{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,9]],"date-time":"2026-05-09T00:49:46Z","timestamp":1778287786356,"version":"3.51.4"},"reference-count":43,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2015,6,2]],"date-time":"2015-06-02T00:00:00Z","timestamp":1433203200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Spanish Government","award":["TEC2012-38630-C04-01"],"award-info":[{"award-number":["TEC2012-38630-C04-01"]}]},{"name":"Colombian Department of Science, Technology and Innovation - COLCIENCIAS","award":["111556933858"],"award-info":[{"award-number":["111556933858"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>This paper addresses the problem of measuring complexity from embedded attractors as a way to characterize changes in the dynamical behavior of different types of systems with a quasi-periodic behavior by observing their outputs. With the aim of measuring the stability of the trajectories of the attractor along time, this paper proposes three new estimations of entropy that are derived from a Markov model of the embedded attractor. The proposed estimators are compared with traditional nonparametric entropy measures, such as approximate entropy, sample entropy and fuzzy entropy, which only take into account the spatial dimension of the trajectory. The method proposes the use of an unsupervised algorithm to find the principal curve, which is considered as the \u201cprofile trajectory\u201d, that will serve to adjust the Markov model. The new entropy measures are evaluated using three synthetic experiments and three datasets of physiological signals. In terms of consistency and discrimination capabilities, the results show that the proposed measures perform better than the other entropy measures used for comparison purposes.<\/jats:p>","DOI":"10.3390\/e17063595","type":"journal-article","created":{"date-parts":[[2015,6,2]],"date-time":"2015-06-02T11:38:36Z","timestamp":1433245116000},"page":"3595-3620","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":15,"title":["Entropies from Markov Models as Complexity Measures of Embedded Attractors"],"prefix":"10.3390","volume":"17","author":[{"given":"Juli\u00e1n","family":"Arias-Londo\u00f1o","sequence":"first","affiliation":[{"name":"Department of Systems Engineering, Universidad de Antioquia, Cll 70 No. 52-21, Medell\u00edn, Colombia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Juan","family":"Godino-Llorente","sequence":"additional","affiliation":[{"name":"Center for Biomedical Technologies, Universidad Polit\u00e9cnica de Madrid, Crta. M40, km. 38, Pozuelode Alarc\u00f3n, 28223, Madrid, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2015,6,2]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Kantz, H., and Schreiber, T. (2004). Nonlinear Time Series Analysis, Cambridge University Press. [2nd].","DOI":"10.1017\/CBO9780511755798"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"431","DOI":"10.1007\/s11071-011-0281-2","article-title":"Entropy measures for biological signal analyses","volume":"68","author":"Gao","year":"2012","journal-title":"Nonlinear Dyn."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"824","DOI":"10.1016\/j.ymssp.2006.02.009","article-title":"Approximate entropy as a diagnostic tool for machine health monitoring","volume":"21","author":"Yan","year":"2007","journal-title":"Mech. Syst. 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