{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,13]],"date-time":"2026-02-13T20:43:07Z","timestamp":1771015387865,"version":"3.50.1"},"reference-count":27,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2024,8,19]],"date-time":"2024-08-19T00:00:00Z","timestamp":1724025600000},"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 embeds time series into m-dimensional spaces and estimates entropy based on the distances between points in these spaces. However, when samples can be considered as missing or invalid, defining distance in the embedding space becomes problematic. Preprocessing techniques, such as deletion or interpolation, can be employed as a solution, producing time series without missing or invalid values. While deletion ignores missing values, interpolation replaces them using approximations based on neighboring points. This paper proposes a novel approach for the computation of sample entropy when values are considered as missing or invalid. The proposed algorithm accommodates points in the m-dimensional space and handles them there. A theoretical and experimental comparison of the proposed algorithm with deletion and interpolation demonstrates several advantages over these other two approaches. Notably, the deviation of the expected sample entropy value for the proposed methodology consistently proves to be lowest one.<\/jats:p>","DOI":"10.3390\/e26080704","type":"journal-article","created":{"date-parts":[[2024,8,20]],"date-time":"2024-08-20T01:38:45Z","timestamp":1724117925000},"page":"704","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Sample Entropy Computation on Signals with Missing Values"],"prefix":"10.3390","volume":"26","author":[{"given":"George","family":"Manis","sequence":"first","affiliation":[{"name":"Department of Computer Science and Engineering, University of Ioannina, 45500 Ioannina, Greece"}]},{"given":"Dimitrios","family":"Platakis","sequence":"additional","affiliation":[{"name":"Department of Computer Science and Engineering, University of Ioannina, 45500 Ioannina, Greece"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9729-2641","authenticated-orcid":false,"given":"Roberto","family":"Sassi","sequence":"additional","affiliation":[{"name":"Dipartimento di Informatica, Universit\u00e0 degli Studi di Milano, 20133 Milano, Italy"}]}],"member":"1968","published-online":{"date-parts":[[2024,8,19]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Chud\u00e1\u010dek, V., Spilka, J., Bur\u0161a, M., Jank\u016f, P., Hruban, L., Huptych, M., and Lhotsk\u00e1, L. (2014). Open access intrapartum CTG database. BMC Pregnancy Childbirth, 14.","DOI":"10.1186\/1471-2393-14-16"},{"key":"ref_2","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_3","doi-asserted-by":"crossref","first-page":"R789","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_4","first-page":"4002412","article-title":"A Low-Cost Implementation of sample entropy in wearable embedded systems: An example of online analysis for sleep EEG","volume":"70","author":"Wang","year":"2021","journal-title":"IEEE Trans. Instrum. Meas."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Cheng, Q., Yang, W., Liu, K., Zhao, W., Wu, L., Lei, L., Dong, T., Hou, N., Yang, F., and Qu, Y. (2019). Increased sample entropy in EEGs during the functional rehabilitation of an injured brain. Entropy, 21.","DOI":"10.3390\/e21070698"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Yan, C., Li, P., Yang, M., Li, Y., Li, J., Zhang, H., and Liu, C. (2022). Entropy analysis of heart rate variability in different sleep stages. Entropy, 24.","DOI":"10.3390\/e24030379"},{"key":"ref_7","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_8","doi-asserted-by":"crossref","first-page":"255","DOI":"10.1016\/j.procs.2022.09.058","article-title":"Approximate entropy and sample entropy algorithms in financial time series analyses","volume":"207","author":"Olbrys","year":"2022","journal-title":"Procedia Comput. Sci."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"2131","DOI":"10.1002\/joc.1357","article-title":"Measurement of climate complexity using sample entropy","volume":"26","author":"Shuangcheng","year":"2006","journal-title":"Int. J. Climatol."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Manis, G., Aktaruzzaman, M., and Sassi, R. (2018). Low Computational Cost for Sample Entropy. Entropy, 20.","DOI":"10.3390\/e20010061"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Liu, W., Jiang, Y., and Xu, Y. (2022). A Super Fast Algorithm for Estimating Sample Entropy. Entropy, 24.","DOI":"10.3390\/e24040524"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"488","DOI":"10.1109\/TITB.2012.2188536","article-title":"Artifact removal in physiological signals\u2014Practices and possibilities","volume":"16","author":"Sweeney","year":"2012","journal-title":"IEEE Trans. Inf. Technol. Biomed."