  We propose localized functional principal component analysis (LFPCA), looking
for orthogonal basis functions with localized support regions that explain most
of the variability of a random process. The LFPCA is formulated as a convex
optimization problem through a novel Deflated Fantope Localization method and
is implemented through an efficient algorithm to obtain the global optimum. We
prove that the proposed LFPCA converges to the original FPCA when the tuning
parameters are chosen appropriately. Simulation shows that the proposed LFPCA
with tuning parameters chosen by cross validation can almost perfectly recover
the true eigenfunctions and significantly improve the estimation accuracy when
the eigenfunctions are truly supported on some subdomains. In the scenario that
the original eigenfunctions are not localized, the proposed LFPCA also serves
as a nice tool in finding orthogonal basis functions that balance between
interpretability and the capability of explaining variability of the data. The
analyses of a country mortality data and a growth curve data reveal interesting
features that cannot be found by standard FPCA methods.
