{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,4]],"date-time":"2026-06-04T08:59:41Z","timestamp":1780563581716,"version":"3.54.1"},"reference-count":19,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2017,4,6]],"date-time":"2017-04-06T00:00:00Z","timestamp":1491436800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The Partition Markov Model characterizes the process by a partition    L    of the state space, where the elements in each part of    L    share the same transition probability to an arbitrary element in the alphabet. This model aims to answer the following questions: what is the minimal number of parameters needed to specify a Markov chain and how to estimate these parameters. In order to answer these questions, we build a consistent strategy for model selection which consist of: giving a size n realization of the process, finding a model within the Partition Markov class, with a minimal number of parts to represent the process law. From the strategy, we derive a measure that establishes a metric in the state space. In addition, we show that if the law of the process is Markovian, then, eventually, when n goes to infinity,    L    will be retrieved. We show an application to model internet navigation patterns.<\/jats:p>","DOI":"10.3390\/e19040160","type":"journal-article","created":{"date-parts":[[2017,4,6]],"date-time":"2017-04-06T09:57:30Z","timestamp":1491472650000},"page":"160","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":32,"title":["Consistent Estimation of Partition Markov Models"],"prefix":"10.3390","volume":"19","author":[{"given":"Jes\u00fas","family":"Garc\u00eda","sequence":"first","affiliation":[{"name":"Department of Statistics, University of Campinas, Rua S\u00e9rgio Buarque de Holanda, 651, Campinas, S\u00e3o Paulo 13083-859, Brazil"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ver\u00f3nica","family":"Gonz\u00e1lez-L\u00f3pez","sequence":"additional","affiliation":[{"name":"Department of Statistics, University of Campinas, Rua S\u00e9rgio Buarque de Holanda, 651, Campinas, S\u00e3o Paulo 13083-859, Brazil"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2017,4,6]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"480","DOI":"10.1214\/aos\/1018031204","article-title":"Variable length Markov chains","volume":"27","author":"Buhlmann","year":"1999","journal-title":"Ann. 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Proceedings of the Fourth Workshop on Information Theoretic Methods in Science and Engineering, Helsinki, Finland."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1766","DOI":"10.1016\/j.spl.2011.07.012","article-title":"Hidden Markov Partition Models","volume":"81","author":"Farcomeni","year":"2011","journal-title":"Stat. Probab. Lett."},{"key":"ref_12","first-page":"1487","article-title":"Copula based model correction for bivariate Bernoulli financial series","volume":"Volume 1558","year":"2013","journal-title":"Proceedings of the 11th International Conference of Numerical Analysis and Applied Mathematics (ICNAAM 2013)"},{"key":"ref_13","unstructured":"Fern\u00e1ndez, M., Garc\u00eda Jes\u00fas, E., and Gonz\u00e1lez-L\u00f3pez, V.A. (2015). Multivariate Markov chain predictions adjusted with copula models. New Trends in Stochastic Modeling and Data Analysis, ISAST."},{"key":"ref_14","first-page":"140005","article-title":"Copula-Based Prediction of Economic Movements","volume":"Volume 1738","author":"Hirsh","year":"2015","journal-title":"Proceedings of the 13th International Conference of Numerical Analysis and Applied Mathematics (ICNAAM 2015)"},{"key":"ref_15","unstructured":"Garc\u00eda, J.E., and Gonz\u00e1lez-L\u00f3pez, V.A. (2013, January 26\u201329). Detecting regime changes in Markov models. Proceedings of The Sixth Workshop on Information Theoretic Methods in Science and Engineering, Tokyo, Japan."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Gusfield, D. (1997). Algorithms on Strings, Trees and Sequences: Computer Science and Computational Biology, Cambridge University Press.","DOI":"10.1017\/CBO9780511574931"},{"key":"ref_17","unstructured":"(2017, April 05). MSNBC.com Anonymous Web Data Data Set. 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