Localized Functional Principal Component Analysis

TL;DR

This paper introduces a new localized functional principal component analysis method that identifies orthogonal basis functions with localized support, improving interpretability and accuracy in explaining data variability.

Contribution

The paper develops a convex optimization framework for localized FPCA using a novel Deflated Fantope Localization method, with proven convergence and efficient algorithms.

Findings

Accurately recovers true eigenfunctions with cross-validated tuning.

Significantly improves estimation accuracy for localized eigenfunctions.

Reveals new insights in mortality and growth data analysis.

Abstract

We propose localized functional principal component analysis (LFPCA), looking for orthogonal basis functions with localized support regions that explain most of the variability of a random process. The LFPCA is formulated as a convex optimization problem through a novel Deflated Fantope Localization method and is implemented through an efficient algorithm to obtain the global optimum. We prove that the proposed LFPCA converges to the original FPCA when the tuning parameters are chosen appropriately. Simulation shows that the proposed LFPCA with tuning parameters chosen by cross validation can almost perfectly recover the true eigenfunctions and significantly improve the estimation accuracy when the eigenfunctions are truly supported on some subdomains. In the scenario that the original eigenfunctions are not localized, the proposed LFPCA also serves as a nice tool in finding orthogonal basis functions that balance between interpretability and the capability of explaining variability of the data. The analyses of a country mortality data and a growth curve data reveal interesting features that cannot be found by standard FPCA methods.