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.

翻译：在多变量函数数据分析中，不同的函数协变量可能具有同质性。这种隐藏的同质性结构能揭示不同协变量之间的关联或连接关系。具有显著同质性的协变量可在同一组内联合分析，从而提供一种简约建模多变量函数数据的方法。本文提出了一种基于"系数形状对齐"新正则化方法的分组多变量函数回归模型，用于处理不同函数协变量的潜在同质性。建模过程包含两个主要步骤：首先通过新正则化方法检测未知的分组结构，将协变量聚合为不相交的组别；随后基于检测到的分组结构建立分组多变量函数回归模型。在此新分组模型中，同一同质组内协变量的系数函数共享相同的形状（尺度不变性）。新正则化方法通过惩罚系数形状的差异来实现。本文深入研究了检测分组结构的一致性性质，并推导了确保揭示真实潜在分组结构的条件，同时建立了模型估计的渐近性质。通过大量仿真研究考察了所提方法的有限样本性质，并在糖质量评估的实际数据分析中验证了其实用价值。本工作为分析函数协变量的潜在同质性以及构建多变量函数数据的简约模型结构提供了新途径。