Functional Principal Component Analysis for Sparse Censored Data

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.

💡 Research Summary

Functional principal component analysis (FPCA) is a cornerstone technique for reducing the infinite‑dimensional nature of functional data to a finite set of interpretable components. Classical FPCA assumes that each observed trajectory is a noisy realization of an underlying L² stochastic process and that the sample mean and covariance can be estimated directly from the data. When measurements are subject to instrument‑induced truncation—i.e., values falling outside a pre‑specified interval