{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,25]],"date-time":"2026-02-25T15:30:18Z","timestamp":1772033418063,"version":"3.50.1"},"reference-count":28,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2018,1,13]],"date-time":"2018-01-13T00:00:00Z","timestamp":1515801600000},"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>Sample Entropy is the most popular definition of entropy and is widely used as a measure of the regularity\/complexity of a time series. On the other hand, it is a computationally expensive method which may require a large amount of time when used in long series or with a large number of signals. The computationally intensive part is the similarity check between points in m dimensional space. In this paper, we propose new algorithms or extend already proposed ones, aiming to compute Sample Entropy quickly. All algorithms return exactly the same value for Sample Entropy, and no approximation techniques are used. We compare and evaluate them using cardiac inter-beat (RR) time series. We investigate three algorithms. The first one is an extension of the     k d    -trees algorithm, customized for Sample Entropy. The second one is an extension of an algorithm initially proposed for Approximate Entropy, again customized for Sample Entropy, but also improved to present even faster results. The last one is a completely new algorithm, presenting the fastest execution times for specific values of m, r, time series length, and signal characteristics. These algorithms are compared with the straightforward implementation, directly resulting from the definition of Sample Entropy, in order to give a clear image of the speedups achieved. All algorithms assume the classical approach to the metric, in which the maximum norm is used. The key idea of the two last suggested algorithms is to avoid unnecessary comparisons by detecting them early. We use the term unnecessary to refer to those comparisons for which we know a priori that they will fail at the similarity check. The number of avoided comparisons is proved to be very large, resulting in an analogous large reduction of execution time, making them the fastest algorithms available today for the computation of Sample Entropy.<\/jats:p>","DOI":"10.3390\/e20010061","type":"journal-article","created":{"date-parts":[[2018,1,15]],"date-time":"2018-01-15T12:30:36Z","timestamp":1516019436000},"page":"61","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":59,"title":["Low Computational Cost for Sample Entropy"],"prefix":"10.3390","volume":"20","author":[{"given":"George","family":"Manis","sequence":"first","affiliation":[{"name":"Department of Computer Science and Engineering, University of Ioannina, Ioannina 45110, Greece"}]},{"given":"Md","family":"Aktaruzzaman","sequence":"additional","affiliation":[{"name":"Department of Computer Science and Engineering, Islamic University Kushtia, Kushtia 7003, Bangladesh"}]},{"given":"Roberto","family":"Sassi","sequence":"additional","affiliation":[{"name":"Dipartimento di Informatica, Universit\u00e0 degli Studi di Milano, Crema 26013, Italy"}]}],"member":"1968","published-online":{"date-parts":[[2018,1,13]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"349","DOI":"10.1007\/s10439-012-0668-3","article-title":"The appropriate use of approximate entropy and sample entropy with short data sets","volume":"41","author":"Yentes","year":"2013","journal-title":"Ann. Biomed. Eng."},{"key":"ref_2","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_3","first-page":"754","article-title":"Entropy per unit time as a metric invariant of automorphism","volume":"124","author":"Kolmogorov","year":"1959","journal-title":"Dokl. Russ. Acad. Sci."},{"key":"ref_4","first-page":"768","article-title":"On the Notion of Entropy of a Dynamical System","volume":"124","author":"Sinai","year":"1959","journal-title":"Dokl. Russ. Acad. Sci."},{"key":"ref_5","unstructured":"Signorini, M.G., Sassi, R., Lombardi, F., and Cerutti, S. (1998, January 1). Regularity patterns in heart rate variability signal: The approximate entropy approach. Proceedings of the 20th International Conference of the IEEE Engineering in Medicine and Biology Society, Hong Kong, China."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"177","DOI":"10.1023\/A:1015212328405","article-title":"Approximate Entropy of Heart Rate Variability: Validation of Methods and Application in Heart Failure","volume":"1","author":"Beckers","year":"2001","journal-title":"Cardiovasc. Eng."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"3","DOI":"10.3389\/fneng.2012.00003","article-title":"Dominant Lyapunov exponent and approximate entropy in heart rate variability during emotional visual elicitation","volume":"5","author":"Valenza","year":"2012","journal-title":"Front. Neuroeng."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"288","DOI":"10.1109\/TITB.2006.884369","article-title":"Approximate Entropy-Based Epileptic EEG Detection Using Artificial Neural Networks","volume":"11","author":"Srinivasan","year":"2007","journal-title":"Trans. Inf. Tech. Biomed."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"2027","DOI":"10.1016\/j.eswa.2007.12.065","article-title":"Automatic detection of epileptic seizures in EEG using discrete wavelet transform and approximate entropy","volume":"36","author":"Ocak","year":"2009","journal-title":"Expert Syst. Appl."