The proliferation of data generation has spurred advancements in functional data analysis. With the ability to analyze multiple variables simultaneously, the demand for working with multivariate functional data has increased. This study proposes a novel formulation of the epigraph and hypograph indexes, as well as their generalized expressions, specifically tailored for the multivariate functional context. These definitions take into account the interrelations between components. Furthermore, the proposed indexes are employed to cluster multivariate functional data. In the clustering process, the indexes are applied to both the data and their first and second derivatives. This generates a reduced-dimension dataset from the original multivariate functional data, enabling the application of well-established multivariate clustering techniques which have been extensively studied in the literature. This methodology has been tested through simulated and real datasets, performing comparative analyses against state-of-the-art to assess its performance.

翻译：数据生成的激增推动了函数型数据分析的进步。随着同时分析多个变量的能力需求增长，处理多元函数型数据的需求日益增加。本研究提出了针对多元函数型背景专门定制的上包络指数与下包络指数的新颖定义及其广义表达式。这些定义充分考虑了分量间的相互关系。此外，所提出的指数被用于对多元函数型数据进行聚类。在聚类过程中，该指数被应用于原始数据及其一阶和二阶导数。这从原始多元函数型数据中生成了一个降维数据集，从而使得文献中已广泛研究的成熟多元聚类技术得以应用。该方法已通过模拟数据集和真实数据集进行测试，并与当前最先进方法进行了对比分析以评估其性能。