{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,18]],"date-time":"2026-05-18T10:58:21Z","timestamp":1779101901223,"version":"3.51.4"},"reference-count":22,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2018,1,11]],"date-time":"2018-01-11T00:00:00Z","timestamp":1515628800000},"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>We propose a definition of entropy for stochastic processes. We provide a reproducing kernel Hilbert space model to estimate entropy from a random sample of realizations of a stochastic process, namely functional data, and introduce two approaches to estimate minimum entropy sets. These sets are relevant to detect anomalous or outlier functional data. A numerical experiment illustrates the performance of the proposed method; in addition, we conduct an analysis of mortality rate curves as an interesting application in a real-data context to explore functional anomaly detection.<\/jats:p>","DOI":"10.3390\/e20010033","type":"journal-article","created":{"date-parts":[[2018,1,11]],"date-time":"2018-01-11T13:36:15Z","timestamp":1515677775000},"page":"33","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["Entropy Measures for Stochastic Processes with Applications in Functional Anomaly Detection"],"prefix":"10.3390","volume":"20","author":[{"given":"Gabriel","family":"Martos","sequence":"first","affiliation":[{"name":"Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires and CONICET, Buenos Aires C1428EGA, Argentina"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Nicol\u00e1s","family":"Hern\u00e1ndez","sequence":"additional","affiliation":[{"name":"Department of Statistics, Universidad Carlos III de Madrid, 28903 Getafe, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Alberto","family":"Mu\u00f1oz","sequence":"additional","affiliation":[{"name":"Department of Statistics, Universidad Carlos III de Madrid, 28903 Getafe, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Javier","family":"Moguerza","sequence":"additional","affiliation":[{"name":"Department of Computer Science and Statistics, University Rey Juan Carlos, 28933 M\u00f3stoles, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2018,1,11]]},"reference":[{"key":"ref_1","unstructured":"R\u00e9nyi, A. 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