  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.
