{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,14]],"date-time":"2026-02-14T06:48:51Z","timestamp":1771051731025,"version":"3.50.1"},"reference-count":24,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2018,1,24]],"date-time":"2018-01-24T00:00:00Z","timestamp":1516752000000},"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>Approximate and sample entropy (AE and SE) provide robust measures of the deterministic or stochastic content of a time series (regularity), as well as the degree of structural richness (complexity), through operations at multiple data scales. Despite the success of the univariate algorithms, multivariate sample entropy (mSE) algorithms are still in their infancy and have considerable shortcomings. Not only are existing mSE algorithms unable to analyse within- and cross-channel dynamics, they can counter-intuitively interpret increased correlation between variates as decreased regularity. To this end, we first revisit the embedding of multivariate delay vectors (DVs), critical to ensuring physically meaningful and accurate analysis. We next propose a novel mSE algorithm and demonstrate its improved performance over existing work, for synthetic data and for classifying wake and sleep states from real-world physiological data. It is furthermore revealed that, unlike other tools, such as the correlation of phase synchrony, synchronized regularity dynamics are uniquely identified via mSE analysis. In addition, a model for the operation of this novel algorithm in the presence of white Gaussian noise is presented, which, in contrast to the existing algorithms, reveals for the first time that increasing correlation between different variates reduces entropy.<\/jats:p>","DOI":"10.3390\/e20020082","type":"journal-article","created":{"date-parts":[[2018,1,24]],"date-time":"2018-01-24T09:47:36Z","timestamp":1516787256000},"page":"82","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":22,"title":["A Novel Multivariate Sample Entropy Algorithm for Modeling Time Series Synchronization"],"prefix":"10.3390","volume":"20","author":[{"given":"David","family":"Looney","sequence":"first","affiliation":[{"name":"Department of Electrical and Electronic Engineering, Imperial College London, London SW7 2AZ, UK"}]},{"given":"Tricia","family":"Adjei","sequence":"additional","affiliation":[{"name":"Department of Electrical and Electronic Engineering, Imperial College London, London SW7 2AZ, UK"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8432-3963","authenticated-orcid":false,"given":"Danilo","family":"Mandic","sequence":"additional","affiliation":[{"name":"Department of Electrical and Electronic Engineering, Imperial College London, London SW7 2AZ, UK"}]}],"member":"1968","published-online":{"date-parts":[[2018,1,24]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.pneurobio.2005.10.003","article-title":"Nonlinear multivariate analysis of neurophysiological signals","volume":"77","author":"Pereda","year":"2005","journal-title":"Prog. Neurobiol."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"2084","DOI":"10.1109\/TPWRS.2011.2120632","article-title":"A study of principal component analysis applied to spatially distributed wind power","volume":"26","author":"Burke","year":"2011","journal-title":"IEEE Trans. Power Syst."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"152","DOI":"10.1016\/j.irfa.2011.02.014","article-title":"Dynamic correlation between stock market and oil prices: The case of oil-importing and oil-exporting countries","volume":"20","author":"Filis","year":"2011","journal-title":"Int. Rev. Financ. Anal."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"424","DOI":"10.2307\/1912791","article-title":"Investigating Causal Relations by Econometric Models and Cross-spectral Methods","volume":"37","author":"Granger","year":"1969","journal-title":"Econometrica"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1016\/S0165-0270(99)00128-4","article-title":"Using partial directed coherence to describe neuronal ensemble interactions","volume":"94","author":"Sameshima","year":"1999","journal-title":"J. Neurosci. Methods"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"461","DOI":"10.1103\/PhysRevLett.85.461","article-title":"Measuring Information Transfer","volume":"85","author":"Schreiber","year":"2000","journal-title":"Phys. Rev. Lett."