  Multivariate functional data from a complex system are naturally
high-dimensional and have complex cross-correlation structure. The complexity
of data structure can be observed as that (1) some functions are strongly
correlated with similar features, while some others may have almost no
cross-correlations with quite diverse features; and (2) the cross-correlation
structure may also change over time due to the system evolution. With this
regard, this paper presents a dynamic subspace learning method for multivariate
functional data modeling. In particular, we consider different functions come
from different subspaces, and only functions of the same subspace have
cross-correlations with each other. The subspaces can be automatically
formulated and learned by reformatting the problem as a sparse regression. By
allowing but regularizing the regression change over time, we can describe the
cross-correlation dynamics. The model can be efficiently estimated by the fast
iterative shrinkage-thresholding algorithm (FISTA), and the features of every
subspace can be extracted using the smooth multi-channel functional PCA.
Numerical studies together with case studies demonstrate the efficiency and
applicability of the proposed methodology.
