  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.
