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.

翻译：在多元函数数据分析中，不同函数型协变量可能在某种意义下具有同质性。这种隐藏的同质性结构蕴含着不同协变量之间的连通性或关联性信息。具有显著同质性的协变量可被联合纳入同一组进行分析，从而为多元函数数据的简约建模提供有效途径。本文通过提出一种名为"系数形状对齐"的新型正则化方法，发展了一种能处理不同函数型协变量潜在同质性的多元函数回归技术。建模过程包含两个主要步骤：首先通过新的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；然后基于检测到的分组结构建立分组多元函数回归模型。在该新型分组模型中，同一同质组内各协变量的系数函数共享形状不变性（允许尺度缩放）。该正则化方法的核心在于对系数形状差异施加惩罚。我们深入研究了所检测分组结构的一致性性质，并给出了保证揭示真实分组结构的条件。同时建立了模型估计的渐近性质。通过大量模拟研究验证了所提方法的有限样本性质。在糖质量评价分析中展示了所提方法的实际应用价值。本研究为分析函数型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途径。