  In multivariate functional data analysis, different functional covariates can
be homogeneous. The hidden homogeneity structure is informative about the
connectivity or association of different covariates. The covariates with
pronounced homogeneity can be analyzed jointly within the same group, which
gives rise to a way of parsimoniously modeling multivariate functional data. In
this paper, a novel grouped multivariate functional regression model with a new
regularization approach termed "coefficient shape alignment" is developed to
tackle the potential homogeneity of different functional covariates. The
modeling procedure includes two main steps: first detect the unknown grouping
structure with the new regularization approach to aggregate covariates into
disjoint groups; and then the grouped multivariate functional regression model
is established based on the detected grouping structure. In this new grouped
model, the coefficient functions of covariates in the same homogeneous group
share the same shape invariant to scaling. The new regularization approach
builds on penalizing the discrepancy of coefficient shape. The consistency
property of the detected grouping structure is thoroughly investigated, and the
conditions that guarantee uncovering the underlying true grouping structure are
developed. The asymptotic properties of the model estimates are also developed.
Extensive simulation studies are conducted to investigate the finite-sample
properties of the developed methods. The practical utility of the proposed
methods is illustrated in the real data analysis on sugar quality evaluation.
This work provides a novel means for analyzing the underlying homogeneity of
functional covariates and developing parsimonious model structures for
multivariate functional data.
