{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,13]],"date-time":"2026-05-13T11:41:35Z","timestamp":1778672495007,"version":"3.51.4"},"reference-count":13,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2012,5,16]],"date-time":"2012-05-16T00:00:00Z","timestamp":1337126400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>This paper presents a novel framework for the complexity analysis of rainfall, runoff, and runoff coefficient (RC) time series using multiscale entropy (MSE). The MSE analysis of RC time series was used to investigate changes in the complexity of rainfall-runoff processes due to human activities. Firstly, a coarse graining process was applied to a time series. The sample entropy was then computed for each coarse-grained time series, and plotted as a function of the scale factor. The proposed method was tested in a case study of daily rainfall and runoff data for the upstream Wu\u2013Tu watershed. Results show that the entropy measures of rainfall time series are higher than those of runoff time series at all scale factors. The entropy measures of the RC time series are between the entropy measures of the rainfall and runoff time series at various scale factors. Results also show that the entropy values of rainfall, runoff, and RC time series increase as scale factors increase. The changes in the complexity of RC time series indicate the changes of rainfall-runoff relations due to human activities and provide a reference for the selection of rainfall-runoff models that are capable of dealing with great complexity and take into account of obvious self-similarity can be suggested to the modeling of rainfall-runoff processes. Moreover, the robustness of the MSE results were tested to confirm that MSE analysis is consistent and the same results when removing 25% data, making this approach suitable for the complexity analysis of rainfall, runoff, and RC time series.<\/jats:p>","DOI":"10.3390\/e14050945","type":"journal-article","created":{"date-parts":[[2012,5,16]],"date-time":"2012-05-16T12:25:29Z","timestamp":1337171129000},"page":"945-957","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":33,"title":["Applying Multiscale Entropy to the Complexity Analysis of Rainfall-Runoff Relationships"],"prefix":"10.3390","volume":"14","author":[{"given":"Chien-Ming","family":"Chou","sequence":"first","affiliation":[{"name":"Department of Design for Sustainable Environment, MingDao University, 369 Wen-Hua Road, Peetow, Changhua 52345, Taiwan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2012,5,16]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"2637","DOI":"10.1029\/93WR00877","article-title":"How much complexity is warranted in a rainfall-runoff model?","volume":"29","author":"Jakeman","year":"1993","journal-title":"Water Resour. Res."},{"key":"ref_2","first-page":"380","article-title":"Multiscale entropy method for analysis of complex geophysical signals (In Chinese)","volume":"4","author":"Xie","year":"2009","journal-title":"Technol. Earthq. Disas. Prev."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"110","DOI":"10.1063\/1.166092","article-title":"Approximate entropy as a complexity measure","volume":"5","author":"Pincus","year":"1995","journal-title":"Chaos"},{"key":"ref_4","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_5","doi-asserted-by":"crossref","first-page":"068102:1","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_6","doi-asserted-by":"crossref","first-page":"137","DOI":"10.1109\/CIC.2002.1166726","article-title":"Multiscale entropy to distinguish between physiologic and synthetic RR time series","volume":"29","author":"Costa","year":"2002","journal-title":"Comput. Cardiolog."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"021906:1","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_8","doi-asserted-by":"crossref","first-page":"454","DOI":"10.1152\/japplphysiol.00244.2004","article-title":"Age-related changes in complexity depend on task dynamics (Letter to the editor)","volume":"97","author":"Vaillancourt","year":"2004","journal-title":"J. Appl. Physiol."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"507","DOI":"10.1007\/s00477-007-0161-y","article-title":"Multi-scale entropy analysis of Mississippi River flow","volume":"22","author":"Li","year":"2008","journal-title":"Stoch. Environ. Res. Risk Assess."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"241","DOI":"10.3390\/e13010241","article-title":"Wavelet-based multi-scale entropy analysis of complex rainfall time series","volume":"13","author":"Chou","year":"2011","journal-title":"Entropy"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Zhou, Y., Zhang, Q., Li, K., and Chen, X. (2012). Hydrological effects of water reservoirs on hydrological processes in the East River (China) basin: Complexity evaluations based on the multi-scale entropy analysis. Hydrolog. Process.","DOI":"10.1002\/hyp.8406"},{"key":"ref_12","first-page":"543","article-title":"Multiscale entropy analysis of complex physiological time series (In Chinese)","volume":"26","author":"Cai","year":"2007","journal-title":"Beijing Biomed. Eng."},{"key":"ref_13","unstructured":"Xu, X.K. (2007). Nonlinear Analysis and Modeling on Mixed Sea Wave (In Chinese). [Ph.D. Thesis, Dalian Maritime Affairs University]."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/14\/5\/945\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T21:50:18Z","timestamp":1760219418000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/14\/5\/945"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,5,16]]},"references-count":13,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2012,5]]}},"alternative-id":["e14050945"],"URL":"https:\/\/doi.org\/10.3390\/e14050945","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2012,5,16]]}}}