Dimensionality reduction when data are density functions

Functional data analysis (FDA) deals with samples where a whole function is observed for each individual. A particular case is when the observed functions are densities, that is also an example of infinite dimensional compositional data. We focus on the dimensionality reduction problem for this particular type of data. Several methods are considered here: functional principal components analysis (PCA), multidimensional scaling (MDS) and variations of both. Special attention is given to the standard graphics used to represent the output of these procedures. We show that some of them are no valid when working with density functions. Alternative graphical outputs are suggested. Both artificial and real data (population pyramids of all countries in the world) illustrate our proposals.