Robust Functional Data Analysis for Discretely Observed Data

TL;DR

This paper develops a unified robust framework for functional data analysis from discretely observed data, addressing issues like heavy tails and contamination, and providing theoretical guarantees for robust mean, covariance, and principal components.

Contribution

It introduces a novel, unified robust approach to functional mean, covariance, and PCA for discretely observed data, extending classical methods with theoretical validation.

Findings

Robust functional mean and covariance estimators are proposed and theoretically validated.

The method provides a robust Karhunen–Loève decomposition and principal components.

Perturbation bounds for eigenfunctions are established, supporting further modeling.

Abstract

This paper examines robust functional data analysis for discretely observed data, where the underlying process encompasses various distributions, such as heavy tail, skewness, or contaminations. We propose a unified robust concept of functional mean, covariance, and principal component analysis, while existing methods and definitions often differ from one another or only address fully observed functions (the ``ideal'' case). Specifically, the robust functional mean can deviate from its non-robust counterpart and is estimated using robust local linear regression. Moreover, we define a new robust functional covariance that shares useful properties with the classic version. Importantly, this covariance yields the robust version of Karhunen--Lo\`eve decomposition and corresponding principal components beneficial for dimension reduction. The theoretical results of the robust functional mean, covariance, and eigenfunction estimates, based on pooling discretely observed data (ranging from sparse to dense), are established and aligned with their non-robust counterparts. The newly-proposed perturbation bounds for estimated eigenfunctions, with indexes allowed to grow with sample size, lay the foundation for further modeling based on robust functional principal component analysis.