Smoothed functional principal components analysis

Unlike classical principal component analysis (PCA) for multivariate data, we need some smoothing or regularizing in estimating functional principal components. There are different approaches for smoothed functional principal components. Kernel-based smoothed PCA performs on smoothed functional data while Silverman's method directly estimates smoothed principal components using the smoothed covariance operator. Silverman's method for smoothed functional principal components has many theoretical and practical advantages. Powerful tools from Hilbert space theory can be used to study the theoretical properties of this method.

This research mainly focuses on studying asymptotic properties of Silverman's method in an abstract Hilbert space by exploiting the more general perturbation results. However, the asymptotic properties of Kernel-based method can also be studied using perturbation theory. We obtain the results using general theory on the perturbation of eigenvalues and eigenvectors of the covariance operator. Consistency and asymptotic distributions are derived under mild conditions. For the sake of simplicity of presentation, we restrict our attention to the first smoothed functional principal component.