Localized Functional Principal Component Analysis Based on Covariance Structure

TL;DR

This paper introduces a novel localized FPCA method that decomposes stochastic processes into disjoint sub-processes, enabling the derivation of interpretable, localized eigenfunctions without explicit sparsity enforcement or over-regularization issues.

Contribution

The proposed approach decomposes processes into disjoint sub-processes to naturally produce localized, orthogonal eigenfunctions, improving interpretability and preserving data structure over penalized methods.

Findings

Localized eigenfunctions improve interpretability.

Method accurately identifies sub-process contributions.

Effective in simulations and real data applications.

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.