.doi.org/urn:doi:10.5565/PUBLMAT6712308
dc.language.none.fl_str_mv Inglés
eng
language_invalid_str_mv Inglés
language eng
dc.rights.none.fl_str_mv open access
http://purl.org/coar/access_right/c_abf2
https://rightsstatements.org/vocab/InC/1.0/
dc.rights.openaire.fl_str_mv info:eu-repo/semantics/openAccess
rights_invalid_str_mv open access
http://purl.org/coar/access_right/c_abf2
https://rightsstatements.org/vocab/InC/1.0/
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.source.none.fl_str_mv reponame:Dipòsit Digital de Documents de la UAB
instname:Universitat Autònoma de Barcelona
instname_str Universitat Autònoma de Barcelona
reponame_str Dipòsit Digital de Documents de la UAB
collection Dipòsit Digital de Documents de la UAB
repository.name.fl_str_mv
repository.mail.fl_str_mv
_version_ 1869415186617597952
score 15,301603

Similares en LA Referencia