In multivariate functional data analysis, different functional covariates can be homogeneous in some sense. The hidden homogeneity structure is informative about the connectivity or association of different covariates. The covariates with pronounced homogeneity can be analyzed jointly in the same group and this gives rise to a way of parsimoniously modeling multivariate functional data. In this paper, we develop a multivariate functional regression technique by a new regularization approach termed "coefficient shape alignment" to tackle the potential homogeneity of different functional covariates. The modeling procedure includes two main steps: first the unknown grouping structure is detected with a new regularization approach to aggregate covariates into disjoint groups; and then a 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 an 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.

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