Data depth functions have been intensively studied for normed vector spaces. However, a discussion on depth functions on data where one specific data structure cannot be presupposed is lacking. In this article, we introduce a notion of depth functions for data types that are not given in statistical standard data formats and therefore we do not have one specific data structure. We call such data in general non-standard data. To achieve this, we represent the data via formal concept analysis which leads to a unified data representation. Besides introducing depth functions for non-standard data using formal concept analysis, we give a systematic basis by introducing structural properties. Furthermore, we embed the generalised Tukey depth into our concept of data depth and analyse it using the introduced structural properties. Thus, this article provides the mathematical formalisation of centrality and outlyingness for non-standard data and therefore increases the spaces centrality is currently discussed. In particular, it gives a basis to define further depth functions and statistical inference methods for non-standard data.

翻译：数据深度函数已在赋范向量空间中得到深入研究。然而，当无法预设特定数据结构时，关于数据深度函数的讨论尚属空白。本文针对非统计标准数据格式的数据类型引入深度函数概念，因此不依赖单一特定数据结构。我们将此类数据统称为非标准数据。为实现这一目标，我们通过形式概念分析对数据进行表示，从而形成统一的数据表征。除了利用形式概念分析引入非标准数据的深度函数外，我们还通过引入结构性质构建了系统化的理论基础。进一步地，我们将广义Tukey深度嵌入到我们的数据深度概念中，并利用所提出的结构性质对其进行分析。因此，本文为非标准数据的中心性和离群性提供了数学形式化描述，拓展了当前中心性研究的空间。特别地，这为定义非标准数据的其他深度函数和统计推断方法奠定了基础。