Compositional splines for representation of density functions
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Compositional splines for representation of density functions Jitka Machalová1 · Renáta Talská1
· Karel Hron1 · Aleš Gába2
Received: 20 December 2019 / Accepted: 12 October 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract In the context of functional data analysis, probability density functions as non-negative functions are characterized by specific properties of scale invariance and relative scale which enable to represent them with the unit integral constraint without loss of information. On the other hand, all these properties are a challenge when the densities need to be approximated with spline functions, including construction of the respective spline basis. The Bayes space methodology of density functions enables to express them as real functions in the standard L 2 space using the centered log-ratio transformation. The resulting functions satisfy the zero integral constraint. This is a key to propose a new spline basis, holding the same property, and consequently to build a new class of spline functions, called compositional splines, which can approximate probability density functions in a consistent way. The paper provides also construction of smoothing compositional splines and possible orthonormalization of the spline basis which might be useful in some applications. Finally, statistical processing of densities using the new approximation tool is demonstrated in case of simplicial functional principal component analysis with anthropometric data. Keywords Spline representation · Constrained approximation · Smoothing spline · Simplicial functional principal component analysis
1 Introduction Probability density functions are non-negative functions satisfying the unit integral constraint. This clearly inhibits their direct processing using standard methods of
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Renáta Talská [email protected]
1
Department of Mathematical Analysis and Applications of Mathematics, Faculty of Science, Palacký University Olomouc, 17. listopadu 12, 77146 Olomouc, Czech Republic
2
Department of Natural Sciences in Kinanthropology, Faculty of Physical Culture, Palacký University Olomouc, Kˇrížkovského 511/8, 77146 Olomouc, Czech Republic
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J. Machalová et al.
functional data analysis (Ramsay and Silverman 2005) since unconstrained functions are assumed there. The same holds also for approximation of the raw input data using splines which is commonly considered to be a key step in functional data analysis. But more severely, in addition to the apparent unit integral constraint of densities which might seem to represent just a kind of numerical obstruction, density functions are rather characterized by deeper geometrical properties that need to be taken into account for any reliable analysis (Egozcue et al. 2006; Van den Boogaart et al. 2010, 2014). Specifically, in contrast to functions in the standard L 2 space, densities obey the scale invariance and relative scale properties (Hron et al. 2016). Scale invariance means that not just the representation of densities with the unit integra
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