# Partial least squares regression on symmetric positive-definite matrices

Partial least squares regression on symmetric positive-definite matrices
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Recently there has been an increased interest in the analysis of differenttypes of manifold-valued data, which include data from symmetric positivedefinitematrices. In many studies of medical cerebral image analysis, amajor concern is establishing the association among a set of covariates andthe manifold-valued data, which are considered as responses for characterizingthe shapes of certain subcortical structures and the differences betweenthem.The manifold-valued data do not form a vector space, and thus, it is notadequate to apply classical statistical techniques directly, as certain operationson vector spaces are not defined in a general Riemannian manifold. Inthis article, an application of the partial least squares regression methodologyis performed for a setting with a large number of covariates in a euclideanspace and one or more responses in a curved manifold, called a Riemanniansymmetric space. To apply such a technique, the Riemannian exponentialmap and the Riemannian logarithmic map are used on a set of symmetricpositive-definite matrices, by which the data are transformed into a vectorspace, where classic statistical techniques can be applied. The methodologyis evaluated using a set of simulated data, and the behavior of the techniqueis analyzed with respect to the principal component regression.

Tipo de documento: Artículo - Article

Palabras clave: Matrix theory, Multicollinearity, Regression, Riemann manifold

Fuente: http://www.bdigital.unal.edu.co

## Introducción

Revista Colombiana de Estadística
Junio 2013, volumen 36, no.

1, pp.

177 a 192
Partial Least Squares Regression on Symmetric
Positive-Definite Matrices
Regresión de mínimos cuadrados parciales sobre matrices simétricas
definidas positiva
Raúl Alberto Pérez1,a , Graciela González-Farias2,b
1 Escuela
de Estadística, Facultad de Ciencias, Universidad Nacional de Colombia,
Medellín, Colombia
2 Departamento
de Probabilidad y Estadística, CIMAT-México Unidad Monterrey,
Monterrey Nuevo León, México
Resumen
Recently there has been an increased interest in the analysis of different
types of manifold-valued data, which include data from symmetric positivedefinite 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.
Palabras clave: Matrix theory, Multicollinearity, Regression, Riemann
manifold.
a Assistant
b Associate
Professor.

E-mail: raper...