Robust functional principal component analysis via a functional pairwise spatial sign operator | JoVE Visualize

Area of Science:

Statistics
Functional Data Analysis
Robust Statistics

Background:

Functional principal component analysis (FPCA) is a standard technique for dimensionality reduction in functional data.
Traditional FPCA methods are sensitive to heavy-tailed distributions and outliers due to reliance on sample covariance estimators.
Existing robust FPCA methods often require restrictive assumptions, such as functional elliptical distributions with inherent symmetry.

Purpose of the Study:

To introduce a novel robust functional principal component analysis (FPCA) method, termed PASS FPCA, designed to overcome limitations of standard FPCA.
To develop robust estimation procedures for eigenfunctions and eigenvalues in the context of functional data.
To extend the methodology to accommodate functional data measured with noise and to relax distributional assumptions.

Main Methods:

A new robust FPCA approach based on a functional pairwise spatial sign (PASS) operator is proposed.
Robust estimation procedures for eigenfunctions and eigenvalues are developed.
The method is extended to handle noisy functional data and is theoretically justified under a new class of weakly functional coordinate symmetry (weakly FCS) distributions.

Main Results:

The PASS operator shares eigenfunctions with the standard covariance operator and recovers eigenvalue ratios.
PASS FPCA demonstrates robustness against heavy-tailedness and outliers, outperforming existing methods, especially with nonelliptical distributions.
The proposed weakly FCS distribution class is more flexible than functional elliptical distributions, allowing for severe asymmetry.

Conclusions:

PASS FPCA provides a robust and flexible alternative to standard FPCA, particularly for datasets with non-ideal distributional properties.
The method's theoretical properties and demonstrated robustness in simulations and real-world applications (accelerometry data) highlight its utility.
This approach advances functional data analysis by enabling reliable analysis under weaker distributional assumptions.