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Title: Classification techniques for probability density functions in the Bayes space framework Authors:  Karel Hron - Palacky University Olomouc (Czech Republic)
Alessandra Menafoglio - Politecnico di Milano (Italy)
Peter Filzmoser - Vienna University of Technology (Austria)
Enea Bongiorno - Universita del Piemonte Orientale (Italy)
Ivana Pavlu - Palacky University Olomouc (Czech Republic) [presenting]
Abstract: Classifying observations into one of the pre-existing groups is one of the frequent tasks in mathematical statistics. With the rising availability of functional data, there is a growing demand for suitable methods needed for their proper statistical analysis. When considering probability density functions (PDFs) as a type of functional data, special care should be given to their specific properties, namely scale invariance, relative scale, and possible unit integral representation. In this sense, the Bayes space methodology serves as a framework that enables the use of standard methods of functional data analysis on properly transformed PDFs. Specifically, the centred log-ratio (clr) transformation plays a key role to represent the PDFs in the standard $L^2$ space. Both parametric (functional logistic regression, functional principal component regression, functional linear discriminant analysis) and nonparametric ($k$-nearest neighbours algorithm) classification methods are considered, together with a semiparametric approach based on a kernel density estimation. These methods are presented on a geochemical dataset of particle size distributions from four measuring sites in the Czech Republic, serving as natural groups for classification.