{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,25]],"date-time":"2026-02-25T21:34:24Z","timestamp":1772055264860,"version":"3.50.1"},"reference-count":27,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2024,2,10]],"date-time":"2024-02-10T00:00:00Z","timestamp":1707523200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Key-Area Research and Development Program of Guangdong Province, China","award":["2021B0101190003"],"award-info":[{"award-number":["2021B0101190003"]}]},{"name":"Key-Area Research and Development Program of Guangdong Province, China","award":["2022A1515010831"],"award-info":[{"award-number":["2022A1515010831"]}]},{"name":"Key-Area Research and Development Program of Guangdong Province, China","award":["12101625"],"award-info":[{"award-number":["12101625"]}]},{"name":"Natural Science Foundation of Guangdong Province, China","award":["2021B0101190003"],"award-info":[{"award-number":["2021B0101190003"]}]},{"name":"Natural Science Foundation of Guangdong Province, China","award":["2022A1515010831"],"award-info":[{"award-number":["2022A1515010831"]}]},{"name":"Natural Science Foundation of Guangdong Province, China","award":["12101625"],"award-info":[{"award-number":["12101625"]}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["2021B0101190003"],"award-info":[{"award-number":["2021B0101190003"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["2022A1515010831"],"award-info":[{"award-number":["2022A1515010831"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12101625"],"award-info":[{"award-number":["12101625"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The two-dimensional sample entropy marks a significant advance in evaluating the regularity and predictability of images in the information domain. Unlike the direct computation of sample entropy, which incurs a time complexity of O(N2) for the series with N length, the Monte Carlo-based algorithm for computing one-dimensional sample entropy (MCSampEn) markedly reduces computational costs by minimizing the dependence on N. This paper extends MCSampEn to two dimensions, referred to as MCSampEn2D. This new approach substantially accelerates the estimation of two-dimensional sample entropy, outperforming the direct method by more than a thousand fold. Despite these advancements, MCSampEn2D encounters challenges with significant errors and slow convergence rates. To counter these issues, we have incorporated an upper confidence bound (UCB) strategy in MCSampEn2D. This strategy involves assigning varied upper confidence bounds in each Monte Carlo experiment iteration to enhance the algorithm\u2019s speed and accuracy. Our evaluation of this enhanced approach, dubbed UCBMCSampEn2D, involved the use of medical and natural image data sets. The experiments demonstrate that UCBMCSampEn2D achieves a 40% reduction in computational time compared to MCSampEn2D. Furthermore, the errors with UCBMCSampEn2D are only 30% of those observed in MCSampEn2D, highlighting its improved accuracy and efficiency.<\/jats:p>","DOI":"10.3390\/e26020155","type":"journal-article","created":{"date-parts":[[2024,2,12]],"date-time":"2024-02-12T03:50:27Z","timestamp":1707709827000},"page":"155","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["A Fast Algorithm for Estimating Two-Dimensional Sample Entropy Based on an Upper Confidence Bound and Monte Carlo Sampling"],"prefix":"10.3390","volume":"26","author":[{"given":"Zeheng","family":"Zhou","sequence":"first","affiliation":[{"name":"School of Computer Science and Engineering, Guangdong Province Key Laboratory of Computational Science, Sun Yat-sen University, Guangzhou 510275, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ying","family":"Jiang","sequence":"additional","affiliation":[{"name":"School of Computer Science and Engineering, Guangdong Province Key Laboratory of Computational Science, Sun Yat-sen University, Guangzhou 510275, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Weifeng","family":"Liu","sequence":"additional","affiliation":[{"name":"School of Computer Science and Engineering, Guangdong Province Key Laboratory of Computational Science, Sun Yat-sen University, Guangzhou 510275, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ruifan","family":"Wu","sequence":"additional","affiliation":[{"name":"School of Computer Science and Engineering, Guangdong Province Key Laboratory of Computational Science, Sun Yat-sen University, Guangzhou 510275, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zerong","family":"Li","sequence":"additional","affiliation":[{"name":"School of Computer Science and Engineering, Guangdong Province Key Laboratory of Computational Science, Sun Yat-sen University, Guangzhou 510275, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Wenchao","family":"Guan","sequence":"additional","affiliation":[{"name":"School of Computer Science and Engineering, Guangdong Province Key Laboratory of Computational Science, Sun Yat-sen University, Guangzhou 510275, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,2,10]]},"reference":[{"key":"ref_1","first-page":"1559","article-title":"A Mathematical Theory of Communication","volume":"5","author":"Shannon","year":"2001","journal-title":"Assoc. Comput. Mach."},{"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":"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_4","doi-asserted-by":"crossref","unstructured":"Tom\u010dala, J. (2020). New fast ApEn and SampEn entropy algorithms implementation and their application to supercomputer power consumption. Entropy, 22.","DOI":"10.3390\/e22080863"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"610","DOI":"10.1109\/LSP.2016.2542881","article-title":"Dispersion entropy: A measure for time-series analysis","volume":"23","author":"Rostaghi","year":"2016","journal-title":"IEEE Signal Process. Lett."