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：在多元函数数据分析中，不同函数协变量可能具有同质性。这种隐藏的同质结构能够揭示不同协变量之间的关联或连接性。具有显著同质性的协变量可在同一组内联合分析，从而为多元函数数据的简约建模提供新途径。本文提出一种新型分组多元函数回归模型，并开发名为"系数形状对齐"的正则化方法，以处理不同函数协变量潜在的异质性。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，将协变量聚合为互不相交的组；随后基于检测到的分组结构建立分组多元函数回归模型。在该新模型中，属于同一同质组的协变量系数函数共享相同的形状（尺度变换不变性）。新正则化方法通过惩罚系数形状差异实现分组。本文深入研究了检测分组结构的一致性性质，并推导了确保发现真实分组结构的条件，同时建立了模型估计的渐近性质。通过大量模拟实验考察了所提方法的有限样本表现，并在糖品质评价的真实数据分析中验证了其实用价值。本研究为分析函数协变量的潜在同质性及构建多元函数数据的简约模型结构提供了新方法。