Functional Principal Component Analysis for Censored Data

Music City Center 

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 censored, 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 naïve application of existing methods, without correction for censoring, introduces bias. We extend the FPCA framework to accommodate noisy, and potentially sparse, censored functional data. Local log-likelihood maximization is used to recover smooth mean and covariance surface estimates that are representative of the latent process's mean and covariance functions. The covariance smoothing 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 an FPC score predictor, 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 lower bias and better predictive performance than existing alternatives. We illustrate its practical value through an application to a study aimed at using censored functional blood glucose data to predict eating disorder diagnoses in type 1 diabetic individuals.

Functional principal component analysis

Scalar-on-function regression

Censored functional data

Censored predictors