A Functional SVD Framework for Regularized Multivariate Functional PCA with Dual Penalization

2026-09-13

AI Summary

本文提出了一种新的正则化多元函数主成分分析框架，通过函数奇异值分解实现对主成分及其得分的同时正则化，并引入稀疏惩罚提高解释性。

Abstract

This paper introduces a novel framework for Regularized Multivariate Functional Principal Component Analysis (ReMFPCA) via Functional Singular Value Decomposition (SVD). The proposed method extends existing MFPCA approaches by incorporating a generalized functional SVD within a Hilbert space framework, enabling simultaneous regularization of both functional principal components (PCs) and their associated PC scores. A key innovation of this framework is the inclusion of a sparsity penalty on the PC scores, which enhances interpretability by filtering out irrelevant subject-specific variations. This dual-penalization strategy represents a significant advancement beyond existing covariance-based eigen decomposition methods, which penalize only the functional PCs. Two power algorithm implementations, sequential and joint, are proposed, together with a cross-validation approach based on iterative regression for optimal smoothing parameter selection. Comprehensive simulation studies and real data applications demonstrate that the proposed framework substantially improves the extraction of informative and interpretable components, offering methodological and practical benefits for analyzing multivariate functional data across diverse domains.

Problem

• Regularized Multivariate Functional PCA
• Functional SVD
• Dual Penalization
• Sparsity Penalty
• Interpretability

Innovation

• Functional SVD
• Dual Penalization
• Sparsity Penalty
• Regularized Multivariate Functional PCA
• Hilbert Space Framework