Dimensionality reduction when data are density functions

Functional Data Analysis deals with samples where a whole function is observed for each individual. A relevant case of FDA is when the observed functions are density functions. Among the particular characteristics of density functions, the most of the fact that they are an example of infinite dimensional compositional data (parts of some whole which only carry relative information) is made. Several dimensionality reduction methods for this particular type of data are compared: functional principal components analysis with or without a previous data transformation, and multidimensional scaling for different inter-density distances, one of them taking into account the compositional nature of density functions. The emphasis is on the steps previous and posterior to the application of a particular dimensionality reduction method: care must be taken in choosing the right density function transformation and/or the appropriate distance between densities before performing dimensionality reduction; subsequently the graphical representation of dimensionality reduction results must take into account that the observed objects are density functions. The different methods are applied1 to artificial and real data (population pyramids for 223 countries in year 2000). As a global conclusion, the use of multidimensional scaling based on compositional distance is recommended.