Functional Principal Component Analysis for Sparse Censored Data

Summary Functional principal component analysis is a key tool in the study of functional data, driving both exploratory analyses and feature construction for use in formal modelling and testing procedures. However, existing methods do not apply when functional observations are censored; for example, when the measurement instrument only supports recordings within a prespecified interval, thereby truncating values outside this range to the nearest boundary. A naïve application of existing methods, without correction for instrument-induced censoring, introduces bias into the estimators of the mean, covariance and functional principal component scores. We extend the functional principal component analysis framework to accommodate noisy and potentially sparse censored functional data. Local loglikelihood maximization is used to recover smooth estimates of the mean and covariance surface that are representative of the latent process’s mean and covariance functions. The covariance-smoothing procedure yields a positive semidefinite covariance surface, computed without the need to retroactively remove negative eigenvalues in the covariance-operator decomposition. Additionally, we construct a predictor of the scores, conditional on the censored functional data, and demonstrate its use in the generalized functional linear model. Convergence rates for the proposed estimators are established. In simulation experiments, the proposed method yields improved predictive performance and lower bias than existing alternatives. We illustrate its practical value in a study aimed at classifying eating disorder diagnoses in individuals with type 1 diabetes, using censored functional blood glucose data.