Estimation of surface area
We study the problem of estimating the surface area of the boundary ∂S of a sufficiently smooth set S ⊂ Rd when the available information is only a finite subset Xn ⊂ S. We propose two estimators. The first makes use of the Devroye–Wise support estimator and is based on Crofton’s formula, which, rou...
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| Other Authors: | , |
| Format: | article |
| Language: | English |
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2022
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| Online Access: | https://hdl.handle.net/20.500.12008/37375 |
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| _version_ | 1868889982854234112 |
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| author | Aaron, Catherine |
| author2 | Cholaquidis, Alejandro Fraiman, Ricardo |
| author2_role | author author |
| author_browse | Aaron, Catherine Cholaquidis, Alejandro Fraiman, Ricardo |
| author_facet | Aaron, Catherine Cholaquidis, Alejandro Fraiman, Ricardo |
| author_role | author |
| collection | COLIBRI |
| dc.contributor.none.fl_str_mv | Aaron Catherine, Université Clermont Auvergne Cholaquidis Alejandro, Universidad de la República (Uruguay). Facultad de Ciencias. Centro de Matemática. Fraiman Ricardo, Universidad de la República (Uruguay). Facultad de Ciencias. Centro de Matemática. |
| dc.creator.none.fl_str_mv | Aaron, Catherine Cholaquidis, Alejandro Fraiman, Ricardo |
| dc.date.none.fl_str_mv | 2022 2023-06-02T14:26:31Z 2023-06-02T14:26:31Z |
| dc.format.none.fl_str_mv | 38 h application/pdf |
| dc.identifier.none.fl_str_mv | Aaron, C, Cholaquidis, A y Fraiman, R. "Estimation of surface area". Electronic Journal of Statistics. [en línea] 2022, 16: 3751–3788. 38 h. 1935-7524 https://hdl.handle.net/20.500.12008/37375 10.1214/22-EJS2031 |
| dc.language.none.fl_str_mv | en eng |
| dc.publisher.none.fl_str_mv | Institute of Mathematical Statistics |
| dc.relation.none.fl_str_mv | Electronic Journal of Statistics, 2022, 16: 3751–3788 |
| dc.rights.none.fl_str_mv | info:eu-repo/semantics/openAccess Licencia Creative Commons Atribución (CC - By 4.0) |
| dc.source.none.fl_str_mv | reponame:COLIBRI instname:Universidad de la República instacron:Universidad de la República |
| dc.subject.none.fl_str_mv | Crofton’s formula Surface estimation α-convex hull Devroye–Wise estimator |
| dc.title.none.fl_str_mv | Estimation of surface area |
| dc.type.none.fl_str_mv | Artículo info:eu-repo/semantics/article info:eu-repo/semantics/publishedVersion |
| description | We study the problem of estimating the surface area of the boundary ∂S of a sufficiently smooth set S ⊂ Rd when the available information is only a finite subset Xn ⊂ S. We propose two estimators. The first makes use of the Devroye–Wise support estimator and is based on Crofton’s formula, which, roughly speaking, states that the (d − 1)-dimensional surface area of a smooth enough set is the mean number of intersections of randomly chosen lines. For that purpose, we propose an estimator of the number of intersections of such lines with support based on the Devroye Wise support estimators. The second surface area estimator makes use of the α-convex hull of Xn, which is denoted by Cα(Xn). More precisely, it is the (d−1)-dimensional surface area of Cα(Xn), as denoted by |Cα(Xn)|d−1, which is proven to converge to the (d − 1)-dimensional surface area of ∂S. Moreover, |Cα(Xn)|d−1 can be computed using Crofton’s formula. Our results depend on the Hausdorff distance between S and Xn for the Devroye–Wise estimator, and the Hausdorff distance between ∂S and ∂Cα(Xn) for the second estimator. |
| eu_rights_str_mv | openAccess |
| format | article |
| id | anni_2248e5340f8ea10314fcc57b15d79a2a |