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"2297","DOI":"10.1073\/pnas.88.6.2297","article-title":"Approximate entropy as a measure of system complexity","volume":"88","author":"Pincus","year":"1991","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_8","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_9","doi-asserted-by":"crossref","first-page":"068102","DOI":"10.1103\/PhysRevLett.89.068102","article-title":"Multiscale entropy analysis of complex physiologic time series","volume":"89","author":"Costa","year":"2002","journal-title":"Phys. Rev. Lett."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"021906","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_11","doi-asserted-by":"crossref","first-page":"257","DOI":"10.1097\/WNP.0b013e3181b2f1e3","article-title":"Sample entropy tracks changes in electroencephalogram power spectrum with sleep state and aging","volume":"26","author":"Bruce","year":"2009","journal-title":"J. Clin. Neurophysiol."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"020901","DOI":"10.1103\/PhysRevE.78.020901","article-title":"Complex dynamics of human red blood cell flickering: Alterations with in vivo aging","volume":"78","author":"Costa","year":"2008","journal-title":"Phys. Rev. E"},{"key":"ref_13","first-page":"317","article-title":"Nonlinear analysis of electroencephalogram and magnetoencephalogram recordings in patients with Alzheimer\u2019s disease","volume":"367","author":"Hornero","year":"2009","journal-title":"Philos. Trans. R. Soc. Lond. A Math. Phys. Eng. Sci."},{"key":"ref_14","unstructured":"Ahmed, M.U., Li, L., Cao, J., and Mandic, D.P. (September, January 30). Multivariate multiscale entropy for brain consciousness analysis. Proceedings of the IEEE Engineering in Medicine and Biology Society (EMBC), Boston, MA, USA."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"061918","DOI":"10.1103\/PhysRevE.84.061918","article-title":"Multivariate multiscale entropy: A tool for complexity analysis of multichannel data","volume":"84","author":"Ahmed","year":"2011","journal-title":"Phys. Rev. E"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"3069","DOI":"10.1016\/j.physd.2008.06.005","article-title":"The effect of time delay on Approximate & Sample Entropy calculations","volume":"237","author":"Kaffashi","year":"2008","journal-title":"Phys. D Nonlinear Phenom."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1804","DOI":"10.1103\/PhysRevLett.76.1804","article-title":"Phase Synchronization of Chaotic Oscillators","volume":"76","author":"Rosenblum","year":"1996","journal-title":"Phys. Rev. Lett."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"045202","DOI":"10.1103\/PhysRevE.64.045202","article-title":"Detecting direction of coupling in interacting oscillators","volume":"64","author":"Rosenblum","year":"2001","journal-title":"Phys. Rev. E"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"980","DOI":"10.1103\/PhysRevE.51.980","article-title":"Generalized synchronization of chaos in directionally coupled chaotic systems","volume":"51","author":"Rulkov","year":"1995","journal-title":"Phys. Rev. E"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"75","DOI":"10.1016\/S0167-2789(98)00151-1","article-title":"Dynamics from multivariate time series","volume":"121","author":"Cao","year":"1998","journal-title":"Phys. D Nonlinear Phenom."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"85","DOI":"10.1109\/TSMCB.2003.808175","article-title":"Effective detection of coupling in short and noisy bivariate data","volume":"33","author":"Bhattacharya","year":"2003","journal-title":"IEEE Trans. Syst. Man Cyberne. Part B Cybern."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"7297","DOI":"10.1523\/JNEUROSCI.22-16-07297.2002","article-title":"Decreased neuronal synchronization during experimental seizures","volume":"22","author":"Netoff","year":"2002","journal-title":"J. Neurosci."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"842","DOI":"10.1161\/01.CIR.94.4.842","article-title":"Respiratory Sinus Arrhythmia","volume":"94","author":"Hayano","year":"1996","journal-title":"Circulation"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"262","DOI":"10.2307\/1403446","article-title":"A simple approximation for bivariate and trivariate normal integrals","volume":"59","author":"Cox","year":"1991","journal-title":"Int. Stat. Rev."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/20\/2\/82\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T14:52:24Z","timestamp":1760194344000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/20\/2\/82"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,1,24]]},"references-count":24,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2018,2]]}},"alternative-id":["e20020082"],"URL":"https:\/\/doi.org\/10.3390\/e20020082","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,1,24]]}}}