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，多变量函数型数据的处理需求也日益增加。本研究提出了针对多变量函数型数据的上包络图指数与下包络图指数的新定义及其广义表达形式，这些定义充分考虑了变量间的相互关联性。进一步地，本文利用所提出的指数对多变量函数型数据进行聚类分析。在聚类过程中，该指数不仅应用于原始数据，还应用于其一阶和二阶导数，从而从原始多变量函数型数据中生成降维数据集，使得能够应用文献中已被广泛研究的成熟多变量聚类技术。该方法通过模拟数据集与真实数据集进行验证，并与当前最先进方法开展对比分析以评估其性能。