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"319","DOI":"10.1016\/j.ymssp.2017.12.008","article-title":"A fault diagnosis scheme for planetary gearboxes using adaptive multi-scale morphology filter and modified hierarchical permutation entropy","volume":"105","author":"Li","year":"2017","journal-title":"Mech. Syst. Signal Proc."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"107182","DOI":"10.1016\/j.ymssp.2020.107182","article-title":"Hierarchical multiscale permutation entropy-based feature extraction and fuzzy support tensor machine with pinball loss for bearing fault identification","volume":"149","author":"Yang","year":"2021","journal-title":"Mech. Syst. Signal Proc."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Li, W., Shen, X., and Yaan, L. (2019). A comparative study of multiscale sample entropy and hierarchical entropy and its application in feature extraction for ship-radiated noise. Entropy, 21.","DOI":"10.3390\/e21080793"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Aboy, M., Cuesta-Frau, D., Austin, D., and Mico-Tormos, P. (2007, January 22\u201326). Characterization of sample entropy in the context of biomedical signal analysis. Proceedings of the 2007 29th Annual International Conference of the IEEE Engineering in Medicine and Biology Society, Lyon, France.","DOI":"10.1109\/IEMBS.2007.4353701"},{"key":"ref_10","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_11","unstructured":"Mao, D. (2008). Biological Time Series Classification via Reproducing Kernels and Sample Entropy. [Ph.D. Thesis, Syracuse University]."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"411","DOI":"10.1016\/0375-9601(91)90237-3","article-title":"A simple noise-reduction method for real data","volume":"160","author":"Schreiber","year":"1991","journal-title":"Phys. Lett. A"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"4456","DOI":"10.1103\/PhysRevA.36.4456","article-title":"Efficient algorithm for estimating the correlation dimension from a set of discrete points","volume":"36","author":"Theiler","year":"1987","journal-title":"Phys. Rev. A Gen. Phys."},{"key":"ref_14","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. Meth. Prog. Bio."},{"key":"ref_15","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_16","first-page":"9312616","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_17","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_18","doi-asserted-by":"crossref","unstructured":"Garivier, A., and Moulines, E. (2011, January 5\u20137). On upper-confidence bound policies for switching bandit problems. Proceedings of the International Conference on Algorithmic Learning Theory, Espoo, Finland.","DOI":"10.1007\/978-3-642-24412-4_16"},{"key":"ref_19","first-page":"109","article-title":"Towards a theory of online learning","volume":"2","author":"Anderson","year":"2004","journal-title":"Theory Pract. Online Learn."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Silva, L.E.V., Senra Filho, A.C.S., Fazan, V.P.S., Felipe, J.C., and Murta, L.O. (2016). Two-dimensional sample entropy: Assessing image texture through irregularity. Biomed. Phys. Eng. Express, 2.","DOI":"10.1088\/2057-1976\/2\/4\/045002"},{"key":"ref_21","unstructured":"da Silva, L.E.V., da Silva Senra Filho, A.C., Fazan, V.P.S., Felipe, J.C., and Murta, L.O. (2014, January 26\u201330). Two-dimensional sample entropy analysis of rat sural nerve aging. Proceedings of the 2014 36th Annual International Conference of the IEEE Engineering in Medicine and Biology Society, Chicago, IL, USA."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1876","DOI":"10.1016\/j.tcs.2009.01.016","article-title":"Exploration\u2013exploitation tradeoff using variance estimates in multi-armed bandits","volume":"410","author":"Audibert","year":"2009","journal-title":"Theor. Comput. Sci."},{"key":"ref_23","unstructured":"Zhou, D., Li, L., and Gu, Q. (2020, January 13\u201318). Neural contextual bandits with ucb-based exploration. Proceedings of the International Conference on Machine Learning, Virtual."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Gupta, N., Granmo, O.-C., and Agrawala, A. (2011, January 18\u201321). Thompson sampling for dynamic multi-armed bandits. Proceedings of the 2011 10th International Conference on Machine Learning and Applications and Workshops, Honolulu, HI, USA.","DOI":"10.1109\/ICMLA.2011.144"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Cheung, W.C., Simchi-Levi, D., and Zhu, R. (2019, January 16\u201318). Learning to optimize under non-stationarity. Proceedings of the 22nd International Conference on Artificial Intelligence and Statistics, Naha, Japan.","DOI":"10.2139\/ssrn.3261050"},{"key":"ref_26","unstructured":"Xu, M., Qin, T., and Liu, T.-Y. (2013, January 5\u201310). Estimation bias in multi-armed bandit algorithms for search advertising. Proceedings of the 26th International Conference on Neural Information Processing Systems, Lake Tahoe, NV, USA."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"423","DOI":"10.1016\/j.procs.2021.01.025","article-title":"Deep learning in image classification using residual network (ResNet) variants for detection of colorectal cancer","volume":"179","author":"Sarwinda","year":"2021","journal-title":"Procedia Comput. Sci."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/26\/2\/155\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T13:58:22Z","timestamp":1760104702000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/26\/2\/155"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,2,10]]},"references-count":27,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2024,2]]}},"alternative-id":["e26020155"],"URL":"https:\/\/doi.org\/10.3390\/e26020155","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,2,10]]}}}