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"iv141","DOI":"10.1093\/europace\/euu262","article-title":"Non-linear regularity of arterial blood pressure variability in patient with atrial fibrillation in tilt-test procedure","volume":"16","author":"Cerutti","year":"2014","journal-title":"Europace"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"2039","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_12","doi-asserted-by":"crossref","first-page":"789","DOI":"10.1152\/ajpregu.00069.2002","article-title":"Sample entropy analysis of neonatal heart rate variability","volume":"283","author":"Lake","year":"2002","journal-title":"Am. J. Physiol. Regul. Integr. Comp. Physiol."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"709","DOI":"10.1016\/j.compbiomed.2008.03.004","article-title":"Effect of mobile phone radiation on heart rate variability","volume":"38","author":"Ahamed","year":"2008","journal-title":"Comput. Biol. Med."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1900","DOI":"10.1109\/TBME.2006.889772","article-title":"Use of sample entropy approach to study heart rate variability in obstructive sleep apnea syndrome","volume":"54","author":"Sahakian","year":"2007","journal-title":"IEEE Trans. Biomed. Eng."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"132","DOI":"10.1016\/j.jneumeth.2012.07.003","article-title":"Automatic epileptic seizure detection in EEGs based on optimized sample entropy and extreme learning machine","volume":"210","author":"Song","year":"2012","journal-title":"J. Neurosci. Methods"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"917","DOI":"10.1016\/j.medengphy.2009.05.002","article-title":"Sample entropy of the main atrial wave predicts spontaneous termination of paroxysmal atrial fibrillation","volume":"31","author":"Alcaraz","year":"2009","journal-title":"Med. Eng. Phys."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1023","DOI":"10.1016\/j.medengphy.2009.06.004","article-title":"On the use of sample entropy to analyze human postural sway data","volume":"31","author":"Ramdani","year":"2009","journal-title":"Med. Eng. Phys."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Manis, G., and Nikolopoulos, S. (2007). Speeding up the computation of approximate entropy. 11th Mediterranean Conference on Medical and Biomedical Engineering and Computing 2007, Springer.","DOI":"10.1007\/978-3-540-73044-6_204"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"48","DOI":"10.1016\/j.cmpb.2008.02.008","article-title":"Fast computation of approximate entropy","volume":"91","author":"Manis","year":"2008","journal-title":"Comput. Methods Programs Biomed."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"382","DOI":"10.1016\/j.cmpb.2010.12.003","article-title":"Fast computation of sample entropy and approximate entropy in biomedicine","volume":"104","author":"Wang","year":"2011","journal-title":"Comput. Methods Programs Biomed."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1142\/S1793536911000775","article-title":"A fast algorithm for computing Sample Entropy","volume":"3","author":"Jiang","year":"2011","journal-title":"Adv. Adapt. Data Anal."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1643","DOI":"10.1152\/ajpheart.1994.266.4.H1643","article-title":"Physiological time-series analysis: What does regularity quantify","volume":"266","author":"Pincus","year":"1994","journal-title":"Am. J. Physiol. Heart Circ. Physiol."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"141","DOI":"10.1016\/j.bspc.2014.07.011","article-title":"Parametric estimation of sample entropy in heart rate variability analysis","volume":"14","author":"Aktaruzzaman","year":"2014","journal-title":"Biomed. Signal Process. Control"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"124","DOI":"10.1016\/j.cmpb.2010.02.009","article-title":"Optimal parameters study for sample entropy-based atrial fibrillation organization analysis","volume":"99","author":"Alcaraz","year":"2010","journal-title":"Comput. Methods Programs Biomed."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"e215","DOI":"10.1161\/01.CIR.101.23.e215","article-title":"PhysioBank, PhysioToolkit, and PhysioNet: Components of a New Research Resource for Complex Physiologic Signals","volume":"101","author":"Goldberger","year":"2000","journal-title":"Circulation"},{"key":"ref_26","unstructured":"Akay, M. (2000). Approximate Entropy and Its Application to Biosignal Analysis. Nonlinear Biomedical Signal Processing: Dynamic Analysis and Modeling, Volume 2, Wiley-IEEE Press."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"63","DOI":"10.1016\/0375-9601(90)90577-B","article-title":"An Optimized Box-Assisted Algorithm for Fractal Dimensions","volume":"148","author":"Grassberger","year":"1990","journal-title":"Phys. Lett. A"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"327","DOI":"10.1016\/0375-9601(89)90629-4","article-title":"Multidimensional Trees, Range Searching, and a Correlation Dimension Algorithm of Reduced Complexity","volume":"140","author":"Stuart","year":"1989","journal-title":"Phys. Lett. A"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/20\/1\/61\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T14:51:10Z","timestamp":1760194270000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/20\/1\/61"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,1,13]]},"references-count":28,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2018,1]]}},"alternative-id":["e20010061"],"URL":"https:\/\/doi.org\/10.3390\/e20010061","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,1,13]]}}}