Adaptive Functional Thresholding for Sparse Covariance Function Estimation in High Dimensions

TL;DR

This paper introduces an adaptive functional thresholding method for estimating sparse covariance functions in high-dimensional functional data, effectively handling partially observed curves and outperforming existing methods in simulations and neuroimaging applications.

Contribution

It proposes a novel adaptive functional thresholding estimator that incorporates variance effects and handles partial observations, advancing high-dimensional covariance estimation techniques.

Findings

The estimator performs significantly better than competitors in simulations.

It effectively handles partially observed functional data.

Applications to neuroimaging datasets demonstrate practical utility.

Abstract

Covariance function estimation is a fundamental task in multivariate functional data analysis and arises in many applications. In this paper, we consider estimating sparse covariance functions for high-dimensional functional data, where the number of random functions p is comparable to, or even larger than the sample size n. Aided by the Hilbert--Schmidt norm of functions, we introduce a new class of functional thresholding operators that combine functional versions of thresholding and shrinkage, and propose the adaptive functional thresholding estimator by incorporating the variance effects of individual entries of the sample covariance function into functional thresholding. To handle the practical scenario where curves are partially observed with errors, we also develop a nonparametric smoothing approach to obtain the smoothed adaptive functional thresholding estimator and its binned implementation to accelerate the computation. We investigate the theoretical properties of our proposals when p grows exponentially with n under both fully and partially observed functional scenarios. Finally, we demonstrate that the proposed adaptive functional thresholding estimators significantly outperform the competitors through extensive simulations and the functional connectivity analysis of two neuroimaging datasets.