{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,4]],"date-time":"2026-03-04T08:50:21Z","timestamp":1772614221495,"version":"3.50.1"},"reference-count":19,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2018,5,31]],"date-time":"2018-05-31T00:00:00Z","timestamp":1527724800000},"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 present a new method for assessing the strength of fuzzy rules with respect to a dataset, based on the measures of the greatest energy and smallest entropy of a fuzzy relation. Considering a fuzzy automaton (relation), in which A is the input fuzzy set and B the output fuzzy set, the fuzzy relation R1 with greatest energy provides information about the greatest strength of the input-output, and the fuzzy relation R2 with the smallest entropy provides information about uncertainty of the input-output relationship. We consider a new index of the fuzziness of the input-output based on R1 and R2. In our method, this index is calculated for each pair of input and output fuzzy sets in a fuzzy rule. A threshold value is set in order to choose the most relevant fuzzy rules with respect to the data.<\/jats:p>","DOI":"10.3390\/e20060424","type":"journal-article","created":{"date-parts":[[2018,6,1]],"date-time":"2018-06-01T03:02:50Z","timestamp":1527822170000},"page":"424","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Energy and Entropy Measures of Fuzzy Relations for Data Analysis"],"prefix":"10.3390","volume":"20","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5690-5384","authenticated-orcid":false,"given":"Ferdinando","family":"Di Martino","sequence":"first","affiliation":[{"name":"Dipartimento di Architettura, Universit\u00e0 degli Studi di Napoli Federico II, Via Toledo 402, 80134 Napoli, Italy"},{"name":"Centro Interdipartimentale di Ricerca A. Calza Bini, Universit\u00e0 degli Studi di Napoli Federico II, Via Toledo 402, 80134 Napoli, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4303-2884","authenticated-orcid":false,"given":"Salvatore","family":"Sessa","sequence":"additional","affiliation":[{"name":"Dipartimento di Architettura, Universit\u00e0 degli Studi di Napoli Federico II, Via Toledo 402, 80134 Napoli, Italy"},{"name":"Centro Interdipartimentale di Ricerca A. Calza Bini, Universit\u00e0 degli Studi di Napoli Federico II, Via Toledo 402, 80134 Napoli, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2018,5,31]]},"reference":[{"key":"ref_1","unstructured":"Gupta, M.M., Ragade, R.K., and Yager, R.R. (1979). Entropy and energy measures of fuzzy sets. 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Symmetry, 9.","DOI":"10.20944\/preprints201709.0043.v1"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/20\/6\/424\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T15:06:43Z","timestamp":1760195203000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/20\/6\/424"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,5,31]]},"references-count":19,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2018,6]]}},"alternative-id":["e20060424"],"URL":"https:\/\/doi.org\/10.3390\/e20060424","relation":{"has-preprint":[{"id-type":"doi","id":"10.20944\/preprints201804.0246.v1","asserted-by":"object"}]},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,5,31]]}}}