  Functional principal component analysis (FPCA) has been widely used to
capture major modes of variation and reduce dimensions in functional data
analysis. However, standard FPCA based on the sample covariance estimator does
not work well in the presence of outliers. To address this challenge, a new
robust functional principal component analysis approach based on the functional
pairwise spatial sign (PASS) operator, termed PASS FPCA, is introduced where we
propose estimation procedures for both eigenfunctions and eigenvalues with and
without measurement error. Compared to existing robust FPCA methods, the
proposed one requires weaker distributional assumptions to conserve the
eigenspace of the covariance function. In particular, a class of distributions
called the weakly functional coordinate symmetric (weakly FCS) is introduced
that allows for severe asymmetry and is strictly larger than the functional
elliptical distribution class, the latter of which has been well used in the
robust statistics literature. The robustness of the PASS FPCA is demonstrated
via simulation studies and analyses of accelerometry data from a large-scale
epidemiological study of physical activity on older women that partly motivates
this work.
