Functional Linear Regression for Partially Observed Functional Data

TL;DR

This paper develops two new methods for functional linear regression when the predictor functions are only partially observed, addressing a gap in existing research and demonstrating their effectiveness through simulations and real DTI data analysis.

Contribution

It introduces two novel approaches for partially observed functional data regression and establishes their asymptotic properties, expanding the applicability of functional linear models.

Findings

Both methods perform well in finite sample simulations.

The methods are successfully applied to Alzheimer's DTI data.

Asymptotic properties are rigorously established for the proposed estimators.

Abstract

In the functional linear regression model, many methods have been proposed and studied to estimate the slope function while the functional predictor was observed in the entire domain. However, works on functional linear regression models with partially observed trajectories have received less attention. In this paper, to fill the literature gap we consider the scenario where individual functional predictor may be observed only on part of the domain. Depending on whether measurement error is presented in functional predictors, two methods are developed, one is based on linear functionals of the observed part of the trajectory and the other one uses conditional principal component scores. We establish the asymptotic properties of the two proposed methods. Finite sample simulations are conducted to verify their performance. Diffusion tensor imaging (DTI) data from Alzheimer's Disease Neuroimaging Initiative (ADNI) study is analyzed.