This paper studies the problem of functional data clustering. Functional data have their own characteristics and contain rich information that cannot be obtained when regarding the data as multivariate data. Functional data are inherently infinite-dimensional, so classical clustering techniques for finite-dimensional data may not be suitable for functional data. There are several clustering methods for functional data based on probabilistic models or basis expansion approaches. However, most of these depend on the symmetric structure of the model or the mean response; hence, these cannot reflect characteristics of the distribution of data beyond the mean, such as behavior at the extremes. In this paper, we propose a new approach for functional data clustering based on the concept of an asymmetric norm. For this purpose, we consider pseudo-quantiles, such as M-quantiles and expectiles, and their corresponding curves that can provide rich distributional information about hidden structures in the data at various levels. Moreover, as a theoretical justification for the proposed method, a strong consistency property is investigated. Results from numerical examples, including real data analysis, demonstrate the promising empirical properties of the proposed approach.

中文翻译：

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伪分位数功能数据聚类

本文研究功能数据聚类问题。功能数据有其自身的特点，包含着将数据视为多元数据所无法获得的丰富信息。函数数据本质上是无限维的，因此用于有限维数据的经典聚类技术可能不适用于函数数据。有几种基于概率模型或基础扩展方法的功能数据聚类方法。但是，其中大部分取决于模型的对称结构或平均响应；因此，这些不能反映超出平均值的数据分布特征，例如极端行为。在本文中，我们提出了一种基于非对称范数概念的功能数据聚类的新方法。以此目的，我们考虑伪分位数，例如 M 分位数和期望分位数，以及它们相应的曲线，这些曲线可以提供有关数据中各个级别隐藏结构的丰富分布信息。此外，作为所提出方法的理论依据，研究了强一致性属性。数值例子的结果，包括真实数据分析，证明了所提出方法的有希望的经验特性。