Comparing FPCA based on conditional quantile functions and FPCA based on conditional mean function

In this work functional principal component analysis (FPCA) based on quantile functions is proposed as an alternative to the classical approach, based on the functional mean. Quantile regression characterizes the conditional distribution of a response variable and, in particular, some features like the tails behavior; smoothing splines have also been usefully applied to quantile regression to allow for a more flexible modelling. This framework finds application in contexts involving multiple high frequency time series, for which the functional data analysis (FDA) approach is a natural choice. Quantile regression is then extended to the estimation of functional quantiles and our proposal explores the performance of the three-mode FPCA as a tool for summarizing information when functional quantiles of different order are simultaneously considered. The methodology is illustrated and compared with the functional mean based FPCA through an application to air pollution data.