{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,26]],"date-time":"2026-06-26T03:21:36Z","timestamp":1782444096546,"version":"3.54.5"},"reference-count":54,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2020,4,1]],"date-time":"2020-04-01T00:00:00Z","timestamp":1585699200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100003593","name":"Conselho Nacional de Desenvolvimento Cient\u00edfico e Tecnol\u00f3gico","doi-asserted-by":"publisher","award":["313326\/2017-7 )"],"award-info":[{"award-number":["313326\/2017-7 )"]}],"id":[{"id":"10.13039\/501100003593","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001807","name":"Funda\u00e7\u00e3o de Amparo \u00e0 Pesquisa do Estado de S\u00e3o Paulo","doi-asserted-by":"publisher","award":["13\/25977-7"],"award-info":[{"award-number":["13\/25977-7"]}],"id":[{"id":"10.13039\/501100001807","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The Fisher\u2013Rao distance is a measure of dissimilarity between probability distributions, which, under certain regularity conditions of the statistical model, is up to a scaling factor the unique Riemannian metric invariant under Markov morphisms. It is related to the Shannon entropy and has been used to enlarge the perspective of analysis in a wide variety of domains such as image processing, radar systems, and morphological classification. Here, we approach this metric considered in the statistical model of normal multivariate probability distributions, for which there is not an explicit expression in general, by gathering known results (closed forms for submanifolds and bounds) and derive expressions for the distance between distributions with the same covariance matrix and between distributions with mirrored covariance matrices. An application of the Fisher\u2013Rao distance to the simplification of Gaussian mixtures using the hierarchical clustering algorithm is also presented.<\/jats:p>","DOI":"10.3390\/e22040404","type":"journal-article","created":{"date-parts":[[2020,4,1]],"date-time":"2020-04-01T15:37:21Z","timestamp":1585755441000},"page":"404","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":29,"title":["The Fisher\u2013Rao Distance between Multivariate Normal Distributions: Special Cases, Bounds and Applications"],"prefix":"10.3390","volume":"22","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0092-4206","authenticated-orcid":false,"given":"Julianna","family":"Pinele","sequence":"first","affiliation":[{"name":"Center of Exact and Technological Sciences, University of Reconcavo of Bahia, Cruz das Almas 44380-000, Brazil"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9079-5860","authenticated-orcid":false,"given":"Jo\u00e3o","family":"Strapasson","sequence":"additional","affiliation":[{"name":"School of Applied Sciences, University of Campinas, Limeira 13484-350, Brazil"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8837-9291","authenticated-orcid":false,"given":"Sueli","family":"Costa","sequence":"additional","affiliation":[{"name":"Institute of Mathematics, University of Campinas, Campinas 13083-859, Brazil"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2020,4,1]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Calin, O., and Udriste, C. 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