{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,3]],"date-time":"2026-07-03T00:12:24Z","timestamp":1783037544660,"version":"3.54.6"},"reference-count":21,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2015,3,12]],"date-time":"2015-03-12T00:00:00Z","timestamp":1426118400000},"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>We introduce a generalization of multiscale entropy (MSE) analysis. The method is termed MSEn, where the subscript denotes the moment used to coarse-grain a time series. MSE\u03bc, described previously, uses the mean value (first moment). Here, we focus on MSE\u03c32 , which uses the second moment, i.e., the variance. MSE\u03c32 quantifies the dynamics of the volatility (variance) of a signal over multiple time scales. We use the method to analyze the structure of heartbeat time series. We find that the dynamics of the volatility of heartbeat time series obtained from healthy young subjects is highly complex. Furthermore, we find that the multiscale complexity of the volatility, not only the multiscale complexity of the mean heart rate, degrades with aging and pathology. The \u201cbursty\u201d behavior of the dynamics may be related to intermittency in energy and information flows, as part of multiscale cycles of activation and recovery. Generalized MSE may also be useful in quantifying the dynamical properties of other physiologic and of non-physiologic time series.<\/jats:p>","DOI":"10.3390\/e17031197","type":"journal-article","created":{"date-parts":[[2015,3,12]],"date-time":"2015-03-12T11:30:24Z","timestamp":1426159824000},"page":"1197-1203","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":142,"title":["Generalized Multiscale Entropy Analysis: Application to Quantifying the Complex Volatility of Human Heartbeat Time Series"],"prefix":"10.3390","volume":"17","author":[{"given":"Madalena","family":"Costa","sequence":"first","affiliation":[{"name":"Beth Israel Deaconess Medical Center, Harvard Medical School, Boston, MA 02215, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ary","family":"Goldberger","sequence":"additional","affiliation":[{"name":"Beth Israel Deaconess Medical Center, Harvard Medical School, Boston, MA 02215, USA"},{"name":"Wyss Institute for Biologically Inspired Engineering at Harvard University, Boston, MA 02215, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2015,3,12]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"277","DOI":"10.1098\/rsta.2008.0232","article-title":"Methods derived from nonlinear dynamics for analysing heart rate variability","volume":"367","author":"Voss","year":"2009","journal-title":"Phil. Trans. R. Soc. A"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"062102-1-4","DOI":"10.1103\/PhysRevLett.89.068102","article-title":"Multiscale entropy analysis of physiologic time series","volume":"89","author":"Costa","year":"2002","journal-title":"Phys. Rev. Lett."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"021906-1-8","DOI":"10.1103\/PhysRevE.71.021906","article-title":"Multiscale entropy analysis of biological signals","volume":"71","author":"Costa","year":"2005","journal-title":"Phys. Rev. E"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"88","DOI":"10.1007\/s10558-007-9049-1","article-title":"Multiscale analysis of heart rate dynamics: Entropy and time irreversibility measures","volume":"8","author":"Costa","year":"2008","journal-title":"Cardiovasc. 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Soc."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"H2039","DOI":"10.1152\/ajpheart.2000.278.6.H2039","article-title":"Physiological time-series analysis using approximate entropy and sample entropy","volume":"278","author":"Richman","year":"2000","journal-title":"Am. J. Physiol. Heart Circ. Physiol."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1093","DOI":"10.1016\/j.jcrc.2013.08.022","article-title":"Combat casualties undergoing lifesaving interventions have decreased heart rate complexity at multiple time scales","volume":"28","author":"Cancio","year":"2013","journal-title":"J. Crit. Care."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"H319","DOI":"10.1152\/ajpheart.00561.2010","article-title":"Accurate estimation of entropy in very short physiological time series: The problem of atrial fibrillation detection in implanted ventricular devices","volume":"300","author":"Lake","year":"2011","journal-title":"Am. J. Physiol. Heart Circ. 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