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"350","DOI":"10.1016\/j.bspc.2011.06.008","article-title":"Using nonlinear features for fetal heart rate classification","volume":"7","author":"Spilka","year":"2012","journal-title":"Biomed. Signal Process. Control"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"225","DOI":"10.1016\/j.bspc.2010.10.002","article-title":"PSD modifications of FHRV due to interpolation and CTG storage rate","volume":"6","author":"Cesarelli","year":"2011","journal-title":"Biomed. Signal Process. Control"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"519","DOI":"10.1080\/713827181","article-title":"An analysis of four missing data treatment methods for supervised learning","volume":"17","author":"Batista","year":"2003","journal-title":"Appl. Artif. Intell."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"520","DOI":"10.1093\/bioinformatics\/17.6.520","article-title":"Missing value estimation methods for DNA microarrays","volume":"17","author":"Troyanskaya","year":"2001","journal-title":"Bioinformatics"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Grzymala-Busse, J.W., Goodwin, L.K., Grzymala-Busse, W.J., and Zheng, X. (September, January 31). Handling missing attribute values in preterm birth data sets. Proceedings of the Rough Sets, Fuzzy Sets, Data Mining, and Granular Computing: 10th International Conference, RSFDGrC 2005, Regina, SK, Canada. Proceedings, Part II 10.","DOI":"10.1007\/11548706_36"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"853","DOI":"10.1175\/1520-0442(2001)014<0853:AOICDE>2.0.CO;2","article-title":"Analysis of Incomplete Climate Data: Estimation of Mean Values and Covariance Matrices and Imputation of Missing Values","volume":"14","author":"Schneider","year":"2001","journal-title":"J. Clim."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Honghai, F., Guoshun, C., Cheng, Y., Bingru, Y., and Yumei, C. (2005, January 14\u201316). A SVM regression based approach to filling in missing values. Proceedings of the International Conference on Knowledge-Based and Intelligent Information and Engineering Systems, Melbourne, Australia.","DOI":"10.1007\/11553939_83"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"187","DOI":"10.1093\/bioinformatics\/bth499","article-title":"Missing value estimation for DNA microarray gene expression data: Local least squares imputation","volume":"21","author":"Kim","year":"2005","journal-title":"Bioinformatics"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"2088","DOI":"10.1093\/bioinformatics\/btg287","article-title":"A Bayesian missing value estimation method for gene expression profile data","volume":"19","author":"Oba","year":"2003","journal-title":"Bioinformatics"},{"key":"ref_22","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_23","doi-asserted-by":"crossref","first-page":"174102","DOI":"10.1103\/PhysRevLett.88.174102","article-title":"Permutation entropy: A natural complexity measure for time series","volume":"88","author":"Bandt","year":"2002","journal-title":"Phys. Rev. Lett."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Humeau-Heurtier, A. (2022). Multiscale entropy approaches and their applications. Entropy, 22.","DOI":"10.3390\/e22060644"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"2711","DOI":"10.1109\/TBME.2017.2664105","article-title":"Bubble entropy: An entropy almost free of parameters","volume":"64","author":"Manis","year":"2017","journal-title":"IEEE Trans. Biomed. Eng."},{"key":"ref_26","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_27","doi-asserted-by":"crossref","unstructured":"Sassi, R., and Mainardi, L. (2008, January 14\u201317). Editing RR Series and Computation of Long-Term Scaling Parameters. Proceedings of the 2008 Computers in Cardiology, Bologna, Italy.","DOI":"10.1109\/CIC.2008.4749104"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/26\/8\/704\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T15:39:12Z","timestamp":1760110752000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/26\/8\/704"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,8,19]]},"references-count":27,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2024,8]]}},"alternative-id":["e26080704"],"URL":"https:\/\/doi.org\/10.3390\/e26080704","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,8,19]]}}}