Numerous problems remain in the construction of statistical depth for functional data. Issues stem largely from the absence of a well-conceived notion of symmetry. The present paper proposes a topologically valid notion of symmetry for distributions on functional metric spaces and a corresponding notion of depth. The latter is shown to satisfy the axiomatic definition of functional depth introduced by Nieto-Reyes and Battey (2016).

中文翻译：

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功能数据深度的拓扑有效构造

在构建功能数据的统计深度时仍然存在许多问题。问题主要是由于缺乏周全的对称性概念。本文针对功能度量空间上的分布提出了一种拓扑上有效的对称概念，以及相应的深度概念。后者被证明满足Nieto-Reyes和Battey（2016）引入的功能深度的公理定义。