Partial Least Squares Regression on Symmetric Positive-Definite Matrices
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| Format: | Artículo científico |
| Language: | en |
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Universidad Nacional de Colombia
2013
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| author | Raúl Alberto Pérez |
| author_facet | Raúl Alberto Pérez |
| contents | Partial Least Squares Regression on Symmetric Positive-Definite Matrices Raúl Alberto Pérez Graciela González-Farias Física, Astronomía y Matemáticas Regression Matrix theory Riemann manifold Multicollinearity Recently there has been an increased interest in the analysis of different types of manifold-valued data, which include data from symmetric positive- definite matrices. In many studies of medical cerebral image analysis, a major concern is establishing the association among a set of covariates and the manifold-valued data, which are considered as responses for characterizing the shapes of certain subcortical structures and the differences between them. The manifold-valued data do not form a vector space, and thus, it is not adequate to apply classical statistical techniques directly, as certain operations on vector spaces are not defined in a general Riemannian manifold. In this article, an application of the partial least squares regression methodology is performed for a setting with a large number of covariates in a euclidean space and one or more responses in a curved manifold, called a Riemannian symmetric space. To apply such a technique, the Riemannian exponential map and the Riemannian logarithmic map are used on a set of symmetric positive-definite matrices, by which the data are transformed into a vector space, where classic statistical techniques can be applied. The methodology is evaluated using a set of simulated data, and the behavior of the technique is analyzed with respect to the principal component regression. 2013 artículo científico 0120-1751 https://www.redalyc.org/articulo.oa?id=89928087010 en http://www.redalyc.org/revista.oa?id=899 Revista Colombiana de Estadística application/pdf Universidad Nacional de Colombia Revista Colombiana de Estadística (Colombia) Num.1 Vol.36 |
| format | Artículo científico |
| id | redalyc_89928087010 |
| institution | Redalyc |
| language | en |
| publishDate | 2013 |
| publisher | Universidad Nacional de Colombia |
| spellingShingle | Partial Least Squares Regression on Symmetric Positive-Definite Matrices Raúl Alberto Pérez Física, Astronomía y Matemáticas Regression Matrix theory Riemann manifold Multicollinearity Partial Least Squares Regression on Symmetric Positive-Definite Matrices Raúl Alberto Pérez Graciela González-Farias Física, Astronomía y Matemáticas Regression Matrix theory Riemann manifold Multicollinearity Recently there has been an increased interest in the analysis of different types of manifold-valued data, which include data from symmetric positive- definite matrices. In many studies of medical cerebral image analysis, a major concern is establishing the association among a set of covariates and the manifold-valued data, which are considered as responses for characterizing the shapes of certain subcortical structures and the differences between them. The manifold-valued data do not form a vector space, and thus, it is not adequate to apply classical statistical techniques directly, as certain operations on vector spaces are not defined in a general Riemannian manifold. In this article, an application of the partial least squares regression methodology is performed for a setting with a large number of covariates in a euclidean space and one or more responses in a curved manifold, called a Riemannian symmetric space. To apply such a technique, the Riemannian exponential map and the Riemannian logarithmic map are used on a set of symmetric positive-definite matrices, by which the data are transformed into a vector space, where classic statistical techniques can be applied. The methodology is evaluated using a set of simulated data, and the behavior of the technique is analyzed with respect to the principal component regression. 2013 artículo científico 0120-1751 https://www.redalyc.org/articulo.oa?id=89928087010 en http://www.redalyc.org/revista.oa?id=899 Revista Colombiana de Estadística application/pdf Universidad Nacional de Colombia Revista Colombiana de Estadística (Colombia) Num.1 Vol.36 |
| title | Partial Least Squares Regression on Symmetric Positive-Definite Matrices |
| topic | Física, Astronomía y Matemáticas Regression Matrix theory Riemann manifold Multicollinearity |
| url | https://www.redalyc.org/articulo.oa?id=89928087010 |