Approximation of density functions using simplicial splines

Abstract

Probability density functions result in practice frequently from aggregation of massive data and their further statistical processing is thus of increasing importance. However, 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 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. Aim of the contribution is to introduce new class of B-splines within the Bayes space methodology which is suitable for representation of density functions. Accordingly, the original densities are expressed as real functions using the centred logratio transformation and optimal smoothing splines with B-spline basis honoring the resulting zero-integral constraint are developed.