Conditional Functional Principal Components Analysis

Abstract.  This work proposes an extension of the functional principal components analysis (FPCA) or Karhunen–Loève expansion, which can take into account non-parametrically the effects of an additional covariate. Such models can also be interpreted as non-parametric mixed effect models for functional data. We propose estimators based on kernel smoothers and a data-driven selection procedure of the smoothing parameters based on a two-step cross-validation criterion. The conditional FPCA is illustrated with the analysis of a data set consisting of egg laying curves for female fruit flies. Convergence rates are given for estimators of the conditional mean function and the conditional covariance operator when the entire curves are collected. Almost sure convergence is also proven when one observes discretized noisy sample paths only. A simulation study allows us to check the good behaviour of the estimators.