Supervised classification of geometrical objects by integrating currents and functional data analysis
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Supervised classification of geometrical objects by integrating currents and functional data analysis S. Barahona1 · P. Centella1 · X. Gual-Arnau2 · M. V. Ibáñez3
· A. Simó3
Received: 22 May 2018 / Accepted: 30 June 2019 © Sociedad de Estadística e Investigación Operativa 2019
Abstract This paper focuses on the application of supervised classification techniques to a set of geometrical objects (bodies) characterized by currents, in particular, discriminant analysis and some nonparametric methods. A current is a relevant mathematical object to model geometrical data, like hypersurfaces, through integration of vector fields over them. As a consequence of the choice of a vector-valued reproducing kernel Hilbert space (RKHS) as a test space to integrate over 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. We present a method to apply supervised classification techniques in the obtained vector-valued RKHS. This method is based on the eigenfunction decomposition of the kernel. The novelty of this paper is therefore the reformulation of a size and shape supervised classification problem in functional data analysis terms using the theory of currents and vector-valued RKHSs. 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. Keywords Currents · Statistical shape and size analysis · Reproducing kernel Hilbert space · Functional data analysis · Supervised classification methods · Discriminant analysis Mathematics Subject Classification 58A25 · 46E22 · 62H30 · 47N30
1 Introduction Supervised classification of geometrical objects (Ripley 2007), i.e., the automated assignation of geometrical objects to pre-defined classes, is a common problem in
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M. V. Ibáñez [email protected]
Extended author information available on the last page of the article
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many scientific fields. This is a difficult task where several challenges have to be addressed. The first challenge is how to handle this kind of data from a mathematical point of view. Several mathematical frameworks have been proposed to deal with geometrical data, and three of them are the most widely used. Firstly, functions can be used to represent closed contours of the objects (curves in 2D and surfaces in 3D) (Younes 1998). Geometrical objects can also be treated as subsets of Rn (Serra 1982; Stoyan and Stoyan 1994). Alternatively, they can be described as sequences of points that are given by certain geometrical or anatomical properties (landmarks) (Kendall et al. 2009; Dryden and Mardia 2016). In our approach, we deal with geometrical objects that have been previously registered (centered and rotated if necessary). The contour of each registered object (a curve if the geometrical objects are in R2 , a surface in R3 , or a hypersurface in Rn ) is represented by a mathematical structure na
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