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Title: Approximation of density functions using compositional splines with optimal knots Authors:  Jitka Machalova - Palacky University (Czech Republic) [presenting]
Karel Hron - Palacky University (Czech Republic)
Abstract: Probability density functions result in practice frequently from the aggregation of massive data, and their further statistical processing is thus of increasing importance. However, the specific properties of density functions prevent from analyzing a sample of densities directly using tools of functional data analysis. Moreover, it is not only about the unit integral constraint, which results from the representation of densities within the equivalence class of proportional positive-valued functions, but also about their relative scale, which emphasizes the effect of small relative contributions of Borel subsets to the overall measure of the support. For practical data processing, it is popular to approximate first the input (discrete) data with a proper spline representation. In this case, the compositional splines, a new class of B-splines in the Bayes space, are suitable for the representation of density functions. The aim is to show the use of the compositional splines, especially the optimal choice of number and position of spline knots is discussed. Accordingly, the original densities are expressed as real functions using the centred log-ratio transformation, and optimal smoothing splines with a new B-spline basis honoring the resulting zero-integral constraint are developed.