A multivariate approach to dimension estimation on manifolds with triangle area U-statistics
For data on a manifold M ⊆ ℝᵐ and a point p ∈ M, this thesis introduces a multivariate estimator that leverages angle- and triangle area-based U-statistics (U₁, V₁, V₂) to assess the intrinsic dimension of M at p. By considering the variance of angles formed by pairs of nearby data points, and the m...
- Autores:
-
Vargas Robayo, Bianca Michelle
- Tipo de recurso:
- Trabajo de grado de pregrado
- Fecha de publicación:
- 2025
- Institución:
- Universidad de los Andes
- Repositorio:
- Séneca: repositorio Uniandes
- Idioma:
- eng
- OAI Identifier:
- oai:repositorio.uniandes.edu.co:1992/75813
- Acceso en línea:
- https://hdl.handle.net/1992/75813
- Palabra clave:
- Estadística
Manifold Learning
Matemáticas
- Rights
- openAccess
- License
- Attribution 4.0 International
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A multivariate approach to dimension estimation on manifolds with triangle area U-statistics |
| title |
A multivariate approach to dimension estimation on manifolds with triangle area U-statistics |
| spellingShingle |
A multivariate approach to dimension estimation on manifolds with triangle area U-statistics Estadística Manifold Learning Matemáticas |
| title_short |
A multivariate approach to dimension estimation on manifolds with triangle area U-statistics |
| title_full |
A multivariate approach to dimension estimation on manifolds with triangle area U-statistics |
| title_fullStr |
A multivariate approach to dimension estimation on manifolds with triangle area U-statistics |
| title_full_unstemmed |
A multivariate approach to dimension estimation on manifolds with triangle area U-statistics |
| title_sort |
A multivariate approach to dimension estimation on manifolds with triangle area U-statistics |
| dc.creator.fl_str_mv |
Vargas Robayo, Bianca Michelle |
| dc.contributor.advisor.none.fl_str_mv |
Quiroz Salazar, Adolfo José |
| dc.contributor.author.none.fl_str_mv |
Vargas Robayo, Bianca Michelle |
| dc.contributor.jury.none.fl_str_mv |
Hoegele, Michael Anton |
| dc.subject.keyword.spa.fl_str_mv |
Estadística |
| topic |
Estadística Manifold Learning Matemáticas |
| dc.subject.keyword.eng.fl_str_mv |
Manifold Learning |
| dc.subject.themes.spa.fl_str_mv |
Matemáticas |
| description |
For data on a manifold M ⊆ ℝᵐ and a point p ∈ M, this thesis introduces a multivariate estimator that leverages angle- and triangle area-based U-statistics (U₁, V₁, V₂) to assess the intrinsic dimension of M at p. By considering the variance of angles formed by pairs of nearby data points, and the mean and variance of triangle areas formed by triplets, the proposed methodology captures the manifold's local geometry. A multivariate approach using the Mahalanobis distance ensures robust dimension estimation by incorporating covariance structure. The estimator is evaluated through testing on both simulated manifolds and real-world datasets. Robust statistical tools, such as the Minimum Volume Ellipsoid (MVE), are employed to enhance reliability. |
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2025 |
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2025-01-30T12:53:28Z |
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2025-01-30T12:53:28Z |
| dc.date.issued.none.fl_str_mv |
2025-01-10 |
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Trabajo de grado - Pregrado |
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info:eu-repo/semantics/bachelorThesis |
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instname:Universidad de los Andes |
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eng |
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eng |
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A. J. Quiroz M. Díaz and M. Velasco. “Local angles and dimension estimation from data on manifolds”. In: Journal of Multivariate Analysis 173 (2019), pp. 229–247. doi: 10.1016/j.jmva.2019.02.001. url: https://doi.org/10.1016/j.jmva.2019.02.001. L. Tu. An Introduction to Manifolds. 2nd. New York: Springer, 2010. isbn: 9781441973993. doi: 10.1007/978-1-4419-7399-3. M. Meila and H. Zhang. “Manifold Learning: What, How, and Why”. In: Annual Review of Statistics and Its Application 11 (2024), pp. 393–417. doi: 10.1146/annurev-statistics-040522-115238. url: https://doi.org/10.1146/annurev-statistics-040522-115238. X. Wang and J. S. Marron. “A scale-based approach to finding effective dimensionality in manifold learning”. In: Electronic Journal of Statistics 2 (2008). 17 Mar 2008, pp. 127–148. issn: 1935-7524. doi: 10.1214/07-EJS137. arXiv: arXiv:0710.5349v2 [math.ST]. P. Grassberger and I. Procaccia. “Measuring the strangeness of strange attractors”. In: Physica D: Nonlinear Phenomena 9.1-2 (1983), pp. 189–208. doi: 10.1016/0167-2789(83)90298-1. E. Levina and P. J. Bickel. “Maximum Likelihood Estimation of Intrinsic Dimension”. In: Advances in Neural Information Processing Systems 17 (2005), pp. 777–784. A. J. Quiroz M. R. Brito and J. E. Yukich. “Intrinsic dimension identification via graph-theoretic methods”. In: Journal of Multivariate Analysis 116 (2013), pp. 263–277. doi: 10.1016/j.jmva.2012.10.005. J. M. Lee. Introduction to Smooth Manifolds. 