  Functional principal component analysis (FPCA) is a key tool in the study of
functional data, driving both exploratory analyses and feature construction for
use in formal modeling and testing procedures. However, existing methods for
FPCA do not apply when functional observations are truncated, e.g., the
measurement instrument only supports recordings within a pre-specified
interval, thereby truncating values outside of the range to the nearest
boundary. A naive application of existing methods without correction for
truncation induces bias. We extend the FPCA framework to accommodate truncated
noisy functional data by first recovering smooth mean and covariance surface
estimates that are representative of the latent process's mean and covariance
functions. Unlike traditional sample covariance smoothing techniques, our
procedure yields a positive semi-definite covariance surface, computed without
the need to retroactively remove negative eigenvalues in the covariance
operator decomposition. Additionally, we construct a FPC score predictor and
demonstrate its use in the generalized functional linear model. Convergence
rates for the proposed estimators are provided. In simulation experiments, the
proposed method yields better predictive performance and lower bias than
existing alternatives. We illustrate its practical value through an application
to a study with truncated blood glucose measurements.
