Nonlinear and additive principal component analysis for functional data

We introduce a nonlinear additive functional principal component analysis (NAFPCA) for vector-valued functional data. This is a generalization of functional principal component analysis and allows the relations among the random functions involved to be nonlinear. The method is constructed via two additively nested Hilbert spaces of functions, in which the first space characterizes the functional nature of the data, and the second space captures the nonlinear dependence. In the meantime, additivity is imposed so that we can avoid high-dimensional kernels in the functional space, which causes the curse of dimensionality. Along with the NAFPCA, we also develop a method of selection of the number of principal components and the tuning parameters that determines the degree of nonlinearity, as well as the asymptotic results for both the fully observed and the incompletely observed functional data. Simulation results show that the new method performs better than functional principal component analysis when the relations among random functions are nonlinear. We apply the new method to online handwritten digits and electroencephalogram (EEG) data sets.