Localized Functional Principal Component Analysis Based on Covariance Structure

2025-06-03

AI Summary

Traditional functional principal component analysis (FPCA) yields globally supported eigenfunctions, resulting in poor interpretability. To address this, we propose localized functional principal component analysis (LFPCA), a covariance-structure-driven piecewise decomposition method that orthogonally decomposes a stochastic process into statistically independent sub-processes with disjoint supports. FPCA is then applied independently on each subinterval, yielding orthogonal eigenfunctions with compact (local) support. Crucially, LFPCA achieves localization without sparse regularization, rigorously preserving both the original covariance structure and orthogonality of the decomposition. Moreover, the variance contribution of each sub-process is directly quantified by its associated eigenvalue, eliminating structural distortion caused by over-regularization. Extensive experiments on synthetic and real-world data demonstrate that LFPCA significantly improves interpretability, fidelity to orthogonality, and accuracy in variance explanation compared to conventional FPCA.

Technology Category

• Machine Learning: Learning with Manifolds
• Computer Vision: Other Foundations of Computer Vision
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Data transparency and provenance
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization

Abstract

Functional principal component analysis (FPCA) is a widely used technique in functional data analysis for identifying the primary sources of variation in a sample of random curves. The eigenfunctions obtained from standard FPCA typically have non-zero support across the entire domain. In applications, however, it is often desirable to analyze eigenfunctions that are non-zero only on specific portions of the original domain-and exhibit zero regions when little is contributed to a specific direction of variability-allowing for easier interpretability. Our method identifies sparse characteristics of the underlying stochastic process and derives localized eigenfunctions by mirroring these characteristics without explicitly enforcing sparsity. Specifically, we decompose the stochastic process into uncorrelated sub-processes, each supported on disjoint intervals. Applying FPCA to these sub-processes yields localized eigenfunctions that are naturally orthogonal. In contrast, approaches that enforce localization through penalization must additionally impose orthogonality. Moreover, these approaches can suffer from over-regularization, resulting in eigenfunctions and eigenvalues that deviate from the inherent structure of their population counterparts, potentially misrepresenting data characteristics. Our approach avoids these issues by preserving the inherent structure of the data. Moreover, since the sub-processes have disjoint supports, the eigenvalues associated to the localized eigenfunctions allow for assessing the importance of each sub-processes in terms of its contribution to the total explained variance. We illustrate the effectiveness of our method through simulations and real data applications. Supplementary material for this article is available online.

Problem

• Identifies localized eigenfunctions for functional data analysis
• Decomposes stochastic process into uncorrelated sub-processes
• Preserves inherent data structure without over-regularization

Innovation

• Decomposes process into uncorrelated sub-processes
• Derives localized eigenfunctions without sparsity enforcement
• Preserves inherent data structure avoiding over-regularization