Classification of geometrical objects by integrating currents and functional data analysis. An application to a 3D database of Spanish child population

Abstract

This paper focuses on the application of Discriminant Analysis to a set of geometrical objects (bodies) characterized by currents. A current is a relevant mathematical object to model geometrical data, like hypersurfaces, through integration of vector fields along them. As a consequence of the choice of a vector-valued Reproducing Kernel Hilbert Space (RKHS) as a test space to integrate hypersurfaces, it is possible to consider that hypersurfaces are embedded in this Hilbert space. This embedding enables us to consider classification algorithms of geometrical objects. A method to apply Functional Discriminant Analysis in the obtained vector-valued RKHS is given. This method is based on the eigenfunction decomposition of the kernel. So, the novelty of this paper is the reformulation of a size and shape classification problem in Functional Data Analysis terms using the theory of currents and vector-valued RKHS. This approach is applied to a 3D database obtained from an anthropometric survey of the Spanish child population with a potential application to online sales of children's wear.