Multivariate Functional Principal Component Analysis (MFPCA) is a valuable tool for exploring
relationships and identifying shared patterns of variation in multivariate functional data. However,
interpreting these functional principal components (PCs) can sometimes be challenging due to
issues such as roughness and sparsity. In this dissertation, we establish the theoretical foundations
of the penalized MFPCA problem within Hilbert space and propose three novel regularized
MFPCA approaches. These approaches utilize eigen decomposition and singular value
decomposition (SVD) techniques to enhance the performance of MFPCA by incorporating multiple
penalty terms, such as roughness and sparsity penalties.
In the first method, a roughness penalty is directly imposed on functional PCs, extending the eigen
decomposition problem to a Hilbert space that specifically accounts for the roughness of the
functions. A parameter vector is employed as a tuning parameter to regulate the smoothness of
each functional variable. Additionally, this method allows for each functional variable to be
smoothed on different domains, providing greater flexibility in handling diverse functional data.
In the other two methods, we establish a mathematical foundation for penalized functional SVD to
address the regularized MFPCA problem. Within the functional SVD framework, we propose
iterative power algorithms that offer both the flexibility to assign unique tuning parameters for
each functional PC and computational efficiency. Moreover, the functional SVD approach allows
for the straightforward and simultaneous incorporation of various penalties, such as smoothing and
sparsity, each serving a distinct purpose. Additionally, our functional SVD approach introduces an
innovative form of sparsity within PC scores, which proves beneficial for obtaining more
informative PCs. Similar to the first approach, these two methods also allow each functional
variable to be defined on different domains, providing