| identifier_str_mv | Aaron, C, Cholaquidis, A y Fraiman, R. "Estimation of surface area". Electronic Journal of Statistics. [en línea] 2022, 16: 3751–3788. 38 h. 1935-7524 10.1214/22-EJS2031 |
| instacron_str | Universidad de la República |
| institution | Universidad de la República |
| instname_str | Universidad de la República |
| language | eng |
| language_invalid_str_mv | en |
| network_acronym_str | anni |
| network_name_str | oai-lr-anni |
| oai_identifier_str | oai:colibri.udelar.edu.uy:20.500.12008/37375 |
| publishDate | 2022 |
| publishDateSort | 2022 |
| publisher.none.fl_str_mv | Institute of Mathematical Statistics |
| reponame_str | COLIBRI |
| repository.mail.fl_str_mv | |
| repository.name.fl_str_mv | |
| repository_id_str | |
| rights_invalid_str_mv | Licencia Creative Commons Atribución (CC - By 4.0) |
| spelling | Estimation of surface areaAaron, CatherineCholaquidis, AlejandroFraiman, RicardoCrofton’s formulaSurface estimationα-convex hullDevroye–Wise estimatorWe study the problem of estimating the surface area of the boundary ∂S of a sufficiently smooth set S ⊂ Rd when the available information is only a finite subset Xn ⊂ S. We propose two estimators. The first makes use of the Devroye–Wise support estimator and is based on Crofton’s formula, which, roughly speaking, states that the (d − 1)-dimensional surface area of a smooth enough set is the mean number of intersections of randomly chosen lines. For that purpose, we propose an estimator of the number of intersections of such lines with support based on the Devroye Wise support estimators. The second surface area estimator makes use of the α-convex hull of Xn, which is denoted by Cα(Xn). More precisely, it is the (d−1)-dimensional surface area of Cα(Xn), as denoted by |Cα(Xn)|d−1, which is proven to converge to the (d − 1)-dimensional surface area of ∂S. Moreover, |Cα(Xn)|d−1 can be computed using Crofton’s formula. Our results depend on the Hausdorff distance between S and Xn for the Devroye–Wise estimator, and the Hausdorff distance between ∂S and ∂Cα(Xn) for the second estimator.ANII: FCE_1_2019_1_156054Institute of Mathematical StatisticsAaron Catherine, Université Clermont AuvergneCholaquidis Alejandro, Universidad de la República (Uruguay). Facultad de Ciencias. Centro de Matemática.Fraiman Ricardo, Universidad de la República (Uruguay). Facultad de Ciencias. Centro de Matemática.2023-06-02T14:26:31Z2023-06-02T14:26:31Z2022Artículoinfo:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersion38 happlication/pdfAaron, C, Cholaquidis, A y Fraiman, R. "Estimation of surface area". Electronic Journal of Statistics. [en línea] 2022, 16: 3751–3788. 38 h.1935-7524https://hdl.handle.net/20.500.12008/3737510.1214/22-EJS2031reponame:COLIBRIinstname:Universidad de la Repúblicainstacron:Universidad de la RepúblicaenengElectronic Journal of Statistics, 2022, 16: 3751–3788Las obras depositadas en el Repositorio se rigen por la Ordenanza de los Derechos de la Propiedad Intelectual de la Universidad de la República.(Res. Nº 91 de C.D.C. de 8/III/1994 – D.O. 7/IV/1994) y por la Ordenanza del Repositorio Abierto de la Universidad de la República (Res. Nº 16 de C.D.C. de 07/10/2014)info:eu-repo/semantics/openAccessLicencia Creative Commons Atribución (CC - By 4.0)oai:colibri.udelar.edu.uy:20.500.12008/373752026-04-14T10:10:05Z |
| spellingShingle | Estimation of surface area Aaron, Catherine Crofton’s formula Surface estimation α-convex hull Devroye–Wise estimator |
| status_str | publishedVersion |
| title | Estimation of surface area |
| title_full | Estimation of surface area |
| title_fullStr | Estimation of surface area |
| title_full_unstemmed | Estimation of surface area |
| title_short | Estimation of surface area |
| title_sort | Estimation of surface area |
| topic | Crofton’s formula Surface estimation α-convex hull Devroye–Wise estimator |
| url | https://hdl.handle.net/20.500.12008/37375 |