Multivariate Functional Principal Component Analysis (MFPCA) is a valuable tool for exploring relationships and identifying shared patterns of variation in multivariate functional data. However, controlling the roughness of the extracted Principal Components (PCs) can be challenging. This paper introduces a novel approach called regularized MFPCA (ReMFPCA) to address this issue and enhance the smoothness and interpretability of the multivariate functional PCs. ReMFPCA incorporates a roughness penalty within a penalized framework, using a parameter vector to regulate the smoothness of each functional variable. The proposed method generates smoothed multivariate functional PCs, providing a concise and interpretable representation of the data. Extensive simulations and real data examples demonstrate the effectiveness of ReMFPCA and its superiority over alternative methods. The proposed approach opens new avenues for analyzing and uncovering relationships in complex multivariate functional datasets.

翻译：多元函数主成分分析（MFPCA）是探索多元函数数据中变量关系并识别其共有变异模式的重要工具。然而，控制所提取主成分（PC）的粗糙度往往具有挑战性。本文提出一种名为正则化MFPCA（ReMFPCA）的新方法以解决该问题，并增强多元函数主成分的光滑性与可解释性。ReMFPCA通过在惩罚框架中引入粗糙度惩罚项，利用参数向量调控每个函数变量的光滑度。所提方法生成光滑化的多元函数主成分，提供数据简洁且可解释的表示。大量模拟实验与真实数据示例验证了ReMFPCA的有效性及其相对于替代方法的优越性。该研究为分析复杂多元函数数据集中的关系开辟了新途径。