Functional Principal Component Analysis for Sparse Censored Data

TL;DR

This paper extends functional principal component analysis (FPCA) to handle sparse, truncated, noisy functional data by developing a new estimation framework that recovers mean and covariance surfaces, ensuring positive semi-definiteness and improving predictive accuracy.

Contribution

The authors introduce a novel FPCA method for truncated noisy data that estimates smooth mean and covariance surfaces, maintaining positive semi-definiteness without eigenvalue correction.

Findings

Improved predictive performance over existing methods.

Lower bias in estimating functional components.

Effective application to truncated blood glucose data.

Abstract

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.