Preprocessing of centred logratio transformed density functions using smoothing splines

TL;DR

This paper introduces a method using optimal smoothing splines to preprocess density functions transformed by centred logratio, addressing their unique geometric constraints for improved statistical analysis.

Contribution

It develops a novel smoothing spline approach tailored for centred logratio transformed density functions, enhancing preprocessing in functional data analysis.

Findings

Effective smoothing of density functions demonstrated on real-world data.

Preprocessing method respects the constant integral constraint of density functions.

Method improves the quality of subsequent statistical analysis.

Abstract

With large-scale database systems, statistical analysis of data, formed by probability distributions, become an important task in explorative data analysis. Nevertheless, due to specific properties of density functions, their proper statistical treatment still represents a challenging task in functional data analysis. Namely, the usual L2 metric does not fully accounts for the relative character of information, carried by density functions; instead, their geometrical features are followed by Bayes spaces of measures. The easiest possibility of expressing density functions in L2 space is to use centred logratio transformation, nevertheless, it results in functional data with a constant integral constraint that needs to be taken into account for further analysis. While theoretical background for reasonable analysis of density functions is already provided comprehensively by Bayes spaces themselves, preprocessing issues still need to be developed. The aim of this paper is to introduce optimal smoothing splines for centred logratio transformed density functions that take all their specific features into account and provide a concise methodology for reasonable preprocessing of raw (discretized) distributional observations. Theoretical developments are illustrated with a real-world data set from official statistics.