The Fisher–Rao Distance between Multivariate Normal Distributions: Special Cases, Bounds and Applications
The Fisher–Rao 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 use...
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| Format: | Article |
| Language: | English |
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MDPI AG
2020-04-01
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| Series: | Entropy |
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| Online Access: | https://www.mdpi.com/1099-4300/22/4/404 |
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| author | Julianna Pinele João E. Strapasson Sueli I. R. Costa |
| author_facet | Julianna Pinele João E. Strapasson Sueli I. R. Costa |
| author_sort | Julianna Pinele |
| collection | DOAJ |
| description | The Fisher–Rao 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–Rao distance to the simplification of Gaussian mixtures using the hierarchical clustering algorithm is also presented. |
| first_indexed | 2024-03-10T20:44:33Z |
| format | Article |
| id | doaj.art-0fc47c07525a4bfaaae7f0aa020f1213 |
| institution | Directory Open Access Journal |
| issn | 1099-4300 |
| language | English |
| last_indexed | 2024-03-10T20:44:33Z |
| publishDate | 2020-04-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Entropy |
| spelling | doaj.art-0fc47c07525a4bfaaae7f0aa020f12132023-11-19T20:25:15ZengMDPI AGEntropy1099-43002020-04-0122440410.3390/e22040404The Fisher–Rao Distance between Multivariate Normal Distributions: Special Cases, Bounds and ApplicationsJulianna Pinele0João E. Strapasson1Sueli I. R. Costa2Center of Exact and Technological Sciences, University of Reconcavo of Bahia, Cruz das Almas 44380-000, BrazilSchool of Applied Sciences, University of Campinas, Limeira 13484-350, BrazilInstitute of Mathematics, University of Campinas, Campinas 13083-859, BrazilThe Fisher–Rao 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–Rao distance to the simplification of Gaussian mixtures using the hierarchical clustering algorithm is also presented.https://www.mdpi.com/1099-4300/22/4/404information geometryFisher–Rao distancemultivariate normal distributionsGaussian mixture simplification |
| spellingShingle | Julianna Pinele João E. Strapasson Sueli I. R. Costa The Fisher–Rao Distance between Multivariate Normal Distributions: Special Cases, Bounds and Applications Entropy information geometry Fisher–Rao distance multivariate normal distributions Gaussian mixture simplification |
| title | The Fisher–Rao Distance between Multivariate Normal Distributions: Special Cases, Bounds and Applications |
| title_full | The Fisher–Rao Distance between Multivariate Normal Distributions: Special Cases, Bounds and Applications |
| title_fullStr | The Fisher–Rao Distance between Multivariate Normal Distributions: Special Cases, Bounds and Applications |
| title_full_unstemmed | The Fisher–Rao Distance between Multivariate Normal Distributions: Special Cases, Bounds and Applications |
| title_short | The Fisher–Rao Distance between Multivariate Normal Distributions: Special Cases, Bounds and Applications |
| title_sort | fisher rao distance between multivariate normal distributions special cases bounds and applications |
| topic | information geometry Fisher–Rao distance multivariate normal distributions Gaussian mixture simplification |
| url | https://www.mdpi.com/1099-4300/22/4/404 |
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