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 that 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：数据生成的迅猛发展推动了函数数据分析领域的进步。随着同时分析多个变量能力的提升，对多元函数数据处理的需求也日益增长。本研究针对多元函数情境，提出了上包络指数与下包络指数及其广义表达式的新定义。这些定义充分考虑了各分量之间的相互关联性。此外，提出的指数被应用于多元函数数据的聚类分析。在聚类过程中，将指数应用于原始数据及其一阶和二阶导数，从而从原始多元函数数据中生成降维数据集，进而能够应用文献中已充分研究的成熟多元聚类技术。该方法已通过模拟数据集和真实数据集进行测试，并与前沿方法进行了比较分析以评估其性能。