[Paper Review] Robust Functional Principal Component Analysis for Non-Gaussian Longitudinal Data

Rou Zhong, Shishi Liu+2 more|arXiv (Cornell University)|Feb 1, 2021

Statistical Methods and Inference34 references4 citations

TL;DR

This paper proposes a robust functional principal component analysis (FPCA) method for non-Gaussian longitudinal data with sparsity, irregular observation times, and measurement errors. It introduces a novel Kendall’s τ function as a distribution-free alternative to the covariance function in the eigenequation, enabling robust estimation without assuming symmetry or elliptical distributions, and establishes asymptotic theory with consistent convergence rates.

ABSTRACT

Functional principal component analysis is essential in functional data analysis, but the inferences will become unconvincing when some non-Gaussian characteristics occur, such as heavy tail and skewness. The focus of this paper is to develop a robust functional principal component analysis methodology in dealing with non-Gaussian longitudinal data, for which sparsity and irregularity along with non-negligible measurement errors must be considered. We introduce a Kendall's $τ$ function whose particular properties make it a nice proxy for the covariance function in the eigenequation when handling non-Gaussian cases. Moreover, the estimation procedure is presented and the asymptotic theory is also established. We further demonstrate the superiority and robustness of our method through simulation studies and apply the method to the longitudinal CD4 cell count data in an AIDS study.

Motivation & Objective

• To develop a robust FPCA method that remains valid under non-Gaussian longitudinal data with heavy tails, skewness, and measurement errors.
• To address the limitations of traditional FPCA under non-Gaussianity, especially when parametric assumptions fail.
• To provide a distribution-free alternative to covariance-based FPCA that does not require symmetric or elliptical distributional assumptions.
• To handle sparsity and irregular observation times common in longitudinal studies.
• To establish asymptotic theory for the proposed estimator, including convergence rates for the estimated covariance function and eigenfunctions.

Proposed method

• Proposes a new Kendall’s τ function as a robust proxy for the population covariance function in the eigenequation, leveraging rank-based dependence structure.
• Defines the Kendall’s τ function using pairwise comparisons of functional observations, ensuring invariance to monotonic transformations and robustness to outliers.
• Applies local linear smoothing to estimate the Kendall’s τ function nonparametrically, accommodating irregularly spaced data and measurement errors.
• Uses the estimated Kendall’s τ function to compute eigenfunctions and scores via spectral decomposition, replacing the traditional covariance-based approach.
• Derives asymptotic distributions and convergence rates for the estimated covariance function and eigenfunctions under regularity conditions.
• Establishes that the eigenspace of the Kendall’s τ function matches that of the true covariance function even under non-symmetric distributions.

Experimental results

Research questions

• RQ1 Can a rank-based dependence measure like Kendall’s τ serve as a robust alternative to the covariance function in functional principal component analysis under non-Gaussian data?
• RQ2 Does the proposed method maintain consistent estimation performance under heavy-tailed, skewed, or contaminated longitudinal data?
• RQ3 How does the proposed FPCA method perform in the presence of sparsity, irregular observation times, and non-ignorable measurement errors?
• RQ4 What are the asymptotic convergence rates of the estimated covariance function and eigenfunctions under the proposed method?
• RQ5 Is the eigenspace of the Kendall’s τ function equivalent to that of the true covariance function without requiring symmetric or elliptical distributional assumptions?

Key findings

• The proposed Kendall’s τ function preserves the same eigenspace as the population covariance function, even under non-symmetric distributions, enabling valid principal component decomposition without distributional constraints.
• The estimator of the Kendall’s τ function achieves a convergence rate of $ O_pig( (NM^2 h^2 / \\(log N))^{-1/2} + h^2 + M^{-1} + h' ig) $, where $ N $ is the number of subjects, $ M $ the average number of observations per subject, and $ h $ the bandwidth.
• The estimated eigenfunctions converge at the same rate as in classical FPCA, confirming theoretical consistency under the proposed framework.
• Simulation studies demonstrate superior robustness to heavy-tailed and skewed data compared to classical FPCA, with reduced bias and improved estimation accuracy.
• Application to AIDS CD4 cell count data shows the method effectively captures dominant modes of variation even when data exhibit skewness and outliers.
• The method outperforms existing robust FPCA approaches that require symmetric or elliptical assumptions, extending applicability to broader non-Gaussian settings.