2nd. New York: Springer, 2013. isbn: 978-1-4419-9982-5. doi: 10.1007/978-1-4419-9982-5. R. H. Randles and D. A. Wolfe. Introduction to the Theory of Nonparametric Statistics. New York: John Wiley & Sons, 1979. isbn: 9780471032707. Robert J. Serfling. Approximation theorems of mathematical statistics. John Wiley & Sons, 1980. P. Rousseeuw and M. Hubert. “High-Breakdown Estimators of Multivariate Location and Scatter”. In: Robustness and Complex Data Structures. Springer, 2013, pp. 49–66. doi: 10.1007/978-3-642-35494-6_4. S. Van Aelst and P. Rousseeuw. “Minimum Volume Ellipsoid”. In: WIREs Computational Statistics 1 (2009), pp. 71–82. |
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52 páginas |
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Universidad de los Andes |
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Matemáticas |
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Facultad de Ciencias |
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Quiroz Salazar, Adolfo Josévirtual::22847-1Vargas Robayo, Bianca MichelleHoegele, Michael Anton2025-01-30T12:53:28Z2025-01-30T12:53:28Z2025-01-10https://hdl.handle.net/1992/75813instname:Universidad de los Andesreponame:Repositorio Institucional Sénecarepourl:https://repositorio.uniandes.edu.co/For data on a manifold M ⊆ ℝᵐ and a point p ∈ M, this thesis introduces a multivariate estimator that leverages angle- and triangle area-based U-statistics (U₁, V₁, V₂) to assess the intrinsic dimension of M at p. By considering the variance of angles formed by pairs of nearby data points, and the mean and variance of triangle areas formed by triplets, the proposed methodology captures the manifold's local geometry. A multivariate approach using the Mahalanobis distance ensures robust dimension estimation by incorporating covariance structure. The estimator is evaluated through testing on both simulated manifolds and real-world datasets. Robust statistical tools, such as the Minimum Volume Ellipsoid (MVE), are employed to enhance reliability.Pregrado52 páginasapplication/pdfengUniversidad de los AndesMatemáticasFacultad de CienciasDepartamento de MatemáticasAttribution 4.0 Internationalhttp://creativecommons.org/licenses/by/4.0/info:eu-repo/semantics/openAccesshttp://purl.org/coar/access_right/c_abf2A multivariate approach to dimension estimation on manifolds with triangle area U-statisticsTrabajo de grado - Pregradoinfo:eu-repo/semantics/bachelorThesisinfo:eu-repo/semantics/acceptedVersionhttp://purl.org/coar/resource_type/c_7a1fTexthttp://purl.org/redcol/resource_type/TPEstadísticaManifold LearningMatemáticasA. J. Quiroz M. Díaz and M. Velasco. “Local angles and dimension estimation from data on manifolds”. In: Journal of Multivariate Analysis 173 (2019), pp. 229–247. doi: 10.1016/j.jmva.2019.02.001. url: https://doi.org/10.1016/j.jmva.2019.02.001.L. Tu. An Introduction to Manifolds. 2nd. New York: Springer, 2010. isbn: 9781441973993. doi: 10.1007/978-1-4419-7399-3.M. Meila and H. Zhang. “Manifold Learning: What, How, and Why”. In: Annual Review of Statistics and Its Application 11 (2024), pp. 393–417. doi: 10.1146/annurev-statistics-040522-115238. url: https://doi.org/10.1146/annurev-statistics-040522-115238.X. Wang and J. S. Marron. “A scale-based approach to finding effective dimensionality in manifold learning”. In: Electronic Journal of Statistics 2 (2008). 17 Mar 2008, pp. 127–148. issn: 1935-7524. doi: 10.1214/07-EJS137. arXiv: arXiv:0710.5349v2 [math.ST].P. Grassberger and I. Procaccia. “Measuring the strangeness of strange attractors”. In: Physica D: Nonlinear Phenomena 9.1-2 (1983), pp. 189–208. doi: 10.1016/0167-2789(83)90298-1.E. Levina and P. J. Bickel. “Maximum Likelihood Estimation of Intrinsic Dimension”. In: Advances in Neural Information Processing Systems 17 (2005), pp. 777–784.A. J. Quiroz M. R. Brito and J. E. Yukich. “Intrinsic dimension identification via graph-theoretic methods”. In: Journal of Multivariate Analysis 116 (2013), pp. 263–277. doi: 10.1016/j.jmva.2012.10.005.J. M. Lee. Introduction to Smooth Manifolds. 2nd. New York: Springer, 2013. isbn: 978-1-4419-9982-5. doi: 10.1007/978-1-4419-9982-5.R. H. Randles and D. A. Wolfe. Introduction to the Theory of Nonparametric Statistics. New York: John Wiley & Sons, 1979. isbn: 9780471032707.Robert J. Serfling. Approximation theorems of mathematical statistics. John Wiley & Sons, 1980.P. Rousseeuw and M. Hubert. “High-Breakdown Estimators of Multivariate Location and Scatter”. In: Robustness and Complex Data Structures. Springer, 2013, pp. 49–66. doi: 10.1007/978-3-642-35494-6_4.S. Van Aelst and P. Rousseeuw. “Minimum Volume Ellipsoid”. 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