Generalized Linear Models for Geometrical Current predictors. An application to predict garment fit

The aim of this paper is to model an ordinal response variable in terms of vector-valued functional data included on a vector-valued RKHS. In particular, we focus on the vector-valued RKHS obtained when a geometrical object (body) is characterized by a current and on the ordinal regression model. A common way to solve this problem in functional data analysis is to express the data in the orthonormal basis given by decomposition of the covariance operator. But our data present very important differences with respect to the usual functional data setting. On the one hand, they are vector-valued functions, and on the other, they are functions in an RKHS with a previously defined norm. We propose to use three different bases: the orthonormal basis given by the kernel that defines the RKHS, a basis obtained from decomposition of the integral operator defined using the covariance function, and a third basis that combines the previous two. The three approaches are compared and applied to an interesting problem: building a model to predict the fit of children’s garment sizes, based on a 3D database of the Spanish child population. Our proposal has been compared with alternative methods that explore the performance of other classifiers (Suppport Vector Machine and k-NN), and with the result of applying the classification method proposed in this work, from different characterizations of the objects (landmarks and multivariate anthropometric measurements instead of currents), obtaining in all these cases worst results

Análisis estadístico de formas 3D con aplicaciones antropométricas

Esta tesis doctoral consta de dos partes claramente diferenciadas. En la primera de ellas se utiliza una herramienta matemática llamada current para caracterizar cuerpos geométricos y poder trabajar con ellos en un espacio de Hilbert con propiedades prácticas: un vector-valued Reproducing Kernel Hilbert Space. Al representar los cuerpos geométricos mediante funciones en este espacio, podemos utilizar la teoría de Análisis de Datos Funcionales para adaptar técnicas estadísticas (como algoritmos de clasificación o métodos de regresión) a un conjunto de cuerpos geométricos. Finalmente, se aplican los modelos teóricos desarrollados a una base de datos formada por escáneres de cuerpos de niños y niñas, para resolver problemas relacionados con el tallaje infantil. La segunda parte del trabajo se desarrolla dentro del ámbito de la Estereología. En él obtenemos fórmulas rotacionales para el área y para las integrales de curvatura media de la superficie frontera de un dominio compacto en un espacio de curvatura constante λ.This doctoral thesis consists of two clearly differentiated parts. In the first one, a mathematical tool called current is used to characterize geometric bodies as functions in a vector-valued Reproducing Kernel Hilbert Space, which is a Hilbert spaces with practical properties. Using Functional Data Analysis Theory we can apply statistical techniques, such us classification algorithms or regression methods, to a set of functions representing geometric bodies. Later, the theoretical models that have been developed are applied to a database consisting of scans of bodies of children to solve problems related to sizing children. The second part of the work is developed within the scope of Stereology. In this part we obtain rotational formulas for the area and for the average curvature integrals of the boundary surface of a compact domain in a space of constant curvature λ.Programa de Doctorat en Cièncie